Algebraic General Topology · Before going to topology, this book studies properties of co-brouwerian lattices and lters. Keywords:algebraic general topology, quasi-uniform spaces,

Algebraic General Topology�Volume 1

Victor Porton

Web: http://www.mathematics21.org

June 6, 2015

�. This document has been written using the GNU TEXMACS text editor (see www.texmacs.org).

Abstract

In this work I introduce and study in details the concepts of funcoids which generalize proximityspaces and reloids which generalize uniform spaces, and generalizations thereof. The concept offuncoid is generalized concept of proximity, the concept of reloid is cleared from super�uous details(generalized) concept of uniformity. Also funcoids and reloids are generalizations of binary relationswhose domains and ranges are �lters (instead of sets).

Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this pro-vides us with a common generalization of calculus and discrete mathematics.

The concept of continuity is de�ned by an algebraic formula (instead of old messy epsilon-deltanotation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category.In one formula continuity, proximity continuity, and uniform continuity are generalized.

Also I de�ne connectedness for funcoids and reloids.Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree

funcoids: multifuncoids. Also I de�ne several kinds of products of funcoids and other morphisms.Before going to topology, this book studies properties of co-brouwerian lattices and �lters.

Keywords: algebraic general topology, quasi-uniform spaces, generalizations of proximity spaces,generalizations of nearness spaces, generalizations of uniform spacesA.M.S. subject classi�cation: 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99

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Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.1 Draft status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.2 Intended audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.3 Reading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.4 Our topic and rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.5 Earlier works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.6 Kinds of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.7 Structure of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.8 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.8.1 Grothendieck universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.8.2 Misc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.9 Unusual notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2 Common knowledge, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2.1 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.1 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2.1.1.1 Intersecting and joining elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.2 Linear order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.3 Meets and joins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.4 Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.5 Lattices and complete lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.6 Distributivity of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.7 Di�erence and complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.8 Boolean lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.9 Center of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.10 Atoms of posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.11 Kuratowski's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.12 Homomorphisms of posets and lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.13 Galois connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.14 Co-Brouwerian lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.15 Dual pseudocomplement on co-Heyting lattices . . . . . . . . . . . . . . . . . . . . . . . ?

2.2 Intro to category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.3 Intro to group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3 More on order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.1 Straight maps and separation subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.1 Straight maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.2 Separation subsets and full stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.3 Atomically Separable Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.2 Free Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.2.1 Starrish posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.3 Quasidi�erence and Quasicomplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.4 Several equal ways to express pseudodi�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.5 Partially ordered categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.5.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

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3.5.2 Dagger categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.5.2.1 Some special classes of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.6 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.7 A proposition about binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8 In�nite associativity and ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.2 Used notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.2.1 Currying and uncurrying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?The customary de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?Currying and uncurrying with a dependent variable . . . . . . . . . . . . . . . . . . ?

3.8.2.2 Functions with ordinal numbers of arguments . . . . . . . . . . . . . . . . . . . . . ?3.8.3 On sums of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4 Ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.2 Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.3 Finite example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.4 The de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.5 De�nition with composition for every multiplier . . . . . . . . . . . . . . . . . . . . ?3.8.4.6 De�nition with shifting arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.7 Associativity of ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

In�nite associativity implies associativity . . . . . . . . . . . . . . . . . . . . . . . . . ?Concatenation is associative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4 Filters and �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.1 Introduction to �lters and �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.1 Filters on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.2 Intro to �lters on a meet-semilattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.3 Intro to �lters on a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.4 Intro to �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.2 Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.1 Core Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.2 Filtrators with Separable Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.3 Intersection and Joining with an Element of the Core . . . . . . . . . . . . . . . . . . . ?4.2.4 Characterization of Finitely Meet-Closed Filtrators . . . . . . . . . . . . . . . . . . . . . ?4.2.5 Stars of Elements of Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.6 Atomic Elements of a Filtrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.7 Prime Filtrator Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.8 Some Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.9 Complements and Core Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.10 Core Part and Atomic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.11 Distributivity of Core Part over Lattice Operations . . . . . . . . . . . . . . . . . . . . ?4.2.12 Co-Separability of Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.13 Filtrators over Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.2.13.1 Distributivity for an Element of Boolean Core . . . . . . . . . . . . . . . . . . . . ?4.3 Filters on a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.1 Filters on posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.2 Filters on meet-semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.3 Order of �lters. Principal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.3.1 Minimal and maximal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.4 Primary �ltrator is �ltered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.5 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.6 Co-separability of Core for Primary Filtrators . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.7 Core Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.8 Intersecting and Joining with an Element of the Core . . . . . . . . . . . . . . . . . . . ?4.3.9 Formulas for Meets and Joins of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.10 Separability of Core for Primary Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . ?

6 Table of contents

4.3.11 Distributivity of the Lattice of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.12 Filters over Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.12.1 Distributivity for an Element of Boolean Core . . . . . . . . . . . . . . . . . . . . ?4.3.13 Generalized Filter Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.14 Stars for �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.14.1 Stars of Filters on Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.15 More about the Lattice of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.16 Atomic Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.17 Some Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.18 Filters and a Special Sublattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.19 Core Part and Atomic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.20 Complements and Core Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.21 Complementive Filters and Factoring by a Filter . . . . . . . . . . . . . . . . . . . . . . ?4.3.22 Pseudodi�erence of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.4 Filters on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.4.1 Fréchet Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.4.2 Number of Filters on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.5 Some Counter-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.5.1 Weak and Strong Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.6 Open problems about �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.2 Quasidi�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.3 Non-Formal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5 Common knowledge, part 2 (topology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.1.1 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.1.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.2 Pretopological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.2.1 Pretopology induced by a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.3.1 Relationships between pretopologies and topologies . . . . . . . . . . . . . . . . . . . . . ?

5.3.1.1 Topological space induced by preclosure space . . . . . . . . . . . . . . . . . . . . . ?5.3.1.2 Preclosure space induced by topological space . . . . . . . . . . . . . . . . . . . . . ?5.3.1.3 Topology induced by a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.4 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6 Funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.1 Informal introduction into funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.2 Basic de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.2.1 Composition of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.3 Funcoid as continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.4 Lattices of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.5 More on composition of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.6 Domain and range of a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.7 Categories of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.8 Specifying funcoids by functions or relations on atomic �lters . . . . . . . . . . . . . . . . . . ?6.9 Direct product of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.10 Atomic funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.11 Complete funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.12 Funcoids corresponding to pretopologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.13 Completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.13.1 More on completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.13.1.1 Open maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.14 Monovalued and injective funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.15 T0-, T1-, T2-, and T3-separable funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

Table of contents 7

6.16 Filters closed regarding a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

7 Reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

7.1 Basic de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.2 Composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.3 Direct product of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.4 Restricting reloid to a �lter. Domain and image . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.5 Categories of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.6 Monovalued and injective reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.7 Complete reloids and completion of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

8 Relationships between funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . ?

8.1 Funcoid induced by a reloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.2 Reloids induced by a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.3 Galois connections between funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.4 Funcoidal reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

9 On distributivity of composition with a principal reloid . . . . . . . . . . . . . . . . . ?

9.1 Decomposition of composition of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . ?9.2 Decomposition of composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.3 Lemmas for the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.4 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.5 Embedding reloids into funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10 Continuous morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10.1 Traditional de�nitions of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.1 Pretopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.2 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.3 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10.2 Our three de�nitions of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.3 Continuity of a restricted morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

11 Connectedness regarding funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . ?

11.1 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.2 Endomorphism series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.3 Connectedness regarding binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.4 Connectedness regarding funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.5 Algebraic properties of S and S� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

12 Total boundness of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

12.1 Thick binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.2 Totally bounded endoreloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.3 Special case of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.4 Relationships with other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.5 Additional predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13 Orderings of �lters in terms of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13.1 Equivalent �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2 Ordering of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13.2.1 Existence of no more than one monovalued injective reloid for a given pair of ultra�lters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2.1.1 The lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2.1.2 The main theorem and its consequences . . . . . . . . . . . . . . . . . . . . . . . . ?

13.3 Rudin-Keisler equivalence and Rudin-Keisler order . . . . . . . . . . . . . . . . . . . . . . . . ?

8 Table of contents

13.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.4.1 Metamonovalued reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

14 Counter-examples about funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . ?

14.1 Second product. Oblique product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

15 Pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

15.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.2 Composition of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.3 Pointfree funcoid as continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.4 The order of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.5 Domain and range of a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.6 Category of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.7 Specifying funcoids by functions or relations on atomic �lters . . . . . . . . . . . . . . . . . ?15.8 More on composition of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.9 Direct product of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.10 Atomic pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.11 Complete pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.12 Completion and co-completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.13 Monovalued and injective pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.14 Elements closed regarding a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.15 Connectedness regarding a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16 Convergence of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?16.2 Relationships between convergence and continuity . . . . . . . . . . . . . . . . . . . . . . . . ?16.3 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?16.4 Generalized limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16.4.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17 Multifuncoids and staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.1 Product of two funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.1.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.1.2 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.2 Function spaces of posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.3 De�nition of staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.4 Upgrading and downgrading a set regarding a �ltrator . . . . . . . . . . . . . . . . . . . . . ?

17.4.1 Upgrading and downgrading staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.5 Principal staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.6 Multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.7 Join of multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.8 In�nite product of poset elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.9 On products of staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.10 Star categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.10.1 Abrupt of quasi-invertible categories with star-morphisms . . . . . . . . . . . . . . . ?17.11 Product of an arbitrary number of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.11.1 Mapping a morphism into a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . ?17.11.2 General cross-composition product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.3 Star composition of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.4 Star composition of Rel-morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.5 Cross-composition product of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.6 Simple product of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.12 Multireloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.12.1 Starred reloidal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.13 Subatomic product of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

Table of contents 9

17.14 On products and projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.1 Staroidal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.2 Cross-composition product of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . ?17.14.3 Subatomic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.4 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.15 Relationships between cross-composition and subatomic products . . . . . . . . . . . . . ?17.16 Coordinate-wise continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.17 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.18 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

17.18.1 Informal questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

18 Identity staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

18.1 Additional propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.2 On pseudofuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.3 Complete staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

18.3.1 Complete free stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?18.3.1.1 Completely starrish posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

18.3.2 More on free stars and complete free stars . . . . . . . . . . . . . . . . . . . . . . . . . 1418.3.3 Complete staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

18.4 Identity staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.1 Identity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.2 General de�nitions of identity staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.3 Identities are staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.4 Special case of sets and �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.5 Relationships between big and small identity staroids . . . . . . . . . . . . . . . . . . 1718.4.6 Identity staroids on principal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.7 Identity staroids represented as meets and joins . . . . . . . . . . . . . . . . . . . . . 17

18.5 Finite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.6 Counter-examples and conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

19 Postface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

19.1 Formalizing this theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10 Table of contents

Chapter 1IntroductionFor related materials, articles, research questions, and erratum consult the Web page of the authorof the book:

http://www.mathematics21.org/algebraic-general-topology.html

1.1 Draft statusThis is a draft.

1.2 Intended audienceThis book is suitable for any math student as well as for researchers.

To make this book be understandable even for �rst grade students, I made a chapter aboutbasic concepts (posets, lattices, topological spaces, etc.), which an already knowledgeable personmay skip reading. It is assumed that the reader knows basic set theory.

But it is also valuable for mature researchers, as it contains much original research which youcould not �nd in any other source except of my work.

Knowledge of the basic set theory is expected from the reader.Despite that this book presents new research, it is well structured and is suitable to be used as

a textbook for a college course.Your comments about this book are welcome to the email [email protected].

1.3 Reading OrderIf you know basic order and lattice theory (including Galois connections and brouwerian lattices)and basics of category theory, you may skip reading the chapter 2 (�Common knowledge, part 1�).

You are recommended to read the rest of this book by the order.

1.4 Our topic and rationaleFrom [38]: Point-set topology, also called set-theoretic topology or general topology, is the studyof the general abstract nature of continuity or "closeness" on spaces. Basic point-set topologicalnotions are ones like continuity, dimension, compactness, and connectedness.

In this work we study a new approach to point-set topology (and pointfree topology).Traditionally general topology is studied using topological spaces (de�ned below in the section

5.3). I however argue that the theory of topological spaces is not the best method of studyinggeneral topology and introduce an alternative theory, the theory of funcoids . Despite of popularityof the theory of topological spaces it has some drawbacks and is in my opinion not the mostappropriate formalism to study most of general topology. Because topological spaces are tailoredfor study of special sets, so called open and closed sets, studying general topology with topologicalspaces is a little anti-natural and ugly. In my opinion the theory of funcoids is more elegant thanthe theory of topological spaces, and it is better to study funcoids than topological spaces. Oneof the main purposes of this work is to present an alternative General Topology based on funcoidsinstead of being based on topological spaces as it is customary. In order to study funcoids theprior knowledge of topological spaces is not necessary. Nevertheless in this work I will considertopological spaces and the topic of interrelation of funcoids with topological spaces.

11

In fact funcoids are a generalization of topological spaces, so the well known theory of topolog-ical spaces is a special case of the below presented theory of funcoids.

But probably the most important reason to study funcoids is that funcoids are a generalizationof proximity spaces (see section 2.2 for the de�nition of proximity spaces). Before this work it waswritten that the theory of proximity spaces was an example of a stalled research, almost nothinginteresting was discovered about this theory. It was so because the proper way to research proximityspaces is to research their generalization, funcoids. And so it was stalled until discovery of funcoids.That generalized theory of proximity spaces will bring us yet many interesting results.

In addition to funcoids I research reloids. Using below de�ned terminology it may be said thatreloids are (basically) �lters on Cartesian product of sets, and this is a special case of uniformspaces. We don't need to de�ne uniform spaces in this work, it is enough for the reader just toknow that uniform spaces are certain �lters on direct product of sets.

Afterward we study some generalizations.Somebody might ask, why to study it? My approach relates to traditional general topology like

complex numbers to real numbers theory. Be sure this will �nd applications.This book has a de�ciency: It does not properly relate my theory with previous research in

general topology and does not consider deeper category theory properties. It is however OK fornow, as I am going to do this study in later volumes (continuation of this book).

Many proofs in this book may seem too easy and thus this theory not sophisticated enough.But it is largely a result of a well structured digraph of proofs, where more di�cult results aremade easy by reducing them to easier lemmas and propositions.

1.5 Earlier works

Some mathematicians were researching generalizations of proximities and uniformities before mebut they have failed to reach the right degree of generalization which is presented in this workallowing to represent properties of spaces with algebraic (or categorical) formulas.

Proximity structures were introduced by Smirnov in [11].Some references to predecessors:

� In [14], [15], [24], [2], [33] generalized uniformities and proximities are studied.

� Proximities and uniformities are also studied in [21], [22], [32], [34], [35].

� [19] and [20] contains recent progress in quasi-uniform spaces. [20] has a very long list ofrelated literature.

Some works ([31]) about proximity spaces consider relationships of proximities and compact topo-logical spaces. In this work the attempt to de�ne or research their generalization, compactness offuncoids or reloids is not done. It seems potentially productive to attempt to borrow the de�nitionsand procedures from the above mentioned works. I hope to do this study in a separate volume.

[10] studies mappings between proximity structures. (In this volume no attempt to researchmappings between funcoids is done.) [25] researches relationships of quasi-uniform spaces andtopological spaces. [1] studies how proximity structures can be treated as uniform structures andcompacti�cation regarding proximity and uniform spaces.

This book is based partially on my articles [29], [27], [28]. [TODO: Add more references to myarticles.]

In [29] I introduced the concept of �lter objects. This was probably not a very good idea. In thiswork I instead use plain �lters (not �lter objects) and t and u notation for joins and meets insteadof [ and \, which may be confused with set theoretic operations, for lattices in consideration (andfor the lattice of �lters the order is reverse to the set theoretic inclusion). Also this work di�ersfrom [29] in using in some formulations the lattice of principal �lters which is isomorphic to thebase poset instead of using the base poset itself (what was possible in [29] thanks to using �lterobjects). I've replaced (F;A) notation for primary �ltrators with (F;Z) for consistency of notationamong sections.

12 Introduction

1.6 Kinds of continuityA research result based on this book but not fully included in this book (and not yet published) isthat the following kinds of continuity are described by the same algebraic (or rather categorical)formulas for di�erent kinds of continuity and have common properties:

� discrete continuity (between digraphs);

� (pre)topological continuity;

� proximal continuity;

� uniform continuity;

� Cauchy continuity;

� (probably other kinds of continuity).

Thus my research justi�es using the same word �continuity� for these diverse kinds of continuity.See http://www.mathematics21.org/algebraic-general-topology.html

1.7 Structure of this bookIn the chapter 2 �Common knowledge, part 1� some well known de�nitions and theories are con-sidered. You may skip its reading if you already know it. That chapter contains info about:

� posets;

� lattices and complete lattices;

� Galois connections;

� co-brouwerian lattices;

� a very short intro into category theory (It is very basic, I even don't de�ne functors as theyhave no use in my theory);

� a very short introduction to group theory.

Afterward there are my little additions to poset/lattice and category theory.Afterward there is the theory of �lters and �ltrators.Then there is �Common knowledge, part 2 (topology)�, which considers brie�y:

� metric spaces;

� topological spaces;

� pretopological spaces;

� proximity spaces.

Despite of the name �Common knowledge� this second common knowledge chapter is recommendedto be read completely even if you know topology well, because it contains some rare theorems notknown to most mathematicians and hard to �nd in literature.

Then the most interesting thing in this book, the theory of funcoids, starts.Afterwards there is the theory of reloids.Then I show relationships between funcoids and reloids.The last I research generalizations of funcoids, pointfree funcoids , staroids and multifuncoids

and some di�erent kinds of products of morphisms.

1.8 Basic notation

1.8.1 Grothendieck universesWe will work in ZFC with an in�nite and uncountable Grothendieck universe.

1.8 Basic notation 13

A Grothendieck universe is just a set big enough to make all usual set theory inside it. Forexample if f is a Grothendieck universe, and sets X; Y 2 f, then also X [ Y 2 f, X \ Y 2 f,X �Y 2f, etc.

A set which is a member of a Grothendieck universe is called a small set (regarding thisGrothendieck universe). We can restrict our consideration to small sets in order to get rid troubleswith proper classes.

De�nition 1.1. Grothendieck universe is a set f such that:

1. If x2f and y 2 x then y 2f.2. If x; y 2f, then fx; yg2f.3. If x2f then Px2f.4. If fxi j i2 I 2fg is a family of elements of f, then

Si2I xi2f.

One can deduce from this also:

1. If x2f, then fxg2f.2. If x is a subset of y 2f, then x2f.3. If x; y 2f then the ordered pair (x; y)= ffx; yg; xg2f.4. If x; y 2f then x[ y and x� y are in f.

5. If fxi j i2 I 2fg is a family of elements of f, then the productQ

i2I xi2f.

6. If x2f, then the cardinality of x is strictly less than the cardinality of f.

1.8.2 MiscIn this book quanti�ers bind tightly. That is 8x 2A: P (x) ^ Q and 8x 2A: P (x)) Q should beread (8x2A:P (x))^Q and (8x2A:P (x)))Q not 8x2A: (P (x)^Q) and 8x2A: (P (x))Q).

The set of functions from a set A to a set B is denoted as BA.I will often skip parentheses and write fx instead of f(x) to denote the result of a function f

acting on the argument x.I will denote hf iX=ff� j �2Xg and X [f ]Y ,9x2X; y2Y :xf y for sets X , Y and a binary

relation f . (Note that functions are a special case of binary relations.)By just hf i and [f ] I will denote the corresponding function and relation on small sets.�x2D: f(x)= f(x; f(x)) j x2Dg for a set D and and a form f depending on the variable x.I will denote source and destination of a morphism f of any category (See �Common knowledge,

part 1� chapter for a de�nition of a category.) as Src f and Dst f correspondingly. Note that belowde�ned domain and image of a funcoid are not the same as it source and destination.

I will denote GR(A;B; f)= f for any morphism (A;B; f) of either Set or Rel.I will denote hf i= hGR f i and [f ]=[GR f ] for any morphism f of either Set or Rel.

1.9 Unusual notation

In the chapter 2 (which you may skip reading if you are already knowledgeable) some non-standardnotation is de�ned. I summarize here this notation for the case if you choose to skip reading thatchapter:

Partial order is denoted as v.Meets and joins are denoted as u, t,

d,F

.I call element b substractive from an elements a (of a distributive lattice A) when the di�erence

a n b exists. I call b complementive to a when there exist c2A such that bu c=0 and bt c= a. Wewill prove that b is complementive to a i� b is substractive from a and bv a.

De�nition 1.2. Call a and b of a poset A intersecting, denoted a�/ b, when there exists a non-least element c such that cv a^ cv b.

14 Introduction

De�nition 1.3. a� b=def:(a�/ b).

De�nition 1.4. I call elements a and b of a poset A joining and denote a� b when there are nonon-greatest element c such that cw a^ cw b.

De�nition 1.5. a�/ b=def:(a� b).

Obvious 1.6. a�/ b i� au b is non-least, for every elements a, b of a meet-semilattice.

Obvious 1.7. a� b if at b is the greatest element, for every elements a, b of a join-semilattice.

I extend the de�nitions of pseudocomplement and dual pseudocomplement to arbitrary posets(not just lattices as it is customary):

De�nition 1.8. Let A be a poset. Pseudocomplement of a is

max fc2A j c� ag:

If z is the pseudocomplement of a we will denote z= a�.

De�nition 1.9. Let A be a poset. Dual pseudocomplement of a is

min fc2A j c� ag:

If z is the dual pseudocomplement of a we will denote z= a+.

1.9 Unusual notation 15

Chapter 2

Common knowledge, part 1

In this chapter we will consider some well known mathematical theories. If you already know themyou may skip reading this chapter (or its parts).

2.1 Order theory

2.1.1 Posets

De�nition 2.1. The identity relation on a set A is idA= f(a; a) j a2Ag.

De�nition 2.2. A preorder on a set A is a binary relation v which is:

� re�exive on A ((v)� idA);

� transitive ((v) � (v)� (v)).

De�nition 2.3. A partial order on a set A is a preorder on A which is antisymmetric ((v) \(v)¡1� (=)).

The reverse relation is denoted w.

De�nition 2.4. a is a subelement of b (or what is the same a is contained in b or b contains a)i� av b.

Obvious 2.5. The reverse of a partial order is also a partial order.

De�nition 2.6. A poset is a set A together with a partial order on it is called a partially orderedset (poset for short).

De�nition 2.7. Strict partial order @ corresponding to the partial order v on a set A is de�nedby the formula (@) = (v) n idA.

De�nition 2.8. A partial order on a set A restricted to a set B �A is (v)\ (B �B).

Obvious 2.9. A partial order on a set A restricted to a set B �A is a partial order on B.

De�nition 2.10.

� The least element 0 of a poset A is de�ned by the formula 8a2A: 0v a.

� The greatest element 1 of a poset A is de�ned by the formula 8a2A: 1w a.

Proposition 2.11. There exist no more than one least element and no more than one greatestelement (for a given poset).

Proof. By antisymmetry. �

De�nition 2.12. The dual order for v is w.

17

Obvious 2.13. Dual of a partial order is a partial order.

De�nition 2.14. The dual poset for a poset (A;v) is the poset (A;w).

Below we will sometimes use duality that is replacement of the partial order and all relatedoperations and relations with their duals. In other words, it is enough to prove a theorem for anorder v and the similar theorem for w follows by duality.

2.1.1.1 Intersecting and joining elements

Let A be a poset.

De�nition 2.15. Call elements a and b of A intersecting, denoted a�/ b, when there exists a non-least element c such that cv a^ cv b.

De�nition 2.16. a� b=def:(a�/ b).

Obvious 2.17. a0�/ b0^ a1w a0^ b1w b0) a1�/ b1.

De�nition 2.18. I call elements a and b of A joining and denote a� b when there is no a non-greatest element c such that cw a^ cw b.

De�nition 2.19. a�/ b=def:(a� b).

Obvious 2.20. Intersecting is the dual of non-joining.

Obvious 2.21. a0� b0^ a1w a0^ b1w b0) a1� b1.

2.1.2 Linear order

De�nition 2.22. A poset A is called linearly ordered set (or what is the same, totally ordered set)if aw b_ bw a for every a; b2A.

Example 2.23. The set of real numbers with the customary order is a linearly ordered set.

De�nition 2.24. A set X 2PA where A is a poset is called a chain if A restricted to X is a totalorder.

2.1.3 Meets and joinsLet A be a poset.

De�nition 2.25. Given a setX 2PA the least element (also called minimum and denoted minX)of X is such a2X that 8x2X: avx.

Least element does not necessarily exists. But if it exists:

Proposition 2.26. For a given X 2PA there exist no more than one least element.

Proof. It follows from anti-symmetry. �

Greatest element is the dual of least element:

De�nition 2.27. Given a set X 2PA the greatest element (also called maximum and denotedmaxX) of X is such a2X that 8x2X : awx.

Remark 2.28. Least and greatest elements of a set X is a trivial generalization of the abovede�ned least and greatest element for the entire poset.

18 Common knowledge, part 1

De�nition 2.29.

� A minimal element of a set X 2PA is such a2A that @x2X : (awx^ x=/ a).� A maximal element of a set X 2PA is such a2A that @x2X : (avx^ x=/ a).

Remark 2.30. Minimal element is not the same as minimum, and maximal element is not thesame as maximum.

Obvious 2.31.

1. The least element (if it exists) is a minimal element.

2. The greatest element (if it exists) is a maximal element.

Exercise 2.1. Show that there may be more than one minimal and more than one maximal element for someposet.

De�nition 2.32. Upper bounds of a set X is the set fy 2A j 8x2X: y wxg.

The dual notion:

De�nition 2.33. Lower bounds of a set X is the set fy 2A j 8x2X: yvxg.

De�nition 2.34. JoinFX (also called supremum and denoted �supX�) of a set X is the least

element of its upper bounds (if it exists).

De�nition 2.35. MeetdX (also called in�mum and denoted �infX�) of a set X is the greatest

element of its lower bounds (if it exists).

We will write b=FX when b2A is the join of X or say that

FX does not exist if there are

no such b2A. (And dually for meets.)

Exercise 2.2. Provide an example ofFX 2/ X for some set X on some poset.

I will denote meets and joins for a speci�c poset A asdA and

FA .

Proposition 2.36.

1. If b is the greatest element of X thenFX = b.

2. If b is the least element of X thendX = b.

Proof. We will prove only the �rst as the second is dual.Let b be the greatest element of X . Then upper bounds of X are fy 2A j yw bg. Obviously b

is the least element of this set, that is the join. �

De�nition 2.37. Binary joins and meets are de�ned by the formulas

xt y=Gfx; yg and xu y=

lfx; yg:

Obvious 2.38. t and u are symmetric operations (whenever these are de�ned for given x and y).

Theorem 2.39.

1. IfFX exists then yw

FX,8x2X : ywx.

2. IfdX exists then yv

dX,8x2X : yvx.

Proof. I will prove only the �rst as the second follows by duality.y w

FX, y is an upper bound for X,8x2X: y wx. �

Corollary 2.40.

1. If at b exists then yw at b, y w a^ y w b.

2.1 Order theory 19

2. If au b exists then yv au b, y v a^ y v b.

2.1.4 Semilattices

De�nition 2.41.

1. A join-semilattice is a poset A such that at b is de�ned for every a; b2A.

2. A meet-semilattice is a poset A such that au b is de�ned for every a; b2A.

Theorem 2.42.

1. The operation t is associative for any join-semilattice.

2. The operation u is associative for any meet-semilattice.

Proof. I will prove only the �rst as the second follows by duality.We need to prove (at b)t c= at (bt c) for every a; b; c2A.Taking into account the de�nition of join, it is enough to prove that

xw (at b)t c,xw at (bt c)

for every x2A. Really, this follows from the chain of equivalences:

xw (at b)t c,xw at b^ xw c,xw a^ xw b^ xw c,xw a^ xw bt c,xw at (bt c): �

Obvious 2.43. a�/ b i� au b is non-least, for every elements a, b of a meet-semilattice.

Obvious 2.44. a� b if at b is the greatest element, for every elements a, b of a join-semilattice.

2.1.5 Lattices and complete lattices

De�nition 2.45. A bounded poset is a poset having both least and greatest elements.

De�nition 2.46. Lattice is a poset which is both join-semilattice and meet-semilattice.

De�nition 2.47. A complete lattice is a poset A such that for every X 2PA bothFX and

dX

exist.

Obvious 2.48. Every complete lattice is a lattice.

Proposition 2.49. Every complete lattice is a bounded poset.

Proof.F; is the least and

d; is the greatest element. �

Theorem 2.50. Let A be a poset.

1. IfFX is de�ned for every X 2PA, then A is a complete lattice.

2. IfdX is de�ned for every X 2PA, then A is a complete lattice.

Proof. See [26] or any lattice theory reference. �

Obvious 2.51. If X �Y for some X;Y 2PA where A is a complete lattice, then

1.FX v

FY ;

2.dX w

dY .

Proposition 2.52. If S 2PPA then for every complete lattice A

1.F S

S=FfFX j X 2Sg;

2.d S

S=dfdX j X 2Sg.


Proof. We will prove only the �rst as the second is dual.By de�nition of joins, it is enough to prove yw

F SS, yw

FfFX j X 2Sg.

Really, y wF S

S , 8x 2S

S: y w x, 8X 2 S8x 2 X: y w x , 8X 2 S: y wF

X ,y w

FfFX j X 2Sg. �

2.1.6 Distributivity of lattices

De�nition 2.53. A distributive lattice is such lattice A that for every x; y; z 2A

1. xu (yt z)= (xu y)t (xu z);

2. xt (yu z)= (xt y)u (xt z).

Theorem 2.54. For a lattice to be distributive it is enough just one of the conditions:

1. xu (yt z)= (xu y)t (xu z);

2. xt (yu z)= (xt y)u (xt z).

Proof. (xt y)u (xtz)=((xt y)ux)t((xt y)uz)=xt ((xuz)t (yuz))=(xt (xuz))t (yuz)=xt (yu z) (applied xu (yt z) = (xu y)t (xu z) twice). �

2.1.7 Di�erence and complement

De�nition 2.55. Let A be a distributive lattice with least element 0. The di�erence (denoteda n b) of elements a and b is such c2A that bu c=0 and at b= bt c. I will call b substractive froma when a n b exists.

Theorem 2.56. If A is a distributive lattice with least element 0, there exists no more than onedi�erence of elements a, b.

Proof. Let c and d be both di�erences a n b. Then bu c= bu d=0 and at b= bt c= bt d. So

c= cu (bt c) = cu (bt d) = (cu b)t (cu d)= 0t (cu d)= cu d:

Similarly d= du c. Consequently c= cu d= du c= d. �

De�nition 2.57. I will call b complementive to a i� there exists c 2 A such that b u c = 0 andbt c= a.

Proposition 2.58. b is complementive to a i� b is substractive from a and bv a.

Proof.

(. Obvious.

). We deduce bv a from bt c= a. Thus at b= a= bt c. �

Proposition 2.59. If b is complementive to a then (a n b)t b= a.

Proof. Because bv a by the previous proposition. �

De�nition 2.60. Let A be a bounded distributive lattice. The complement (denoted a�) of anelement a2A is such b2A that au b=0 and at b=1.

Proposition 2.61. If A is a bounded distributive lattice then a�=1 n a.

Proof. b= a�, bu a=0^ bt a=1, bu a=0^ 1t a= at b, b=1 n a. �

Corollary 2.62. If A is a bounded distributive lattice then exists no more than one complementof an element a2A.

2.1 Order theory 21

De�nition 2.63. An element of bounded distributive lattice is called complemented when itscomplement exists.

De�nition 2.64. A distributive lattice is a complemented lattice i� every its element is comple-mented.

Proposition 2.65. For a distributive lattice (a n b)n c=a n (bt c) if a n b and (a n b)n c are de�ned.

Proof. ((a n b) n c)u c=0; ((a n b) n c)t c=(a n b)t c; (a n b)u b=0; (a n b)t b= at b.We need to prove ((a n b) n c)u (bt c)= 0 and ((a n b) n c)t (bt c)= at (bt c).In fact,

((a n b) n c)u (bt c) =

(((a n b) n c)u b)t (((a n b) n c)u c) =

(((a n b) n c)u b)t 0 =

((a n b) n c)u b v(a n b)u b = 0;

so ((a n b) n c)u (bt c)= 0;

((a n b) n c)t (bt c) =

(((a n b) n c)t c)t b =

(a n b)t ct b =

((a n b)t b)t c =

at bt c:

�

2.1.8 Boolean lattices

De�nition 2.66. A boolean lattice is a complemented distributive lattice.

The most important example of a boolean lattice is PA where A is a set, ordered by setinclusion.

Theorem 2.67. (De Morgan's laws) For every elements a, b of a boolean lattice

1. at b= a�u b�;2. au b= a�t b�.

Proof. We will prove only the �rst as the second is dual.It is enough to prove that at b is a complement of a�u b�. Really:

(at b)u (a�u b�)v au (a�u b�)= (au a�)u b�=0u b�=0;

(at b)t (a�u b�)= ((at b)t a�)u ((at b)t b�)w (at a�)u (bt b�)=1u 1=1:

Thus (at b)u (a�u b�)=0 and (at b)t (a�u b�)=1. �

De�nition 2.68. A complete lattice A is join in�nite distributive when x uFS =

Fhx u iS;

complete lattice is meet in�nite distributive when xtdS=

dhxt iS for all x2A and S 2PA.

De�nition 2.69. In�nite distributive complete lattice is a complete lattice which is both joinin�nite distributive and meet in�nite distributive.

Theorem 2.70. Every complete boolean lattice is both join in�nite distributive and meet in�nitedistributive.

Proof. We will prove only join in�nitely distributivity, as the other is dual.Let S be a subset of a complete boolean lattice.


xuFS w

Fhxu iS is obvious. Now let u be any upper bound of hxu iS, that is xu yvu for

all y 2S. Theny= y u (xt x�)= (yu x)t (yu x�)vutx�;

and soFS vutx�. Thus

xuG

S vxu (ut x�)= (xu u)t (xux�)= (xuu)t 0=xuuvu;

that is xuFS is the least upper bound of hxu iS. �

Theorem 2.71. (in�nite De Morgan's laws) For every subset S of a complete boolean lattice

1.FS=

dx2S x�;

2.dS=

Fx2S x�.

Proof. It's enough to prove thatFS is a complement of

dx2S x� (the second follows from duality).

Really, using the previous theorem:GS t

l

x2Sx�=

l

x2S

GS t

�x�=

l

x2S

�GS t x� j x2S

wl

x2Sfxt x� j x2Sg=1;

GS u

l

x2Sx�=

Gy2S

*lx2S

x�u+y=

Gy2S

(lx2S

x�u y j y 2S)vGy2S

fy�u y j y 2Sg=0:

SoFS t

dx2S x�=1 and

FS u

dx2S x�=0. �

2.1.9 Center of a lattice

De�nition 2.72. The center Z(A) of a bounded distributive lattice A is the set of its comple-mented elements.

Remark 2.73. For a de�nition of center of non-distributive lattices see [5].

Remark 2.74. In [23] the word center and the notation Z(A) are used in a di�erent sense.

De�nition 2.75. A sublattice K of a complete lattice L is a closed sublattice of L if K containsthe meet and the join of any its nonempty subset.

Theorem 2.76. Center of an in�nitely distributive lattice is its closed sublattice.

Proof. See [16]. �

Remark 2.77. See [17] for a more strong result.

Theorem 2.78. The center of a bounded distributive lattice constitutes its sublattice.

Proof. Let A be a bounded distributive lattice and Z(A) be its center. Let a; b 2 Z(A). Conse-quently a�; b�2Z(A). Then a�t b� is the complement of au b because

(au b)u (a�t b�)= (au bu a�)t (au bu b�)=0t 0=0 and(au b)t (a�t b�)= (at a�t b�)u (bt a�t b�)=1u 1=1:

So au b is complemented. Similarly at b is complemented. �

Theorem 2.79. The center of a bounded distributive lattice constitutes a boolean lattice.

Proof. Because it is a distributive complemented lattice. �

2.1.10 Atoms of posets

De�nition 2.80. An atom of a poset is an element which has no non-least subelements.

2.1 Order theory 23

Remark 2.81. This de�nition is valid even for posets without least element.

I will denote (atomsAa) or just (atomsa) the set of atoms contained in an element a of a posetA. I will denote atomsA the set of all atoms of a poset A.

De�nition 2.82. A poset A is called atomic i� atoms a=/ ; for every non-least element a of theposet A.

De�nition 2.83. Atomistic poset is such a poset that a=F

atoms a for every non-least elementa of this poset.

Obvious 2.84. Every atomistic poset is atomic.

Proposition 2.85. Let A be a poset. If a is an atom of A and B 2A then avB, a�/ B.

Proof.

). avB) av a^ avB, thus a�/ B because a is not least.

(. a�/ B implies existence of non-least element x such that xvB and xv a. Because a is anatom, we have x= a. So avB. �

Theorem 2.86. atomsdS=

ThatomsiS whenever

dS is de�ned for every S 2PA where A is

a poset.

Proof. For any atom c

c2 atomsl

S ,cv

lS ,

8a2S: cv a ,8a2S: c2 atoms a ,c2\hatomsiS:

�

Corollary 2.87. atoms(au b) = atoms a\ atoms b for an arbitrary meet-semilattice.

Theorem 2.88. A complete boolean lattice is atomic i� it is atomistic.

Proof.

(. Obvious.

). Let A be an atomic boolean lattice. Let a2A. Suppose b=F

atomsa@a. If x2atoms(a nb)then x v a n b and so x v a and hence x v b. But we have x= x u b v (a n b) u b= 0 whatcontradicts to our supposition. �

2.1.11 Kuratowski's lemma

Theorem 2.89. (Kuratowski lemma) Any chain in a poset is contained in a maximal chain (if weorder chains by inclusion).

I will skip the proof of Kuratowski lemma as this proof can be found in any set theory or ordertheory reference.

2.1.12 Homomorphisms of posets and lattices

De�nition 2.90. A monotone function (also called order homomorphism) from a poset A to aposet B is such a function f that xv y) fxv fy.


De�nition 2.91. Order embedding is a monotone injective function whose inverse is alsomonotone.

De�nition 2.92. Order isomorphism is a surjective order embedding.

Order isomorphism preserves properties of posets, such as order, joins and meets, etc.

De�nition 2.93.

1. Join semilattice homomorphism is a function f from a join semilattice A to a join semilatticeB, such that f(xt y)= fxt fy for every x; y 2A.

2. Meet semilattice homomorphism is a function f from a meet semilattice A to a meet semi-lattice B, such that f(xu y)= fxu fy for every x; y 2A.

Obvious 2.94.

1. Join semilattice homomorphisms are monotone.

2. Meet semilattice homomorphisms are monotone.

De�nition 2.95. A lattice homomorphism is a function from a lattice to a lattice, which is bothjoin semilattice homomorphism and meet semilattice homomorphism.

De�nition 2.96. Complete lattice homomorphism from a complete lattice A to a complete latticeB is a function f from A to B which preserves all meets and joins, that is f

FS =

Fhf iS and

fdS=

dhf iS for every S 2PA.

2.1.13 Galois connectionsSee [3] and [12] for more detailed treatment of Galois connections.

De�nition 2.97. Let A and B be two posets. A Galois connection between A and B is a pair offunctions f =(f�; f�) with f�:A!B and f�:B!A such that:

8x2A; y 2B: (f�xv y, xv f� y):

f� is called the upper adjoint of f� and f� is called the lower adjoint of f�.

Theorem 2.98. A pair (f�; f�) of functions f�:A!B and f�:B!A is a Galois connection i�both of the following:

1. f� and f� are monotone.

2. xv f� f�x and f� f� y v y for every x2A and y 2B.

Proof.

).2. xv f� f

�x since f�xv f�x; f� f� y v y since f� yv f� y.1. Let a; b2A and av b. Then av bv f� f

� b. So by de�nition f� av f� b that is f� ismonotone. Analogously f� in monotone.

(. f� xv y) f� f�xv f� y)xv f� y. The other direction is analogous. �

Theorem 2.99.

1. f� � f� � f�= f�.

2. f� � f� � f�= f�.

Proof.

1. Let x 2 A. We have x v f� f� x; consequently f� x v f� f� f

� x. On the other hand,f� f� f

�xv f� x. So f� f� f�x= f�x.

2.1 Order theory 25

2. Similar. �

De�nition 2.100. A function f is called idempotent i� f(f(X))= f(X) for every argument X .

Proposition 2.101. f� � f� and f� � f� are idempotent.

Proof. f� � f� is idempotent because f� f� f� f� y= f� f� y. f� � f� is similar. �

Theorem 2.102. Each of two adjoints is uniquely determined by the other.

Proof. Let p and q be both upper adjoints of f . We have for all x2A and y 2B:

xv p(y), f(x)v y,xv q(y):

For x= p(y) we obtain p(y)v q(y) and for x= q(y) we obtain q(y)v p(y). So q(y)= p(y). �

Theorem 2.103. Let f be a function from a poset A to a poset B.

1. Both:

1. If f is monotone and g(b) =max fx 2A j fx v bg is de�ned for every b 2B then gis the upper adjoint of f .

2. If g:B!A is the upper adjoint of f then g(b)=maxfx2A j fxv bg for every b2B.

2. Both:

1. If f is monotone and g(b)=min fx2A j fxw bg is de�ned for every b2B then g isthe lower adjoint of f .

2. If g:B!A is the lower adjoint of f then g(b)=minfx2A j fxw bg for every b2B.

Proof. We will prove only the �rst as the second is its dual.

1. Let g(b) =max fx2A j fxv bg for every b2B. Then

xv gy, xvmax fx2A j fxv yg) fxv y

(because f is monotone) and

xv gy, xvmax fx2A j fxv yg( fxv y:

So fxv y,xv gy that is f is the lower adjoint of g.

2. We have

g(b)=max fx2A j fxv bg ,fgbv b^8x2A: (fxv b)xv gb)

what is true by properties of adjoints. �

Theorem 2.104. Let f be a function from a poset A to a poset B.

1. If f is an upper adjoint, f preserves all existing in�ma in A.

2. If A is a complete lattice and f preserves all in�ma, then f is an upper adjoint of a functionB!A.

3. If f is a lower adjoint, f preserves all existing suprema in A.

4. If A is a complete lattice and f preserves all suprema, then f is a lower adjoint of a functionB!A.

Proof. We will prove only �rst two items because the rest items are similar.

1. Let S 2PA anddS exists. f

dS is a lower bound for hf iS because f is order-preserving.

If a is a lower bound for hf iS then 8x2S:av fx that is 8x2S: gavx where g is the loweradjoint of f . Thus ga v

dS and hence f

dS w a. So f

dS is the greatest lower bound

for hf iS.


2. Let A be a complete lattice and f preserves all in�ma. Let

g(a) =lfx2A j fxw ag:

Since f preserves in�ma, we have

f(g(a))=lff(x) j x2A; fxw agw a:

g(f(b))=dfx2A j fxw fbgv b.

Obviously f is monotone and thus g is also monotone.So f is the upper adjoint of g. �

Corollary 2.105. Let f be a function from a complete lattice A to a poset B. Then:

1. f is an upper adjoint of a function B!A i� f preserves all in�ma in A.

2. f is an lower adjoint of a function B!A i� f preserves all suprema in A.

2.1.14 Co-Brouwerian lattices

De�nition 2.106. Let A be a poset. Pseudocomplement of a2A is

max fc2A j c� ag:

If z is the pseudocomplement of a we will denote z= a�.

De�nition 2.107. Let A be a poset. Dual pseudocomplement of a2A is

min fc2A j c� ag:

If z is the dual pseudocomplement of a we will denote z= a+.

Proposition 2.108. If a is a complemented element of a bounded distributive lattice, then a� isboth pseudocomplement and dual pseudocomplement of a.

Proof. Because of duality it is enough to prove that a� is pseudocomplement of a.We need to prove c�a)cva� for every element c of our poset, and a��a. The second is obvious.

Let's prove c� a) cv a�.Really, let c� a. Then cu a=0; a�t (cu a)= a�; (a�t c)u (a�t a) = a�; a�t c= a�; cv a�. �

De�nition 2.109. Let A be a join-semilattice. Let a; b2A. Pseudodi�erence of a and b is

min fz 2A j av bt zg:

If z is a pseudodi�erence of a and b we will denote z= a n� b.

Remark 2.110. I do not require that a� is unde�ned if there are no pseudocomplement of a andlikewise for dual pseudocomplement and pseudodi�erence. In fact below I will de�ne quasicomple-ment, dual quasicomplement, and quasidi�erence which generalize pseudo-* counterparts. I willdenote a� the more general case of quasicomplement than of pseudocomplement, and likewise forother notation.

Obvious 2.111. Dual pseudocomplement is the dual of pseudocomplement.

De�nition 2.112. Co-brouwerian lattice is a lattice for which pseudodi�erence of any two itselements is de�ned.

Proposition 2.113. Every non-empty co-brouwerian lattice A has least element.

Proof. Let a be an arbitrary lattice element. Then

a n� a=min fz 2A j av at zg=minA:

2.1 Order theory 27

So minA exists. �

De�nition 2.114. Co-Heyting lattice is co-brouwerian lattice with greatest element.

Theorem 2.115. For a co-brouwerian lattice at¡ is an upper adjoint of ¡n�a for every a2A.

Proof. g(b)=minfx2A j atxw bg= b n�a exists for every b2A and thus is the lower adjoint ofat¡. �

Corollary 2.116. 8a; x; y 2A: (x n� av y,xv at y) for a co-brouwerian lattice.

De�nition 2.117. Let a; b2A where A is a complete lattice. Quasidi�erence a n� b is de�ned bythe formula:

a n� b=lfz 2A j av bt zg:

Remark 2.118. A more detailed theory of quasidi�erence (as well as quasicomplement and dualquasicomplement) will be considered below.

Lemma 2.119. (a n� b)t b=at b for elements a, b of a meet in�nite distributive complete lattice.

Proof.

(a n� b)t b =lfz 2A j av bt zgt b =l

fz t b j z 2A; av bt zg =lft2A j tw b; av tg =

at b:�

Theorem 2.120. The following are equivalent for a complete lattice A:

1. A is meet in�nite distributive.

2. A is a co-brouwerian lattice.

3. A is a co-Heyting lattice.

4. at¡ has lower adjoint for every a2A.

Proof.

(2),(3). Obvious (taking into account completeness of A).

(4))(1). Let ¡n�a be the lower adjoint of a t ¡. Let S 2 PA. For every y 2 S we havey w (a t y) n� a by properties of Galois connections; consequently y w (

dha t iS) n� a;d

S w (dhat iS) n� a. So

atl

S w¡¡l

hat iS�n� a

�t aw

lhat iS:

But atdS v

dhat iS is obvious.

(1))(2). Let a n� b=dfz 2 A j a v b t zg. To prove that A is a co-brouwerian lattice it is

enough to prove av bt (a n� b). But it follows from the lemma.

(2))(4). a n� b=min fz 2A j av bt zg. So at¡ is the upper adjoint of ¡n�a.

(1))(4). Because at¡ preserves all meets. �

Corollary 2.121. Co-brouwerian lattices are distributive.

The following theorem is essentially borrowed from [18]:


Theorem 2.122. A lattice A with least element 0 is co-brouwerian with pseudodi�erence n� i�n� is a binary operation on A satisfying the following identities:

1. a n� a=0;

2. at (b n� a)= at b;

3. bt (b n� a)= b;

4. (bt c) n� a=(b n� a)t (c n� a).

Proof.

(. We have

cw b n� a) ct aw at (b n� a)= at bw b;

ct aw b) c= ct (c n� a)w (a n� a)t (c n� a)= (at c) n� aw b n� a.So cw b n� a, ct aw b that is at¡ is an upper adjoint of ¡n�a. By a theorem above

our lattice is co-brouwerian. By another theorem above n� is a pseudodi�erence.

).

1. Obvious.

2.

at (b n� a) =

atlfz 2A j bv at zg =l

fat z j z 2A; bv at zg =

at b:

3. bt (b n� a)= btdfz 2A j bv at zg=

dfbt z j z 2A; bv at zg= b.

4. Obviously (b t c) n� a w b n� a and (b t c) n� a w c n� a. Thus (b t c) n� a w(b n� a)t (c n� a). We have

(b n� a)t (c n� a)t a =

((b n� a)t a)t ((c n� a)t a) =

(bt a)t (ct a) =

at bt c wbt c:

From this by de�nition of adjoints: (b n� a)t (c n� a)w (bt c) n� a. �

Theorem 2.123. (FS) n� a =

Ffx n� a j x 2 Sg for all a 2 A and S 2 PA where A is a co-

brouwerian lattice andFS is de�ned.

Proof. Because lower adjoint preserves all suprema. �

Theorem 2.124. (a n� b)n� c=a n� (bt c) for elements a, b, c of a complete co-brouwerian lattice.

Proof. a n� b=dfz 2A j av bt zg.

(a n� b) n� c=dfz 2A j a n� bv ct zg.

a n� (bt c)=dfz 2A j av bt ct zg.

It is left to prove a n� bv ct z, av bt ct z.Let a n� bv ct z. Then at bv bt ct z by the lemma and consequently av bt ct z.Let av bt ct z. Then a n� bv (bt ct z) n� bv ct z by a theorem above. �

2.1.15 Dual pseudocomplement on co-Heyting lattices

Proposition 2.125. For co-Heyting algebras 1 n� b= b+.

2.1 Order theory 29

Proof. 1 n� b=min fz 2A j 1v bt zg=min fz 2A j 1= bt zg=min fz 2A j b� zg= b+. �

Theorem 2.126. (au b)+= a+t b+ for every elements a, b of a co-Heyting algebra.

Proof. at (au b)+w (au b)t (au b)+w 1. So at (au b)+w 1; (au b)+w 1 n� a= a+.We have (au b)+w a+. Similarly (au b)+w b+. Thus (au b)+w a+t b+.On the other hand, a+ t b+ t (a u b) = (a+ t b+ t a) u (a+ t b+ t b). Obviously a+ t b+ t a=

a+t b+t b=1. So a+t b+t (au b)w 1 and thus a+t b+w 1 n� (au b)= (au b)+.So (au b)+= a+t b+. �

2.2 Intro to category theory

I recall that this is a very basic introduction to category theory, I even do not de�ne functors asthey have no use in my theory.

De�nition 2.127. A directed multigraph is:

1. a set O (vertices);

2. a set M (edges);

3. functions Src and Dst (source and destination) from M to O.

Note that in category theory vertices are called objects and edges are called morphisms .

De�nition 2.128. A precategory is a directed multigraph together with a partial binary operation� on the setM such that g � f is de�ned i� Dst f =Src g (for every morphisms f and g) such that

1. Src(g � f) = Src f and Dst(g � f) =Dst g whenever the composition g � f of morphisms fand g is de�ned.

2. (h� g) � f =h � (g � f) whenever compositions in this equation are de�ned.

De�nition 2.129. The set Mor(A; B) (morphisms from an object A to an object B) is exactlymorphisms which have A as the source and B as the destination.

De�nition 2.130. Identity morphism is such a morphism e that e� f = f and g � e= g whenevercompositions in these formulas are de�ned.

De�nition 2.131. A category is a precategory with additional requirement that for every objectX there exists identity morphism 1X.

Proposition 2.132. For every object X there exist no more than one identity morphism.

Proof. Let p and q be both identity morphisms for a object X . Then p= p � q= q. �

De�nition 2.133. An isomorphism is such a morphism f of a category that there exists amorphism f¡1 (inverse of f) such that f � f¡1=1Dst f and f¡1 � f =1Src f.

Proposition 2.134. An isomorphism has exactly one inverse.

Proof. Let g and h be both inverses of f . Then h=h � 1Dst f =h � f � g=1Src f � g= g. �

De�nition 2.135. A groupoid is a category all of whose morphisms are isomorphisms.

Some important examples of categories:

Exercise 2.3. Prove that the below examples of categories are really categories.

De�nition 2.136. The category Set is:

� Objects are small sets.


� Morphisms from an object A to an object B are triples (A;B; f) where f is a function fromA to B.

� Composition of morphisms is de�ned by the formula: (B; C; g) � (A;B; f) = (A; C; g � f)where g � f is function composition.

De�nition 2.137. The category Rel is:


� Morphisms from an object A to an object B are triples (A;B; f) where f is a binary relationbetween A and B.

� Composition of morphisms is de�ned by the formula: (B; C; g) � (A;B; f) = (A; C; g � f)where g � f is relation composition.

I will denote GR(A;B; f)= f for any morphism (A;B; f) of either Set or Rel.I will denote hf i= hGR f i and [f ]=[GR f ] for any morphism f of either Set or Rel.

De�nition 2.138. A morphism whose source is the same as destination is called endomorphism.

De�nition 2.139. Wide subcategory of a category (O;M) is a category (O;M0) where M0�Mand the composition on (O;M0) is a restriction of composition of (O;M). (Similarly wide sub-precategory can be de�ned.)

2.3 Intro to group theoryDe�nition 2.140. A semigroup is a pair of a set G and an associative binary operation on G.

De�nition 2.141. A group is a pair of a set G and a binary operation � on G such that:

1. (h � g) � f =h � (g � f) for every f ; g; h2G.

2. There exists an element e (identity) of G such that f � e= e � f = f for every f 2G.

3. For every element f there exists an element f¡1 such that f � f¡1= f¡1 � f = e.

Obvious 2.142. Every group is a semigroup.

Proposition 2.143. In every group there exist exactly one identity element.

Proof. If p and q are both identities, then p= p � q= q. �

Proposition 2.144. Every group element has exactly one inverse.

Proof. Let p and q be both inverses of f 2G. Then f � p= p � f = e and f � q = q � f = e. Thenp= p � e= p � f � q= e � q= q. �

Proposition 2.145. (g � f)¡1= f¡1 � g¡1 for every group elements f and g.

Proof. (f¡1 � g¡1) � (g � f)= f¡1 � g¡1 � g � f= f¡1 �e � f= f¡1 � f=e. Similarly (g � f) � (f¡1 � g¡1)=e.So f¡1 � g¡1 is the inverse of g � f . �

De�nition 2.146. A permutation group on a set D is a group whose elements are functions onD and whose composition is function composition.

Obvious 2.147. Elements of a permutation group are bijections.

De�nition 2.148. A transitive permutation group on a set D is such a permutation group G onD that for every x; y 2D there exists r 2G such that y= r(x).

A groupoid with single (arbitrarily chosen) object corresponds to every group. The morphismsof this category are elements of the group and the composition of morphisms is the group operation.

2.3 Intro to group theory 31

Chapter 3

More on order theory

3.1 Straight maps and separation subsets

3.1.1 Straight maps

De�nition 3.1. Let f be a monotone map from a meet-semilattice A to some poset B. I call fa straight map when

8a; b2A: (fav fb) fa= f(au b)):

Proposition 3.2. The following statements are equivalent for a monotone map f :

1. f is a straight map.

2. 8a; b2A: (fav fb) fav f(au b)).

3. 8a; b2A: (fav fb) faAf(au b)).4. 8a; b2A: (faA f(au b)) favfb).

Proof.

(1),(2),(3). Due faw f(au b).

(3),(4). Obvious. �

Remark 3.3. The de�nition of straight map can be generalized for any poset A by the formula

8a; b2A: (fav fb)9c2A: (cv a^ cv b^ fa= fc)):

This generalization is not yet researched however.

Proposition 3.4. Let f be a monotone map from a meet-semilattice A to a meet-semilattice B. If

8a; b2A: f(au b)= fau fbthen f is a straight map.

Proof. Let fav fb. Then f(au b)= fau fb= fa. �

Proposition 3.5. Let f be a monotone map from a meet-semilattice A to some poset B. If

8a; b2A: (fav fb) av b)then f is a straight map.

Proof. fav fb) av b) a= au b) fa= f(au b). �

Theorem 3.6. If f is a straight monotone map from a meet-semilattice A then the followingstatements are equivalent:

1. f is an injection.

33

2. 8a; b2A: (fav fb) av b).

3. 8a; b2A: (a@ b) fa@ fb).

4. 8a; b2A: (a@ b) fa=/ fb).

5. 8a; b2A: (a@ b) fawfb).

6. 8a; b2A: (fav fb) aAb).

Proof.

(1))(3). Let a; b2A. Let fa= fb)a= b. Let a@ b. fa=/ fb because a=/ b. fav fb becauseav b. So fa@ fb.

(2))(1). Let a; b2A. Let fav fb)avb. Let fa= fb. Then av b and bva and consequentlya= b.

(3))(2). Let 8a; b2A: (a@ b) fa@ fb). Let avb. Then aAau b. So faA f(au b). If fav fbthen fav f (au b) what is a contradiction.

(3))(5))(4). Obvious.

(4))(3). Because a@ b) av b) fav fb.

(5),(6). Obvious. �

3.1.2 Separation subsets and full stars

De�nition 3.7. @Y a= fx2Y j x�/ ag for an element a of a poset A and Y 2PA.

De�nition 3.8. Full star of a2A is ?a= @A a.

Proposition 3.9. If A is a meet-semilattice, then ? is a straight monotone map.

Proof. Monotonicity is obvious. Let ?av ? (au b). Then it exists x2?a such that x2/ ?(au b). Soxu a2/ ?b but xu a2 ?a and consequently ?av ? b. �

De�nition 3.10. A separation subset of a poset A is such its subset Y that

8a; b2A: (@Y a=@Y b) a= b):

De�nition 3.11. I call separable such poset that ? is an injection.

Obvious 3.12. A poset is separable i� it has a separation subset.

De�nition 3.13. A poset A has disjunction property of Wallman i� for any a; b2A either bv aor there exists a non-least element cv b such that a� c.

Theorem 3.14. For a meet-semilattice with least element the following statements are equivalent:

1. A is separable.

2. 8a; b2A: (?av ?b) av b).

3. 8a; b2A: (a@ b) ?a@ ?b).4. 8a; b2A: (a@ b) ?a=/ ?b).

5. 8a; b2A: (a@ b) ?aw ? b).

6. 8a; b2A: (?av ?b) aAb).7. A conforms to Wallman's disjunction property.

8. 8a; b2A: (a@ b)9c2A n f0g: (c� a^ cv b)).

34 More on order theory

Proof.

(1),(2),(3),(4),(5),(6). By the above theorem.

(8))(4). Let property (8) hold. Let a@ b. Then it exists element c v b such that c=/ 0 andcu a=0. But cu b=/ 0. So ?a=/ ?b.

(2))(7). Let property (2) hold. Let avb. Then ?av ? b that is it there exists c 2 ?a suchthat c2/ ?b, in other words cu a=/ 0 and cu b=0. Let d= cu a. Then dv a and d=/ 0 anddu b=0. So disjunction property of Wallman holds.

(7))(8). Obvious.

(8))(7). Let bva. Then au b@ b that is a0@ b where a0= au b. Consequently 9c 2A n f0g:(c� a0 ^ c v b). We have c u a= c u b u a= c u a0. So c v b and c u a= 0. Thus Wallman'sdisjunction property holds. �

Proposition 3.15. Every boolean lattice is separable.

Proof. Let a; b 2 A where A is a boolean lattice an a =/ b. Then a u b� =/ 0 or a� u b =/ 0 becauseotherwise au b�=0 and at b�=1 and thus a= b. Without loss of generality assume au b�=/ 0. Thenau c=/ 0 and bu c=0 for c= au b�=/ 0. �

3.1.3 Atomically Separable Lattices

Proposition 3.16. �atoms� is a straight monotone map (for any meet-semilattice).

Proof. Monotonicity is obvious. The rest follows from the formula

atoms(au b)= atoms a\ atoms b

(the corollary 2.87). �

De�nition 3.17. I will call atomically separable such a poset that �atoms� is an injection.

Proposition 3.18. 8a; b2A: (a@b)atomsa�atoms b) i� A is atomically separable for a poset A.

Proof.

(. Obvious.

). Let a=/ b for example avb. Then au b@ a; atoms a� atoms(au b)= atomsa\ atoms b andthus atoms a=/ atoms b. �

Proposition 3.19. Any atomistic poset is atomically separable.

Proof. We need to prove that atoms a= atoms b) a= b. But it is obvious because

a=G

atoms a and b=G

atoms b: �

Theorem 3.20. If a lattice with least element is atomic and separable then it is atomistic.

Proof. Suppose the contrary that is aAF atoms a. Then, because our lattice is separable, thereexists c2A such that cu a=/ 0 and cu

Fatoms a=0. There exists atom dv c such that dv cu a.

duF

atoms av cuF

atoms a=0. But d2 atoms a. Contradiction. �

Theorem 3.21. Let A be an atomic meet-semilattice with least element. Then the followingstatements are equivalent:

1. A is separable.

2. A is atomically separable.

3. A conforms to Wallman's disjunction property.

3.1 Straight maps and separation subsets 35

4. 8a; b2A: (a@ b)9c2A n f0g: (c� a^ cv b)).

Proof.

(1),(3),(4). Proved above.

(2))(4). Let our semilattice be atomically separable. Let a@ b. Then atomsa� atoms b andso there exists c 2 atoms b such that c 2/ atoms a. c=/ 0 and c v b, from which (taking intoaccount that c is an atom) cvb and cua=0. So our semilattice conforms to the formula (4).

(4))(2). Let formula (4) hold. Then for any elements a@ b there exists c=/ 0 such that cv band cu a=0. Because A is atomic there exists atom dv c. d2 atoms b and d2/ atoms a. Soatoms a=/ atoms b and atoms a� atoms b. Consequently atoms a� atoms b. �

Theorem 3.22. Any atomistic meet-semilattice with least element is separable.

Proof. From the above. �

3.2 Free Stars

De�nition 3.23. An upper set is such a set F 2PZ that

8X 2F ; Y 2Z: (Y wX)Y 2F ):

De�nition 3.24. Let A be a poset. Free stars on A are such S 2PA that the least element (ifit exists) is not in S and for every X;Y 2A

8Z 2A: (Z wX ^Z wY )Z 2S),X 2S _Y 2S:

Proposition 3.25. S 2PA where A is a poset is a free star i� all of the following:

1. The least element (if it exists) is not in S.

2. 8Z 2A: (Z wX ^Z wY )Z 2S))X 2S _Y 2S for every X;Y 2A.

3. S is an upper set.

Proof.

). (1) and (2) are obvious. Let prove that S is an upper set. Let X 2S and X vY 2A. ThenX 2S _X 2S and thus 8Z 2A: (Z wX ^Z wX)Z 2S) that is 8Z 2A: (Z wX)Z 2S),and so Y 2S.

(. We need to prove that

8Z 2A: (Z wX ^Z wY )Z 2S)(X 2S _Y 2S:

LetX 2S_Y 2S. Then ZwX ^ZwY )Z2S for every Z 2A because S is an upper set. �

Proposition 3.26. Let A be a join-semilattice. S 2PA is a free star i� all of the following:


2. X tY 2S)X 2S _Y 2S for every X;Y 2A.


Proof.

). We need to prove only X tY 2S)X 2S _Y 2S. Let X tY 2S. Because S is an upperset, we have 8Z 2A: (Z wX tY )Z 2S) and thus 8Z 2A: (Z wX ^Z wY )Z 2S) fromwhich we conclude X 2S _Y 2S.

(. We need to prove 8Z 2A: (Z wX ^Z wY )Z 2S)(X 2S _Y 2S.But it trivially follows from that S is an upper set. �


Proposition 3.27. Let A be a join-semilattice. S 2PA is a free star i� the least element (if itexists) is not in S and for every X;Y 2A

X tY 2S,X 2S _Y 2S:

Proof.

). We need to prove only X tY 2S(X 2S_Y 2S what follows from that S is an upper set.

(. We need to prove only that S is an upper set. Let X 2 S and X v Y 2 A. ThenX 2S)X 2S _Y 2S,X tY 2S)Y 2S. So S is an upper set. �

3.2.1 Starrish posets

De�nition 3.28. I will call a poset starrish when the full star ?a is a free star for every elementa of this poset.

Proposition 3.29. Every distributive lattice is starrish.

Proof. Let A be a distributive lattice, a 2 A. Obviously 0 2/ ?a (if 0 exists); obviously ?a is anupper set. If xt y 2 ?a, then (x t y) u a is non-least that is (xu a) t (y u a) is non-least what isequivalent to xu a or yu a being non-least that is x2 ?a_ y 2 ?a. �

Theorem 3.30. If A is a starrish join-semilattice lattice then

atoms(at b)= atoms a[ atoms b

for every a; b2A.

Proof. For every atom c we have: c2atoms(at b), c�/ at b,at b2?c,a2?c_ b2?c, c�/ a_c�/ b, c2 atoms a_ c2 atoms b. �

3.3 Quasidi�erence and Quasicomplement

I've got quasidi�erence and quasicomplement (and dual quasicomplement) replacing max and minin the de�nition of pseudodi�erence and pseudocomplement (and dual pseudocomplement) withF

andd

. Thus quasidi�erence and (dual) quasicomplement are generalizations of their pseudo-counterparts.

Remark 3.31. Pseudocomplements and pseudodi�erences are standard terminology. Quasi - coun-terparts are my neologisms.

De�nition 3.32. Let A be a poset, a2A. Quasicomplement of a is

a�=Gfc2A j c� ag:

De�nition 3.33. Let A be a poset, a2A. Dual quasicomplement of a is

a+=lfc2A j c� ag:

I will denote quasicomplement and dual quasicomplement for a speci�c poset A as a�(A) anda+(A).

De�nition 3.34. Let a; b2A where A is a distributive lattice. Quasidi�erence of a and b is

a n� b=lfz 2A j av bt zg:

De�nition 3.35. Let a; b2A where A is a distributive lattice. Second quasidi�erence of a and b is

a#b=Gfz 2A j z v a^ z� bg:

3.3 Quasidifference and Quasicomplement 37

Theorem 3.36. a n� b=dfz 2A j zvaâv bt zg where A is a distributive lattice and a; b2A.

Proof. Obviously fz 2 A j z v a ^ a v b t zg � fz 2 A j a v b t zg. Thusdfz 2 A j z v a ^

av bt zgw a n� b.Let z 2A and z 0= z u a.a v b t z ) a v (b t z) u a, a v (b u a) t (z u a) , a v (b u a) t z 0 ) a v b u z 0 and

av bt z( av bu z 0. Thus av bt z, av bu z 0.If z 2fz 2A j av bt zg then av bt z and thus

z 02fz 2A j z v a^ av bt zg:

But z 0v z thus havingdfz 2A j z v a^ av bt zgv

dfz 2A j av bt zg. �

Remark 3.37. If we drop the requirement that A is distributive, two formulas for quasidi�erence(the de�nition and the last theorem) fork.

Obvious 3.38. Dual quasicomplement is the dual of quasicomplement.

Obvious 3.39.

� Every pseudocomplement is quasicomplement.

� Every dual pseudocomplement is dual quasicomplement.

� Every pseudodi�erence is quasidi�erence.

Below we will stick to the more general quasies than pseudos. If needed, one can check that aquasicomplement a� is a pseudocomplement by the equation a� � a (and analogously with otherquasies).

Next we will express quasidi�erence through quasicomplement.

Proposition 3.40.

1. a n� b= a n� (au b) for any distributive lattice;

2. a#b= a#(au b) for any distributive lattice with least element.

Proof.

1. a v (a u b) t z , a v (a t z) u (b t z) , a v a t z ^ a v b t z , a v b t z. Thusa n� (au b)=

dfz 2A j av (au b)t zg=

dfz 2A j av bt zg= a n� b.

2. a#(a u b) =Ffz 2 A j z v a ^ z u a u b = 0g =

Ffz 2 A j z v a ^ (z u a) u a u b = 0g =F

fz u a j z 2A; z u au b=0g=Ffz 2A j z v a; z u b=0g= a#b. �

I will denote Da the lattice fx2A j xv ag.

Theorem 3.41. For a; b2A where A is a distributive lattice with least element

1. a n� b=(au b)+(Da); [TODO: least element is not required?]

2. a#b=(au b)�(Da).

Proof.

1.

(au b)+(Da) =lfc2Da j ct (au b)= ag =lfc2Da j ct (au b)w ag =l

fc2Da j (ct a)u (ct b)w ag =lfc2A j cv a^ ct bw ag =

a n� b:


2.

(au b)�(Da) =Gfc2Da j cu au b=0g =G

fc2A j cv a^ cu au b=0g =Gfc2A j cv a^ cu b=0g =

a#b:

�

Proposition 3.42. (at b) n� bv a for an arbitrary complete lattice.

Proof. (at b) n� b=dfz 2A j at bv bt zg.

But av z) at bv bt z. So fz 2A j at bv bt zg� fz 2A j av zg.Consequently, (at b) n� bv

dfz 2A j av zg= a. �

3.4 Several equal ways to express pseudodi�erence

Theorem 3.43. For an atomistic co-brouwerian lattice A and a; b 2A the following expressionsare always equal:

1. a n� b=dfz 2A j av bt zg (quasidi�erence of a and b);

2. a#b=Ffz 2A j z v a^ z u b=0g (second quasidi�erence of a and b);

3.F(atoms a n atoms b).

Proof. Proof of (1)=(3):

a n� b =¡Gatoms a

�n� b = (theorem 2.123)G

fA n� b j A2 atoms ag =G ��A if A2/ atoms b0 if A2 atoms b

�j A2 atoms a

�=G

fA j A2 atomsa;A2/ atoms bg =G(atoms a n atoms b):

Proof of (2)=(3):a n� b is de�ned because our lattice is co-brouwerian. Taking the above into account, we have

a n� b =G(atoms a n atoms b) =G

fz 2 atoms a j z u b=0Ag:

SoFfz 2 atoms a j z u b=0Ag is de�ned.

If z v a ^ z u b= 0A then z 0=Ffx 2 atoms z j xu b= 0Ag is de�ned. z 0 is a lower bound for

fz 2 atoms a j z u b=0Ag.Thus z 02fz 2A j z v a^ z u b=0Ag and so

Ffz 2 atoms a j z u b=0Ag is an upper bound of

fz 2A j z v a^ z u b=0Ag.If y is above every z 02fz 2A j z v a^ z u b=0Ag then y is above every z 2 atoms a such that

z u b=0A and thus y is aboveFfz 2 atomsa j z u b=0Ag.

ThusFfz 2 atomsa j z u b=0Ag is least upper bound of

fz 2A j z v a^ z u b=0Ag;

3.4 Several equal ways to express pseudodifference 39

that isGfz 2A j z v a^ z u b=0Ag=

Gfz 2 atoms a j z u b=0Ag=

G(atoms a n atoms b): �

3.5 Partially ordered categories

3.5.1 De�nition

De�nition 3.44. I will call a partially ordered (pre)category a (pre)category together with partialorder v on each of its Mor-sets with the additional requirement that

f1v f2^ g1v g2) g1 � f1v g2 � f2for every morphisms f1, g1, f2, g2 such that Src f1= Src f2 ^ Dst f1= Dst f2 = Src g1 = Src g2 ^Dst g1=Dst g2.

3.5.2 Dagger categories

De�nition 3.45. I will call a dagger precategory a precategory together with an involutive con-travariant identity-on-objects prefunctor x 7!xy.

In other words, a dagger precategory is a precategory equipped with a function x 7! xy on itsset of morphisms which reverses the source and the destination and is subject to the followingidentities for every morphisms f and g:

1. f yy= f ;

2. (g � f)y= f y � gy.

De�nition 3.46. I will call a dagger category a category together with an involutive contravariantidentity-on-objects functor x 7!xy.

In other words, a dagger category is a category equipped with a function x 7! xy on its set ofmorphisms which reverses the source and the destination and is subject to the following identitiesfor every morphisms f and g and object A:

1. f yy= f ;

2. (g � f)y= f y � gy;

3. (1A)y=1A.

Theorem 3.47. If a category is a dagger precategory then it is a dagger category.

Proof. We need to prove only that (1A)y=1A. Really,

(1A)y=(1A)

y � 1A=(1A)y � (1A)yy=((1A)y � 1A)y=(1A)

yy=1A: �

For a partially ordered dagger (pre)category I will additionally require (for every morphisms fand g with the same source and destination)

f yv gy, f v g:

An example of dagger category is the category Rel whose objects are sets and whose morphisms arebinary relations between these sets with usual composition of binary relations and with f y= f¡1.

De�nition 3.48. A morphism f of a dagger category is called unitary when it is an isomorphismand f y= f¡1.

De�nition 3.49. Symmetric (endo)morphism of a dagger precategory is such a morphism f thatf = f y.


De�nition 3.50. Transitive (endo)morphism of a precategory is such a morphism f that f= f � f .

Theorem 3.51. The following conditions are equivalent for a morphism f of a dagger precategory:

1. f is symmetric and transitive.

2. f = f y � f .

Proof.

(1))(2). If f is symmetric and transitive then f y � f = f � f = f .

(2))(1). f y= (f y � f)y= f y � f yy= f y � f = f , so f is symmetric. f = f y � f = f � f , so f istransitive. �

3.5.2.1 Some special classes of morphisms

De�nition 3.52. For a partially ordered dagger category I will call monovalued morphism sucha morphism f that f � f yv 1Dst f.

De�nition 3.53. For a partially ordered dagger category I will call entirely de�ned morphismsuch a morphism f that f y � f w 1Src f.

De�nition 3.54. For a partially ordered dagger category I will call injective morphism such amorphism f that f y � f v 1Src f.

De�nition 3.55. For a partially ordered dagger category I will call surjective morphism such amorphism f that f � f yw 1Dst f.

Remark 3.56. It is easy to show that this is a generalization of monovalued, entirely de�ned,injective, and surjective functions as morphisms of the category Rel.

Obvious 3.57. �Injective morphism� is a dual of �monovalued morphism� and �surjective mor-phism� is a dual of �entirely de�ned morphism�.

De�nition 3.58. For a given partially ordered dagger category C the category of monovalued(entirely de�ned , injective, surjective) morphisms of C is the category with the same set of objectsas of C and the set of morphisms being the set of monovalued (entirely de�ned, injective, surjective)morphisms of C with the composition of morphisms the same as in C.

We need to prove that these are really categories, that is that composition of monovalued(entirely de�ned, injective, surjective) morphisms is monovalued (entirely de�ned, injective, sur-jective) and that identity morphisms are monovalued, entirely de�ned, injective, and surjective.

Proof. We will prove only for monovalued morphisms and entirely de�ned morphisms, as injectiveand surjective morphisms are their duals.

Monovalued. Let f and g be monovalued morphisms, Dst f = Src g. (g � f) � (g � f)y =g� f � f y� gyv g�1Dst f � gy= g�1Src g� gy= g� gyv1Dst g=1Dst(g�f). So g� f is monovalued.

That identity morphisms are monovalued follows from the following: 1A � (1A)y = 1A �1A=1A=1Dst 1Av 1Dst 1A.

Entirely de�ned. Let f and g be entirely de�ned morphisms, Dst f=Src g. (g� f)y� (g� f)=f y � gy � g � f w f y � 1Src g � f = f y � 1Dst f � f = f y � f w 1Src f =1Src(g�f). So g � f is entirelyde�ned.

That identity morphisms are entirely de�ned follows from the following:(1A)y � 1A=1A� 1A=1A=1Src 1Aw 1Src 1A. �

De�nition 3.59. I will call a bijective morphism a morphism which is entirely de�ned, mono-valued, injective, and surjective.

Proposition 3.60. If a morphism is bijective then it is an isomorphism.

3.5 Partially ordered categories 41

Proof. Let f be bijective. Then f � f yv1Dst f, f y� f w1Src f, f y� f v1Src f, f � f yw1Dst f. Thusf � f y=1Dst f and f y � f =1Src f that is f y is an inverse of f . �

[TODO: Below require that Mor-sets are complete lattices.]

De�nition 3.61. A morphism f of a partially ordered category is metamonovalued when (dG)�

f =dg2G (g � f) whenever G is a set of morphisms with a suitable domain and image.

De�nition 3.62. A morphism f of a partially ordered category is metainjective when f � (dG)=d

g2G (f � g) whenever G is a set of morphisms with a suitable domain and image.

Obvious 3.63. Metamonovaluedness and metainjectivity are dual to each other.

De�nition 3.64. A morphism f of a partially ordered category is metacomplete when f � (FG)=d

g2G (f � g) whenever G is a set of morphisms with a suitable domain and image.

De�nition 3.65. A morphism f of a partially ordered category is co-metacomplete when (FG) �

f =dg2G (g � f) whenever G is a set of morphisms with a suitable domain and image.

3.6 PartitioningDe�nition 3.66. Let A be a complete lattice. Torning of an element a2A is a set S 2PA nf0gsuch that G

S= a and 8x; y 2S: (x=/ y) x� y):

De�nition 3.67. Let A be a complete lattice. Weak partition of an element a 2 A is a setS 2PA n f0g such that G

S= a and 8x2S:x�G

(S n fxg):

De�nition 3.68. Let A be a complete lattice. Strong partition of an element a 2 A is a setS 2PA n f0g such thatG

S= a and 8A; B 2PS:¡A�B)

GA�

GB�:

Obvious 3.69.

1. Every strong partition is a weak partition.

2. Every weak partition is a torning.

3.7 A proposition about binary relationsProposition 3.70. Let f , g, h be binary relations. Then g � f �/ h, g�/ h � f¡1.

Proof.

g � f �/ h ,9a; c: a ((g � f)\h) c ,

9a; c: (a (g � f) c^ ah c) ,9a; b; c: (a f b^ b g c^ ah c) ,9b; c: (b g c^ b (h � f¡1) c) ,9b; c: (b (g \ (h� f¡1)) c) ,

g�/ h� f¡1:�


3.8 In�nite associativity and ordinated product

3.8.1 IntroductionWe will consider some function f which takes an arbitrary ordinal number of arguments. That isf can be taken for arbitrary (small, if to be precise) ordinal number of arguments. More formally:Let x=xi2n be a family indexed by an ordinal n. Then f(x) can be taken. The same function fcan take di�erent number of arguments. (See below for the exact de�nition.)

Some of such functions f are associative in the sense de�ned below. If a function is associativein the below de�ned sense, then the binary operation induced by this function is associative in theusual meaning of the word �associativity� as de�ned in basic algebra.

I also introduce and research an important example of in�nitely associative function, which Icall ordinated product .

Note that my searching about in�nite associativity and ordinals in Internet has provided nouseful results. As such there is a reason to assume that my research of generalized associativity interms of ordinals is novel.

3.8.2 Used notationWe identify natural numbers with �nite Von Neumann's ordinals (further just ordinals or ordinalnumbers).

For simplicity we will deal with small sets (members of a Grothendieck universe). We will denotethe Grothendieck universe (aka universal set) as f.

I will denote a tuple of n elements like Ja0; :::; an¡1K. By de�nition

Ja0; :::; an¡1K= f(0; a0); :::; (n¡ 1; an¡1)g:

Note that an ordered pair (a; b) is not the same as the tuple Ja; bK of two elements.

De�nition 3.71. An anchored relation is a tuple Jn; rK where n is an index set and r is an n-aryrelation.

For an anchored relation arityJn;rK=n. The graph3.1 of Jn;rK is de�ned as follows: GRJn;rK=r.

De�nition 3.72. Pri is a function de�ned by the formula

Pri f = fxi j x2 f g

for every small n-ary relation f where n is an ordinal number and i2n. Particularly for every n-ary relation f and i2n where n2N

Pri f = fxi j Jx0; :::; xn¡1K2 f g:

Recall that Cartesian product is de�ned as follows:Ya=

�z 2¡[

im a�dom a j 8i2dom a: z(i)2 ai

:

Obvious 3.73. If a is a small function, thenQ

a= fz 2fdom a j 8i2dom a: z(i)2 aig.

3.8.2.1 Currying and uncurrying

The customary de�nitionLet X , Y , Z be sets.We will consider variables x2X and y 2Y .Let a function f 2 ZX�Y . Then curry(f) 2 (ZY )X is the function de�ned by the formula

(curry(f)x) y= f(x; y).Let now f 2 (ZY )X. Then uncurry(f) 2 ZX�Y is the function de�ned by the formula

uncurry(f)(x; y)= (fx) y.

3.1. It is unrelated with graph theory.

3.8 Infinite associativity and ordinated product 43

Obvious 3.74.

1. uncurry(curry(f))= f for every f 2ZX�Y .2. curry(uncurry(f))= f for every f 2 (ZY )X.

Currying and uncurrying with a dependent variableLet X, Z be sets and Y be a function with the domain X. (Vaguely saying, Y is a variable

dependent on X.)The disjoint union

`Y =

Sffig�Yi j i2domY g= f(i; x) j i2domY ; x2Yig.

We will consider variables x2X and y 2Yx.Let a function f 2 Z

ì2XYi (or equivalently f 2 Z

`Y ). Then curry(f) 2

Qi2X ZYi is the

function de�ned by the formula (curry(f)x) y= f(x; y).Let now f 2

Qi2X ZYi. Then uncurry(f) 2 Z

ì2XYi is the function de�ned by the formula

uncurry(f)(x; y)= (fx) y.

Obvious 3.75.

1. uncurry(curry(f))= f for every f 2Zì2XYi.

2. curry(uncurry(f))= f for every f 2Q

i2X ZYi.

3.8.2.2 Functions with ordinal numbers of arguments

Let Ord be the set of small ordinal numbers.If X and Y are sets and n is an ordinal number, the set of functions taking n arguments on

the set X and returning a value in Y is Y Xn.

The set of all small functions taking ordinal numbers of arguments is YSn2OrdX

n

.I will denote OrdVar(X) = f

Sn2OrdX

n

and call it ordinal variadic. (�Var� in this notation istaken from the word variadic in the collocation variadic function used in computer science.)

3.8.3 On sums of ordinalsLet a be an ordinal-indexed family of ordinals.

Proposition 3.76.`

a with lexicographic order is a well-ordered set.

Proof. Let S be non-empty subset of`

a.Take i0=minPr0S and x0=minfPr1 y j y2S; y(0)= i0g (these exist by properties of ordinals).

Then (i0;x0) is the least element of S. �

De�nition 3.77.P

a is the unique ordinal order-isomorphic to`

a. [TODO: For �nite ordinalsit is just a sum of natural numbers.]

This ordinal exists and is unique because our set is well-ordered.

Remark 3.78. An in�nite sum of ordinals is not customary de�ned.

The structured sumL

a of a is an order isomorphism from lexicographically ordered set`

ainto

Pa.

There exists (for a given a) exactly one structured sum, by properties of well-ordered sets.

Obvious 3.79.P

a= imL

a.

Theorem 3.80. (L

a)(n;x)=P

i2n ai+x.

Proof. We need to prove that it is an order isomorphism. Let's prove it is an injection that ism>n)

Pi2m ai+x>

Pi2n ai+x and y >x)

Pi2n ai+ y >

Pi2n ai+x.

Really, if m>n thenP

i2m ai+x>Pi2n+1 ai+x>P

i2n ai+x. The second formula is trueby properties of ordinals.

Let's prove that it is a surjection. Let r 2P

a. There exist n 2 dom a and x 2 an such thatr = (

La)(n; x). Thus r = (

La)(n; 0) + x=

Pi2n ai + x because (

La)(n; 0) =

Pi2n ai since

(n; 0) hasP

i2n ai predecessors. �


3.8.4 Ordinated product

3.8.4.1 Introduction

Ordinated product de�ned below is a variation of Cartesian product, but is associative unlike Carte-sian product. However, ordinated product unlike Cartesian product is de�ned not for arbitrarysets, but only for relations having ordinal numbers of arguments.

Let F indexed by an ordinal number be a small family of anchored relations.

3.8.4.2 Concatenation

De�nition 3.81. Let z be an indexed by an ordinal number family of functions each taking anordinal number of arguments. The concatenation of z is

concat z= uncurry(z) �¡M

(dom � z)�¡1:

Obvious 3.82. If z is a �nite family of �nitary typles, it is concatenation of dom z tuples in theusual sense (as it is commonly used in computer science).

Proposition 3.83. If z 2Q

(GR �F ) then concat z= uncurry(z) � (L

(arity�F ))¡1.

Proof. If z 2Q

(GR �F ) then dom z(i)= dom (GR �F )i= domFi= arityFi for every i2 domF .Thus dom � z= arity�F . �

Proposition 3.84. dom concat z=P

i2dom z dom zi.

Proof. Because dom (L

(dom � z))¡1=P

i2domF dom zi, it is enough to prove that

domuncurry(z)= domM

(dom � z):Really,

domM

(dom � z)= f(i;x) j i2dom (dom � z); x2dom zig= f(i;x) j i2dom z; x2dom zig=a

z

and domuncurry(z) =`

i2X zi=`

z. �

3.8.4.3 Finite example

If F is a �nite family (indexed by a natural number domF ) of anchored �nitary relations, then byde�nition GR

Q(ord) F = fJa0;0; :::; a0;arityF0¡1; :::; adomF¡1;0; :::; adomF¡1;arityFdomF¡1¡1K j Ja0;0;:::; a0;arityF0¡1K2GRF0^ ::: ^ JadomF¡1;arityFdomF¡1¡1K2GRFdomF¡1g and

arityY(ord)

F = arityF0+ :::+ arityFdomF¡1:

The above formula can be shortened to

GRY(ord)

F=�concat z j z 2

Y(GR �F )

:

3.8.4.4 The de�nition

De�nition 3.85. The anchored relation (which I call ordinated product )Q(ord) F is de�ned by

the formulas:

arityY(ord)

F =X

(arity�F );

GRY(ord)

F=�concat z j z 2

Y(GR �F )

:

Proposition 3.86.Q(ord) F is a properly de�ned anchored relation.


Proof. dom concat z=P

i2domF dom zi=P

i2domF arityFi=P

(arity�F ). �

3.8.4.5 De�nition with composition for every multiplier

q(F )i=def(curry(

L(arity�F )))i.

Theorem 3.87. GRQ(ord) F =

�L2f

P(arity�F ) j 8i2domF :L� q(F )i2GRFi

.

Proof. GRQ(ord) F=fconcat z j z 2

Q(GR �F )g;

GRQ(ord)

F =�uncurry(z) � (

L(arity � F ))¡1 j z 2

Qi2domF f

arityFi; 8i 2 dom F : z(i) 2GRFi

.

Let L= uncurry(z). Then z= curry(L).GR

Q(ord)F =

�L � (

L(arity �F ))¡1 j curry(L)2

Qi2domF f

arityFi; 8i2 domF : curry(L)i2GRFi

;

GRQ(ord)

F =nL� (

L(arity�F ))¡1 j L2f

ì2domF

arityFi; 8i2domF : curry(L)i2GRFi

o;

GRQ(ord) F =

�L2f

P(arity�F ) j 8i2domF : curry(L �

L(arity�F ))i2GRFi

;

(curry(L�L

(arity�F ))i)x=L((curry(L

(arity�F ))i)x)=L(q(F )ix) = (L� q(F )i)x;curry(L �

L(arity�F ))i=L� q(F )i;

GRQ(ord) F =

�L2f

P(arity�F ) j 8i2domF :L � q(F )i2GRFi

. �

Corollary 3.88. GRQ(ord) F =

�L2 (

Sim(GR �F ))

P(arity�F ) j 8i2domF :L� q(F )i2GRFi

.

Corollary 3.89. GRQ(ord)

F is small if F is small.

3.8.4.6 De�nition with shifting arguments

Let Fi0= fL �Pr1jfig�arityFi j L2GRFig.

Proposition 3.90. Fi0= fL �Pr1jfig�f j L2GRFig.

Proof. If L2GRFi then domL= arityFi. Thus

L �Pr1jfig�arityFi=L�Pr1jfig�domL=L �Pr1jfig�f: �

Proposition 3.91. Fi0 is an (fig� arityFi)-ary relation.

Proof. We need to prove that dom(L � Pr1jfig�arityFi) = fig � arityFi for L 2GR Fi, but that'sobvious. �

Obvious 3.92.`

(arity�F )=Si2domF (fig� arityFi) =

Si2domF domFi

0.

Lemma 3.93. P 2Q

i2domF Fi0, curry(

Sim P ) 2

Q(GR � F ) for a dom F indexed family P

where Pi2ffig�arityFi for every i2 domF , that is for P 2`

i2domF ffig�arityFi.

Proof. For every P 2`

i2domF ffig�arityFi we have:

P 2Q

i2domF Fi0 , P 2 fz 2 fdomF j 8i 2 dom F : z(i) 2 Fi

0g , P 2 fdomF ^ 8i 2 dom F :

P (i) 2 Fi0 , P 2 fdomF ^ 8i 2 dom F9L 2 GR Fi: Pi = L � (Pr1jfig�f) , P 2 fdomF ^

8i 2 dom F9L 2 GR Fi:¡Pi 2 ffig�arityFi ^ 8x 2 arity Fi: Pi(i; x) = Lx

�, P 2 fdomF ^

8i 2 dom F9L 2 GR Fi:¡Pi 2 ffig�arityFi ^ curry(Pi)i = L

�, P 2 fdomF ^ 8i 2 dom F :¡

Pi 2 ffig�arityFi ^ curry(Pi)i 2 GR Fi�, 8i 2 dom F9Qi 2 (farityFi)fig: (Pi = uncurry(Qi) ^

(Qi)i2farityFi^Qi i2GRFi),8i2domF9Qi2(farityFi)fig¡Pi=uncurry(Qi)^

¡Si2domFQi

�i2

GR Fi�, 8i 2 dom F9Qi 2 (farityFi)fig

¡Pi = uncurry(Qi) ^

Si2domF Qi 2

Q(GR � F )

�,

8i 2 dom F :Si2domF curry(Pi) 2

Q(GR � F ) , curry

¡Si2domF Pi

�2Q

(GR � F ) ,curry(

SimP )2

Q(GR �F ). �


Lemma 3.94.ncurry(f) �

L(arity�F ) j f 2GR

Q(ord) Fo=Q

(GR �F ).

Proof. First GRQ(ord) F = funcurry(z) � (

L(dom � z))¡1 j z 2

Q(GR �F )g, that isn

f j f 2GRQ(ord) F

o= funcurry(z) � (

L(arity�F ))¡1 j z 2

Q(GR �F )g.

SinceL

(arity�F ) is a bijection, we havenf �L

(arity�F ) j f 2GRQ(ord) F

o= funcurry(z) j z 2

Q(GR �F )g what is equivalent ton

curry(f) �L


o= fz j z 2

Q(GR �F )g that isn

curry(f) �L


o=Q

(GR �F ). �

Lemma 3.95.�S

im P j P 2`

i2domF ffig�arityFi ^ curry(

Sim P ) 2

Q(GR � F )

=n

L2fì2domFarityFi j curry(L)2

Q(GR �F )

o.

Proof. Let L0 2nL 2 f

ì2XarityFi j curry(L) 2

Q(GR � F )

o. Then L0 2 f

ì2domFarityFi and

curry(L0)2Q

(GR �F ).Let P = �i 2 dom F : L0jfig�arityFi. Then P 2

ì2domF f

fig�arityFi andS

im P = L0. SoL02

�SimP j P 2

ì2domF f

fig�arityFi^ curry(S

imP )2Q

(GR �F ).

Let now L02�S

imP j P 2`

i2domF ffig�arityFi^curry(

SimP )2

Q(GR�F )

. Then there

exists P 2`

i2domF ffig�arityFi such that L0=

SimP and curry(L0)2

Q(GR�F ). Evidently L02

fì2domFarityFi. So L02

�SimP j P 2

ì2domF f

fig�arityFi^ curry(S

imP )2Q

(GR�F ). �

Lemma 3.96.nf �L


o=�S

imP j P 2Q

i2domF Fi0.

Proof. L 2�S

im P j P 2Q

i2domF Fi0 , L 2

�Sim P j P 2

ì2domF f

fig�arityFi ^curry(

Sim P ) 2

Q(GR � F )

, L 2 f

ì2domFarityFi ^ curry(L) 2

Q(GR � F ) , L 2

fì2domFarityFi ^ curry(L) 2

ncurry(f) �

L(arity � F ) j f 2 GR

Q(ord) Fo, (becauseL

(arity � F ) is a bijection),curry(L) � (L

(arity � F ))¡1 2ncurry(f) j f 2 GR

Q(ord) Fo,

L � (L

(arity � F ))¡1 2nf j f 2 GR

Q(ord) Fo, (because

L(arity � F ) is a bijection),

L2nf �L


o. �

Theorem 3.97.

GRY(ord)

F =

(¡[imP

��¡M

(arity�F )�¡1 j P 2 Y

i2domF

Fi0):

Proof. From the lemma, becauseL

(arity�F ) is a bijection. �

Theorem 3.98.

GRY(ord)

F =

( [i2domF

¡Pi �

¡M(arity�F )

�¡1� j P 2 Yi2domF

Fi0):

Proof. From the previous theorem. �

Theorem 3.99.

GRY(ord)

F =

([imP j P 2

Yi2domF

�f �¡M

(arity�F )�¡1 j f 2Fi0):


Proof. From the previous. �

Remark 3.100. Note that the above formulas contain bothSi2domF domFi

0 andSi2domF Fi

0.These forms are similar but di�erent.

3.8.4.7 Associativity of ordinated product

Let f be an ordinal variadic function.Let S be an ordinal indexed family of functions of ordinal indexed families of functions each

taking an ordinal number of arguments in a set X.I call f in�nite associative when

1. f(f �S) = f(concatS) for every S;

2. f(JxK)=x for x2X.

In�nite associativity implies associativity

Proposition 3.101. Let f be an in�nitely associative function taking an ordinal number ofarguments in a set X. De�ne x? y= fJx; yK for x; y2X. Then the binary operation ? is associative.

Proof. Let x; y; z 2X . Then (x ? y) ? z = fJfJx; yK; zK= f(fJx; yK; fJzK) = fJx; y; zK. Similarlyx? (y ? z)= fJx; y; zK. So (x? y) ? z=x? (y ? z). �

Concatenation is associativeFirst we will prove some lemmas.Let a and b be functions on a poset. Let a� b i� there exist an order isomorphism f such that

a= b� f . Evidently � is an equivalence relation.

Obvious 3.102. concat a= concat b,uncurry(a)�uncurry(b) for every ordinal indexed familiesa and b of functions taking an ordinal number of arguments.

Thank to the above, we can reduce properties of concat to properties of uncurry.

Lemma 3.103. a�b)uncurrya�uncurry b for every ordinal indexed families a and b of functionstaking an ordinal number of arguments.

Proof. There exist an order isomorphism f such that a= b� f .uncurry(a)(x; y)= (ax)y=(bfx)y=uncurry(b)(fx; y)=uncurry(b)g(x; y) where g(x; y)= (fx;

y).g is an order isomorphism because g(x0; y0) > g(x1; y1), (x0; y0) > (x1; y1). (Injectivity and

surjectivity are obvious.) �

Lemma 3.104. Let ai� bi for some fi for every i. Then uncurry a� uncurry b for every ordinalindexed families a and b of ordinal indexed families of functions taking an ordinal number ofarguments.

Proof. Let ai= bi � fi where fi is an order isomorphism for every i.uncurry(a)(i; y) = ai y = bi fi y = uncurry(b)(i; fi y) = uncurry(b)g(i; y) = (uncurry(b) � g)(i; y)

where g(i; y)= (i; fi y).g is an order isomorphism because g(i; y0)> g(i; y1), fi y0> fi y1, y0> y1 and i0>i1) g(i0;

y0)> g(i1; y1). (Injectivity and surjectivity are obvious.) �

Let now S be an ordinal indexed family of ordinal indexed families of functions taking an ordinalnumber of arguments.

Lemma 3.105. uncurry(uncurry �S)�uncurry(uncurryS).

Proof. uncurry �S=�i2S: uncurry(Si);uncurry(uncurry �S)(i; (x; y))= (uncurry Si)(x; y)= (Six)y;(uncurry(uncurryS))((i;x); y) = ((uncurryS)(i;x))y=(Si x)y.


Thus uncurry(uncurry � S)(i; (x; y)) = (uncurry(uncurry S))((i; x); y) and thus evidentlyuncurry(uncurry �S)� uncurry(uncurryS). �

Theorem 3.106. concat is an in�nitely associative function.

Proof. concat(JxK) = x for a function x taking an ordinal number of argument is obvious. It isremained to prove

concat(concat �S) = concat(concatS);

We have, using the lemmas, concat(concat � S) � uncurry(concat � S) � (by lemma 3.104)�uncurry(uncurry �S)� uncurry(uncurryS)� uncurry(concatS)� concat(concatS). �

Corollary 3.107. Ordinated product is an in�nitely associative function.


Chapter 4

Filters and �ltrators

This chapter is based on my article [29].This chapter is grouped in the following way:

� First it goes a short introduction in pedagogical order (�rst less general stu� and examples,last the most general stu�):

� �lters on a set;

� �lters on a meet-semilattice;

� �lters on a poset;

� �ltrators.

� Then it goes the formal part in the order from the most general to the least general:

� �ltrators;

� �lters on a poset;

� �lters on a set.

Most theorems about �ltrators (and also some theorems about �lters on posets) have the formA)B where A is the speci�c theorem condition and B is the main theorem statement. To mostsuch theorems correspond simple B when we restrict to consideration only to the �ltrator of �lterson a �xed set. In some sense only B here is important, A here is a technical condition. So readingtheorems about �ltrators concentrate on the theorem statement rather than on theorem conditions.

4.1 Introduction to �lters and �ltrators

4.1.1 Filters on a setWe sometimes want to de�ne something resembling an in�nitely small (or in�nitely big) set, forexample the in�nitely small interval near 0 on the real line. Of course there is no such set, justlike as there is no natural number which is the di�erence 2 ¡ 3. To overcome this shortcomingwe introduce whole numbers, and 2¡ 3 becomes well de�ned. In the same way to consider thingswhich are like in�nitely small (or in�nitely big) sets we introduce �lters.

An example of a �lter is the in�nitely small interval near 0 on the real line. To come to in�nitelysmall, we consider all intervals (¡"; ") for all " > 0. This �lter consists of all intervals (¡"; ") forall " > 0 and also all subsets of R containing such intervals as subsets. Informally speaking, thisis the greatest �lter contained in every interval (¡"; ") for all "> 0.

De�nition 4.1. A �lter on a set f is a F 2PPf such that:

1. 8A; B 2F :A\B 2F ;

2. 8A; B 2Pf: (A2F ^B �A)B 2F).

Exercise 4.1. Verify that the above introduced in�nitely small interval near 0 on the real line is a �lter on R.

51

Exercise 4.2. Describe �the neighborhood of positive in�nity� �lter on R.

De�nition 4.2. A �lter not containing empty set is called a proper �lter .

Obvious 4.3. The non-proper �lter is Pf.

Remark 4.4. Some other authors require that all �lters are proper. This is a stupid idea and weallow non-proper �lters, in the same way as we allow to use the number 0.

4.1.2 Intro to �lters on a meet-semilatticeA trivial generalization of the above:

De�nition 4.5. A �lter on a meet-semilattice Z is a F 2PZ such that:

1. 8A; B 2F :AuB 2F ;

2. 8A; B 2Z: (A2F ^B wA)B 2F).

4.1.3 Intro to �lters on a poset

De�nition 4.6. A �lter on a poset Z is a F 2PZ such that:

1. 8A; B 2F9C 2F :C vA; B;

2. 8A; B 2Z: (A2F ^B wA)B 2F).

It is easy to show (and there is a proof of it somewhere below) that this coincides with theabove de�nition in the case if Z is a meet-semilattice.

4.1.4 Intro to �ltrators

De�nition 4.7. Filter "x= fc2Z j cwxg is called the principal �lter induced by the element x.A �lter is principal i� it is a principal �lter induced by some element.

I denote P the set of all principal �lters (for a given poset Z).Now let (only in this paragraph) F is an arbitrary poset and P is its subset. I call pairs (F;P)

of a poset with its subset �ltrators . And when F is the set of �lters and P is the set of principal�lters on some poset I call them primary �ltrators.

Filtrators are a more general case than the special case of �ltrators on powersets.

4.2 Filtrators

De�nition 4.8. I will call a �ltrator a pair (A;Z) of a poset A and its subset Z�A. I call A the baseof the �ltrator and Z the core of the �ltrator. I will also say that (A;Z) is a �ltrator over poset Z.

De�nition 4.9. I will call a lattice �ltrator a pair (A;Z) of a lattice A and its subset Z�A.

De�nition 4.10. I will call a complete lattice �ltrator a pair (A; Z) of a complete lattice A andits subset Z�A.

De�nition 4.11. I will call a central �ltrator a �ltrator (A;Z(A)) where Z(A) is the center of abounded lattice A.

De�nition 4.12. I will call element of a �ltrator an element of its base.

De�nition 4.13. up a= fc2Z j cw ag for an element a of a �ltrator.

52 Filters and filtrators

De�nition 4.14. downa= fc2Z j cv ag for an element a of a �ltrator.

Obvious 4.15. �up� and �down� are dual.

Our main purpose here is knowing properties of the core of a �ltrator to infer properties of thebase of the �ltrator, speci�cally properties of up a for every element a.

De�nition 4.16. I call a �ltrator with join-closed core such a �ltrator (A;Z) thatFZ S =

FA Swhenever

FZ S exists for S 2PZ.

De�nition 4.17. I call a �ltrator with meet-closed core such a �ltrator (A;Z) thatdZ

S=dA

Swhenever

dZS exists for S 2PZ.

De�nition 4.18. I call a �ltrator with �nitely join-closed core such a �ltrator (A;Z) that atZ b=atA b whenever atZ b exists for a; b2Z.

De�nition 4.19. I call a �ltrator with �nitely meet-closed core such a �ltrator (A;Z) that auZ b=auA b whenever auZ b exists for a; b2Z.

De�nition 4.20. Filtered �ltrator is a �ltrator (A;Z) such that 8a2A: a=dA up a.

De�nition 4.21. Pre�ltered �ltrator is a �ltrator (A;Z) such that �up� is injective.

De�nition 4.22. Semi�ltered �ltrator is a �ltrator (A;Z) such that

8a; b2A: (up a�up b) av b):

Obvious 4.23.

� Every �ltered �ltrator is semi�ltered.

� Every semi�ltered �ltrator is pre�ltered.

Obvious 4.24. �up� is a straight map from A to the dual of the poset PZ if (A;Z) is a semi�ltered�ltrator.

Theorem 4.25. Each semi�ltered �ltrator is a �ltrator with join-closed core.

Proof. Let (A;Z) be a semi�ltered �ltrator. Let S 2PZ andFZ S be de�ned. We need to proveFA S =

FZ S. ThatFZ S is an upper bound for S is obvious. Let a 2A be an upper bound for

S. It's enough to prove thatFZ

S v a. Really,

c2up a) cw a)8x2S: cwx) cwGZ

S) c2 upGZ

S;

so up a�upFZ

S and thus awFZ

S because it is semi�ltered. �

4.2.1 Core Part

De�nition 4.26. The core part of an element a2A is Cor a=dZ up a.

De�nition 4.27. The dual core part of an element a2A is Cor0 a=FZ downa.

Obvious 4.28. Cor0 is dual of Cor.

Theorem 4.29. Cor av a whenever Cor a exists for any element a of a �ltered �ltrator.

Proof. Cor a=dZ up av

dA up a= a. �

Corollary 4.30. Cor a2downa whenever Cor a exists for any element a of a �ltered �ltrator.

4.2 Filtrators 53

Theorem 4.31. Cor0ava whenever Cor0a exists for any element a of a �ltrator with join-closedcore.

Proof. Cor0 a=FZ downa=

FA downav a. �

Corollary 4.32. Cor0a2downa whenever Cor0a exists for any element a of a �ltrator with join-closed core.

Proposition 4.33. Cor0 a v Cor a whenever both Cor a and Cor0 a exist for any element a of a�ltrator with join-closed core.

Proof. Cor a=dZ up awCor0 a because 8A2up a:Cor0 avA. �

Theorem 4.34. Cor0a=Cora whenever both Cora and Cor0a exist for any element of a �ltered�ltrator.

Proof. It is with join-closed core because it is semi�ltered. So Cor0 avCor a. Cor a2downa. SoCor av

FZ downa=Cor0 a. �

Obvious 4.35. Cor0 a=maxdowna for an element a of a �ltrator with join-closed core.

4.2.2 Filtrators with Separable Core

De�nition 4.36. Let (A;Z) be a �ltrator. It is a �ltrator with separable core when

8x; y 2A: (x�A y)9X 2 upx:X �A y):

Proposition 4.37. Let (A;Z) be a �ltrator. It is a �ltrator with separable core i�

8x; y 2A: (x�A y)9X 2up x; Y 2up y:X �AY ):

Proof.

). Apply the de�nition twice.

(. Obvious. �

De�nition 4.38. Let (A;Z) be a �ltrator. It is a �ltrator with co-separable core when

8x; y 2A: (x�A y)9X 2downx:X �A y):

Obvious 4.39. Co-separability is the dual of separability.

Proposition 4.40. Let (A;Z) be a �ltrator. It is a �ltrator with co-separable core i�

8x; y 2A: (x�A y)9X 2 downx; Y 2down y:X �AY ):

Proof. By duality. �

4.2.3 Intersection and Joining with an Element of the Core

De�nition 4.41. I call down-aligned �ltrator such a �ltrator (A; Z) that A and Z have commonleast element. (Let's denote it 0.)

De�nition 4.42. I call up-aligned �ltrator such a �ltrator (A; Z) that A and Z have commongreatest element. (Let's denote it 1.)

Theorem 4.43. For a �ltrator (A;Z) where Z is a boolean lattice, for every B 2Z, A2A:

1. B�AA,B wA if it is down-aligned, with �nitely meet-closed and separable core;


2. B�AA,B vA if it is up-aligned, with �nitely join-closed and co-separable core.

Proof. We will prove only the �rst as the second is dual.

B�AA ,9A2upA:B�AA ,

9A2upA:B uAA=0 ,9A2upA:B uZA=0 ,

9A2 upA:B wA ,B 2upA ,B wA:

�

4.2.4 Characterization of Finitely Meet-Closed Filtrators

Theorem 4.44. The following are equivalent for a �ltrator (A;Z) whose core is a meet semilatticesuch that 8a2A: up a=/ ;:

1. The �ltrator is �nitely meet-closed.

2. up a is a �lter for every a2A.

Proof.

(1))(2). Let X;Y 2up a. Then X uZY =X uAY w a. That up a is an upper set is obvious.So taking into account that up a=/ ;, up a is a �lter.

(2))(1). It is enough to prove that avA; B) avAuZB for every A; B 2A. Really:

avA; B)A; B 2 up a)AuZB 2up a) avAuZB: �

4.2.5 Stars of Elements of Filtrators

De�nition 4.45. Let (A;Z) be a �ltrator. Core star of an element a of a �ltrator is

@a= fx2Z j x�/A ag:

Proposition 4.46. up a� @a for any non-least element a of a �ltrator.

Proof. For any element X 2Z

X 2 up a) avX ^ av a)X �/A a)X 2 @a: �

Theorem 4.47. Let (A; Z) be a distributive lattice �ltrator with least element and �nitely join-closed core which is a join semilattice. Then @a is a free star for each a2A.

Proof. For every A; B 2Z

AtZB 2 @a ,AtAB 2 @a ,

(AtAB)uA a=/ 0A ,(AuA a)tA (B uA a)=/ 0A ,AuA a=/ 0A_B uA a=/ 0A ,

A2 @a_B 2 @a:

That @a doesn't contain 0A is obvious. �

4.2 Filtrators 55

De�nition 4.48. I call a �ltrator star-separable when its core is a separation subset of its base.

4.2.6 Atomic Elements of a FiltratorSee [4] and [9] for more detailed treatment of ultra�lters and prime �lters.

Theorem 4.49. Let (A;Z) be a semi�ltered down-aligned �ltrator with �nitely meet-closed coreZ which is a meet-semilattice. Then a is an atom of Z i� a2Z and a is an atom of A.

Proof.

(. Obvious.

). We need to prove that if a is an atom of Z then a is an atom of A. Suppose the contrarythat a is not an atom of A. Then there exists x2A such that 0=/ x@ a. Because �up� is astraight monotone map to the dual of the poset PZ (obvious 4.24), up a� up x. So thereexists K 2upx such thatK 2/ upa. Also a2upx. We have KuZa=KuAa2upx; KuZa=/ 0and K uZ a@ a. So a is not an atom of Z. �

Theorem 4.50. Let (A; Z) be a semi�ltered down-aligned �ltrator and A is a meet-semilattice.Then a2A is an atom of A i� up a= @a.

Proof.

). Let a be an atom of A. up a� @a because a=/ 0. up a� @a because for any K 2A

K 2up a,K w a,K uA a=/ 0,K 2 @a:

(. Let up a= @a. Then a=/ 0. Consequently for every x2A we have

0@x@ a )xuA a=/ 0 )

8K 2up x:K 2 @a )8K 2upx:K 2 up a )

up x� up a )xw a:

So a is an atom of A. �

4.2.7 Prime Filtrator Elements

De�nition 4.51. Let (A; Z) be a down-aligned �ltrator. Prime �ltrator elements are such a2Athat up a is a free star.

Proposition 4.52. Let (A;Z) be a down-aligned �ltrator with �nitely join-closed core, whereA is astarrish join-semilattice and Z is a join-semilattice. Then atomic elements of this �ltrator are prime.

Proof. Let a be an atom of the lattice A. We have for every X;Y 2Z

X tZY 2up a ,X tAY 2up a ,X tAY w a ,X tAY �/A a ,

X �/A a_Y �/A a ,X w a_Y w a ,

X 2up a_Y 2up a:�


4.2.8 Some Criteria

Theorem 4.53. For a semi�ltered, star-separable, down-aligned �ltrator (A;Z) with �nitely meetclosed and separable core where Z is a complete boolean lattice and both Z and A are atomisticlattices the following conditions are equivalent for any F 2A:

1. F 2Z;

2. 8S 2PA:¡F uA

FAS=/ 0)9K2S:F uAK=/ 0

�;

3. 8S 2PZ:¡F uA

FA S=/ 0)9K 2S:F uAK =/ 0�.

Proof. Our �ltrator is with join-closed core (theorem 4.25).

(1))(2). Let F 2Z. Then (taking into account the proposition 4.43)

F uAGA

S=/ 0,FwGA

S)9K2S:FwK,9K2S:F uAK=/ 0:

(2))(3). Obvious.

(3))(1). Let the formula (3) be true. Then for L2Z and S= atomsZL it takes the form

F uAGA

atomsZL=/ 0)9K 2S:F uAK =/ 0

that is F uAL=/ 0)9K 2 S:F uAK =/ 0 becauseFA atomsZL=

FZ atomsZL=L. Thatis F uAL=/ 0)F uAKL=/ 0 where KL2S. Thus KL is an atom of both A and Z (see thetheorem 4.49), so having F uAL=/ 0)F wKL. Let

F =GZfKL j L2Z;F uAL=/ 0g:

Then

F =GAfKL j L2Z;F uAL=/ 0g:

Obviously F vF . We have LuAF =/ 0)KLuZF =/ 0)LuZF =/ 0)LuAF =/ 0, thus bystar separability of our �ltrator F vF and so F =F 2Z. �

De�nition 4.54. Let S be a subset of a meet-semilattice. The �lter base generated by S is the set

[S]u= fa0u ::: u an j ai2S; i=0; 1; :::g:

Lemma 4.55. The set of all �nite subsets of an in�nite set A has the same cardinality as A.

Proof. Let denote the number of n-element subsets of A as sn. Obviously sn6 cardAn= cardA.Then the number S of all �nite subsets of A is equal to s0+s1+ :::6 cardA+ cardA+ :::= cardA.That S> cardA is obvious. So S= cardA. �

Lemma 4.56. A �lter base generated by an in�nite set has the same cardinality as that set.

Proof. From the previous lemma. �

De�nition 4.57. Let A be a complete lattice. A set S 2PA is �lter-closed when for every �lterbase T 2PS we have

dT 2S.

Theorem 4.58. A subset S of a complete lattice is �lter-closed i� for every nonempty chainT 2PS we have

dT 2S.

Proof. (proof sketch by Joel David Hamkins)

). Because every nonempty chain is a �lter base.

4.2 Filtrators 57

(. We will assume that cardinality of a set is an ordinal de�ned by von Neumann cardinalassignment (what is a standard practice in ZFC). Recall that �<�,�2 � for ordinals �, �.

We will take it as given that for every nonempty chain T 2PS we havedT 2S.

We will prove the following statement: If cardS=n then S is �lter closed, for any cardinaln.

Instead we will prove it not only for cardinals but for wider class of ordinals: If cardS=nthen S is �lter closed, for any ordinal n.

We will prove it using trans�nite induction by n.For �nite n we have

dT 2S because T �S has minimal element.

Let cardT =n be an in�nite ordinal.Let the assumption hold for every m2 cardT .We can assign T = fa� j �2 cardT g for some a� because card cardT = cardT .Consider � 2 cardT .Let P�= fa� j �2 �g. Let b�=

dP�. Obviously b�=

d[P�]u. We have

card [P�]u= cardP�= card � < cardT

(used the lemma and von Neumann cardinal assignment). By the assumption of inductionb� 2S.8� 2 cardT :P� �T and thus b� w

dT .

It is easy to see that the set fP� j � 2 cardT g is a chain. Consequently fb� j �2 cardT gis a chain.

By the theorem conditions b=dfb� j � 2 card T g 2 S (taken into account that b� 2 S

by the assumption of induction).Obviously bw

dT .

bv b� and so 8� 2 card T ; � 2 �: bv a�. Let � 2 card T . Then (because card T is limitordinal, see [41]) there exists � 2 card T such that � 2 � 2 card T . So b v a� for every�2 cardT . Thus bv

dT .

FinallydT = b2S. �

4.2.9 Complements and Core Parts

Lemma 4.59. If (A;Z) is a �ltered, up-aligned �ltrator with co-separable core which is a completelattice, then for any a; c2A

c�A a, c�ACor a:

Proof.

). If c�A a then by co-separability of the core exists K 2downa such that c�AK. To �nishthe proof we will show that K v Cor a. To show this is enough to show that 8X 2 up a:K vX what is obvious.

(. Cor av a (by the theorem 4.29 using that our �ltrator is �ltered). �

Theorem 4.60. If (A;Z) is a �ltered up-aligned complete lattice �ltrator with co-separable corewhich is a complete boolean lattice, then a+=Cor a for every a2A.

Proof. Our �ltrator is with join-closed core (theorem 4.25).

a+ =lAfc2A j ctA a=1Ag =

lAfc2A j ctACor a=1Ag =

lAfc2A j cwCor ag =

Cor a


(used the lemma and theorem 4.43). �

Corollary 4.61. If (A;Z) is a �ltered up-aligned complete lattice �ltrator with co-separable corewhich is a complete boolean lattice, then a+2Z for every a2A.

Theorem 4.62. If (A; Z) is a �ltered complete lattice �ltrator with down-aligned, �nitely meet-closed, separable core which is a complete boolean lattice, then a�=Cora=Cor0a for every a2A.

Proof. Our �ltrator is with join-closed core (theorem 4.25). a�=FA fc 2A j c uA a= 0Ag. But

cuA a=0A)9C 2up c:C uA a=0A. So

a� =GAfC 2Z j C uA a=0Ag =GA

fC 2Z j avCg =GAfC j C 2Z; avCg =GA

fC j C 2 up ag =GZfC j C 2 up ag =

lZfC j C 2 up ag =

lZup a =

Cor a

(used theorem 4.43).Cor a=Cor0 a by theorem 4.34. �

Corollary 4.63. If (A;Z) is a �ltered down-aligned and up-aligned complete lattice �ltrator with�nitely meet-closed, separable and co-separable core which is a complete boolean lattice, thena�= a+ for every a2A.

Proof. Comparing two last theorems. �

Theorem 4.64. If (A;Z) is a complete lattice �ltrator with join-closed separable core which is acomplete lattice, then a�2Z for every a2A.

Proof. fc2A j cuA a=0Ag�fA2Z j AuA a=0Ag; consequently a�wFA fA2Z j AuA a=0Ag.

But if c2fc2A j cuA a=0Ag then there exists A2Z such that Aw c and AuA a=0A that isA2fA2Z j AuA a=0Ag. Consequently a�v

FA fA2Z j AuA a=0Ag.We have a�=

FA fA2Z j AuA a=0Ag=FZ fA2Z j AuA a=0Ag2Z. �

Theorem 4.65. If (A;Z) is an up-aligned �ltered complete lattice �ltrator with co-separable corewhich is a complete boolean lattice, then a+ is dual pseudocomplement of a, that is

a+=min fc2A j ctA a=1Agfor every a2A.

Proof. Our �ltrator is with join-closed core (theorem 4.25). It's enough to prove that a+tAa=1A.But a+ tA a=Cor a tA a wCor a tACor a=Cor a tZCor a= 1A (used the theorem 4.29 and thefact that our �ltrator is �ltered). �

De�nition 4.66. The edge part of an element a 2 A is Edg a = a n Cor a, the dual edge part isEdg0 a= a nCor0 a.

4.2 Filtrators 59

Knowing core part and edge part or dual core part and dual edge part of a �lter, the �lter canbe restored by the formulas:

a=Cor atAEdg a and a=Cor0 atAEdg0 a:

4.2.10 Core Part and Atomic Elements

Proposition 4.67. Let (A; Z) be a �ltrator with join-closed core and Z be an atomistic lattice.Then for every a2A such that Cor0 a exists we have

Cor0 a=GZfx j x is an atom of Z; xv ag:

Proof.

Cor0 a =GZfA2Z j Av ag =GZ (GZ

atomsZA j A2Z; Av a

)=

GZ [fatomsZA j A2Z; Av ag =GZ

fx j x is an atom of Z; xv ag:�

4.2.11 Distributivity of Core Part over Lattice Operations

Theorem 4.68. If (A;Z) is a join-closed �ltrator and A is a meet-semilattice and Z is a completelattice, then for every a; b2A

Cor0(auA b)=Cor0 auZCor0 b:

Proof. From theorem conditions it follows that Cor0(auA b) exists.We have Cor0 pv p for every p2A because our �ltrator is with join-closed core.Obviously Cor0(auA b)vCor0 a and Cor0(auA b)vCor0 b.If x v Cor0 a and x v Cor0 b for some x 2 Z then x v a and x v b, thus x v a uA b and

xvCor0(auA b). �

Theorem 4.69. If (A; Z) is a join-closed �ltrator and both A and Z are complete lattices, thenfor every S 2PA

Cor0lA

S=lZhCor0iS:

Proof. From theorem conditions it follows that Cor0dA

S exists.We have Cor0 pv p for every p2A because our �ltrator is with join-closed core.Obviously Cor0

dA S vCor0 a for every a2S.If xvCor0 a for every a2S for some x2Z then xv a, thus xv

dA S and xvCor0dA S. �

Corollary 4.70. If (A; Z) is a join-closed �ltrator and both A and Z are complete lattices, thenfor every S 2PZ

Cor0lA

S=lZ

S:

Theorem 4.71. Let (A;Z) be a semi�ltered down-aligned �ltrator with �nitely meet-closed coreZ which is a complete atomistic lattice and A is a complete starrish lattice, then Cor0(a tA b) =Cor0 atZCor0 b for every a; b2A.


Proof. From theorem conditions it follows that Cor0(atA b) exists.Cor0(atA b)=

FZ fx j x is an atom of Z; xv atA bg (used proposition 4.67).By theorem 4.49 we have Cor0(atA b)=

FZ(atomsA(atA b)\Z)=

FZ((atomsAa[atomsA b)\

Z) =FZ ((atomsA a \ Z) [ (atomsA b \ Z)) =

FZ (atomsA a \ Z) tZFZ (atomsA b \ Z) (used the

theorem 3.30). Again using theorem 4.49, we getCor0(a tA b) =

FZ fx j x is an atom of Z; x v ag tZFZ fx j x is an atom of Z; x v bg =

Cor0 atZCor0 b (again used proposition 4.67). �

Theorem 4.72. Let (A;Z) be a �ltered starrish down-aligned complete lattice �ltrator with �nitelymeet-closed, separable core which is a complete atomistic boolean lattice. Then (atAb)�=a�uZb�.

Proof. (atA b)�=Cor0(atA b)=Cor0atZCorb=Cor0auZCor0b=a�uZb� (used theorem 4.62). �

4.2.12 Co-Separability of Core

Theorem 4.73. Let (A;Z) be an up-aligned �ltered �ltrator whose core is a meet in�nite distrib-utive complete lattice. Then this �ltrator is with co-separable core.

Proof. Our �ltrator is with join-closed core (theorem 4.25).Let a; b2A. Cor a and Cor b exist since Z is a complete lattice.Cora2downa and Cor b2downb by the corollary 4.30 since our �ltrator is �ltered. So we have

9x2downa; y 2down b:xtA y=1 (Cor atACor b=1 , (by �nite join-closedness of the core)Cor atZCor b=1 ,

lZup atZ

lZup b=1 , (by in�nite distributivity)

lZfxtZ y j x2 up a; y 2 up bg=1 ,8x2 up a; y 2up b:xtZ y=1 , (by �nite join-closedness of the core)8x2up a; y 2 up b:xtA y=1 (

atA b=1:

�

4.2.13 Filtrators over Boolean Lattices

Proposition 4.74. Let (A;Z) be a down-aligned and up-aligned �nitely meet-closed and �nitelyjoin-closed distributive lattice �ltrator and Z be a boolean lattice. Then a nAB=auAB for everya2A, B 2Z.

Proof. (auAB)tAB=(atAB)uA (B tAB)= (atAB)uA (B tZB)= (atAB)uA 1= atAB.(auAB)uAB= auA (B uAB)= auA (B uZB) = auA 0=0.So auAB is the di�erence of a and B. �

4.2.13.1 Distributivity for an Element of Boolean Core

Lemma 4.75. Let (A;Z) be an up-aligned �nitely join-closed and �nitely meet-closed distributivelattice �ltrator over a boolean lattice. Then AuA is a lower adjoint of AtA for every A2Z.

Proof. We will use the theorem 2.98.That AuA and AtA are monotone is obvious.We need to prove (for every x; y 2A) that

xvAtA (AuA x) and AuA (AtA y)v y:

4.2 Filtrators 61

Really, AtA (AuA x)= (AtAA)uA (AtA x)= (AtZA)uA (AtA x) = 1uA (AtA x) =AtA xwxand AuA (AtA y)= (AuAA)tA (AuA y) = (AuZA)tA (AuA y) = 1tA (AuA y)=AuA yv y. �

Theorem 4.76. Let (A; Z) be an up-aligned �nitely join-closed and �nitely meet-closed distrib-utive lattice �ltrator over a boolean lattice. Then AuA

FAS =

FA hAuA iS for every A2 Z andevery set S 2PA.

Proof. Direct consequence of the lemma. �

4.3 Filters on a poset

4.3.1 Filters on posetsLet Z be a poset.

De�nition 4.77. Filter base is a nonempty subset F of Z such that

8X;Y 2F9Z 2F : (Z vX ^Z vY ):

Obvious 4.78. A nonempty chain is a �lter base.

De�nition 4.79. Filter is a subset of Z which is both a �lter base and an upper set.

I will denote the set of �lters (for a given or implied poset Z) as F and call F the set of �ltersover the poset Z.

Proposition 4.80. If 1 is the maximal element of Z then 12F for every �lter F .

Proof. If 12/ F then 8K 2Z:K 2/ F and so F is empty what is impossible. �

Proposition 4.81. Let S be a �lter base on a poset. If A0; :::; An2S (n2N), then

9C 2S: (C vA0^ ::: ^C vAn):

Proof. It can be easily proved by induction. �

Dual of �lters is called ideals . We do not use ideals in this work however.

4.3.2 Filters on meet-semilattices

Theorem 4.82. If Z is a meet-semilattice and F is a nonempty subset of Z then the followingconditions are equivalent:

1. F is a �lter.

2. 8X;Y 2F :X uY 2F and F is an upper set.

3. 8X;Y 2Z: (X;Y 2F,X uY 2F ).

Proof.

(1))(2). Let F be a �lter. Then F is an upper set. If X;Y 2F then Z vX ^Z vY for someZ 2F . Because F is an upper set and Z vX uY then X uY 2F .

(2))(1). Let 8X; Y 2 F :X u Y 2 F and F be an upper set. We need to prove that F is a�lter base. But it is obvious taking Z=X uY (we have also taken into account that F =/ ;).

(2))(3). Let 8X;Y 2F :X uY 2F and F be an upper set. Then

8X;Y 2Z: (X;Y 2F)X uY 2F ):


Let X uY 2F ; then X;Y 2F because F is an upper set.

(3))(2). Let

8X;Y 2Z: (X;Y 2F,X uY 2F ):

Then 8X;Y 2F :X uY 2F . Let X 2F and X vY 2Z. Then X uY =X 2F . ConsequentlyX;Y 2F . So F is an upper set. �

Proposition 4.83. Let S be a �lter base on a meet-semilattice. If A0; :::; An2S (n2N), then

9C 2S:C vA0u ::: uAn:

Proof. It can be easily proved by induction. �

Proposition 4.84. If Z is a meet-semilattice and S is a �lter base on it, A 2 Z, then hA u iS isalso a �lter base.

Proof. hAu iS=/ ; because S=/ ;.Let X; Y 2 hAu iS. Then X =AuX 0 and Y =A u Y 0 where X 0; Y 0 2 S. There exists Z 0 2 S

such that Z 0vX 0uY 0. So X uY =AuX 0uY 0wAuZ 02 hAu iS. �

4.3.3 Order of �lters. Principal �ltersI will make the set of �lters F into a poset by the order de�ned by the formula: av b, a� b.

De�nition 4.85. The principal �lter corresponding to an element a2Z is

"a= fx2Z j xw ag:

Elements of P= h"iZ are called principal �lters .

Obvious 4.86. Principal �lters are �lters.

Obvious 4.87. " is an order embedding from Z to F.

Corollary 4.88. " is an order isomorphism between Z and P.

De�nition 4.89. For every poset Z I call (F;P) the primary �ltrator (for the base Z).

Proposition 4.90. "K wA,K 2A.

Proof. "K wA,"K �A,K 2A. �

Proposition 4.91. up a= h"ia for an element a of a primary �ltrator.

Proof. For every L2P we have L="K for some K 2Z and L2upa,Lwa,"K wa,K 2a,L2 h"ia. �

4.3.3.1 Minimal and maximal �lters

Obvious 4.92. The �lter 0F=Z (equal to the principal �lter for the least element of Z if it exists)is the least element of the poset of �lters.

Proposition 4.93. If there exists greatest element 1Z of the poset Z then 1F=f1Zg is the greatestelement of F.

Proof. Take into account that �lters are nonempty. �

4.3.4 Primary �ltrator is �ltered[TODO: Can the proof be simpli�ed using the fact that ��ltered� is the same as �semi�ltered�?]

4.3 Filters on a poset 63

Theorem 4.94. A=dF h"iA for every �lter A on a poset.

Proof. A is obviously a lower bound for h"iA. Let B be a lower bound for h"iA that is

8K 2A:B v"K

that is 8K 2A:K 2B that is A�B that is B vA. So A is the greatest lower bound for h"iA. �

Corollary 4.95. Every primary �ltrator is �ltered.

Corollary 4.96. Every primary �ltrator is with join-closed core.

Proof. Theorem 4.25. �

Proposition 4.97. The �ltrator (F;P) is with �nitely meet-closed core if Z is a meet-semilattice.


4.3.5 Alignment

Obvious 4.98.

1. If Z has least element, the primary �ltrator is down-aligned.

2. If Z has greatest element, the primary �ltrator is up-aligned.

4.3.6 Co-separability of Core for Primary Filtrators

Proposition 4.99. Every primary �ltrator over a meet in�nite distributive complete lattice iswith co-separable core.

Proof. It is up-aligned, �ltered. So we can apply the theorem 4.73. �

4.3.7 Core Part

Proposition 4.100. Cor0 a=Cor a for every �lter a on a complete lattice.

Proof. By the theorem 4.34 and corollary 4.95. �

Proposition 4.101. Cor av a for every �lter a on a complete lattice.

Proof. By the theorem 4.29 and corollary 4.95. �

Proposition 4.102. Cor a=maxdowna for every �lter a on a complete lattice.

Proof. Proposition 4.100, obvious 4.35, corollary 4.96. �

4.3.8 Intersecting and Joining with an Element of the Core

Theorem 4.103. For a �ltrator (F;P) where Z is a boolean lattice, for every B 2P, A2F:

1. B�FA,B wA;

2. B�FA,B vA if Z is a complete lattice.

Proof.

1. Using theorem 4.43, obvious 4.98, proposition 4.97, theorem 4.112.

2. Using theorem 4.43, obvious 4.98, corollary 4.96, theorem 4.73. �


4.3.9 Formulas for Meets and Joins of Filters

Lemma 4.104. If f is an order embedding from a poset A to a complete lattice B and S 2PA

and there exists such F 2A that fF =FB hf iS, then

FA S exists and fFA S=

FB hf iS.

Proof. f is an order isomorphism from A to Bjhf iA. fF 2Bjhf iA.Consequently,

FB hf iS 2Bjhf iA andFBjhf iA hf iS=

FB hf iS.fFA

S=FBjhf iA hf iS because f is an order isomorphism.

Combining, fFA S=

FB hf iS. �

Corollary 4.105. If B is a complete lattice and A is its subset and S 2PA andFB S 2A, thenFA

S exists andFA

S=FB

S.

Theorem 4.106. If Z is a meet-semilattice with greatest element 1 thenFF

S exists andGFS=

\S

for every S 2PF.

Proof. Taking into account the corollary of the lemma, it is enough to prove that there existsF 2F such that F =

TS, that is that R=

TS is a �lter.

R is nonempty because 1 2 R. Let A; B 2 R; then 8F 2 S: A; B 2 F , consequently 8F 2 S:AuZB 2F . Consequently AuZB 2

TS=R. So R is a �lter base. Let X 2R and X vY 2Z; then

8F 2S:X 2F ; 8F 2S:Y 2F ; Y 2R. So R is an upper set. �

Corollary 4.107. If Z is a meet-semilattice with greatest element then F is a complete lattice.

Corollary 4.108. If Z is a meet-semilattice with greatest element then for any A;B 2F

AtFB=A\B:

We will denote meets and joins on the lattice of �lters just as u and t.

Theorem 4.109. If Z is a join-semilattice then F is a join-semilattice and for any A;B 2F

AtFB=A\B:

Proof. Taking into account the corollary of the lemma, it is enough to prove R=A\B is a �lter.R is nonempty because there exists X 2A and Y 2B and R3X tZY .Let A; B 2 R. Then A; B 2 A; so there exists C 2 A such that C v A ^ C vB. Analogously

there exists D 2B such that DvA^DvB. Let E=C tZD. Then E 2A and E 2B; E 2R andE vAÊ vB. So R is a �lter base.

That R is an upper set is obvious. �

Theorem 4.110. If Z is a distributive lattice then for S 2PF n f;g

lFS=

�K0uZ ::: uZKn j Ki2

[S where i=0; :::; n for n2N

:

Proof. Let's denote the right part of the equality to be proven as R. First we will prove that Ris a �lter. R is nonempty because S is nonempty.

Let A; B 2R. Then A=X0uZ ::: uZXk, B=Y0uZ ::: uZYl where Xi; Yj 2S

S. So

AuZB=X0uZ ::: uZXkuZ Y0uZ ::: uZYl2R:

Let �lter C wA2R. Consequently (distributivity used)

C =C tZA=(C tZX0)uZ ::: uZ (C tZXk):


Xi2Pi for some Pi2S; C tZXi2Pi; C tZXi2S

S; consequently C 2R.We have proved that that R is a �lter base and an upper set. So R is a �lter.Let A2S. Then A�

SS;

R�fK0uZ ::: uZKn j Ki2A where i=0; :::; n for n2N g=A:

Consequently AwR.Let now B 2 F and 8A2 S:AwB. Then 8A 2 S:A�B; B �

SS. From this B � T for every

�nite set T �S

S. Consequently B 3dZ

T . Thus B �R; B vR.Comparing we get

dF S=R. �

Theorem 4.111. If Z is a distributive lattice then for any F0; :::;Fm2F (m2N)

F0uF ::: uFFm= fK0uZ ::: uZKm j Ki2F i where i=0; :::; mg:

Proof. Let's denote the right part of the equality to be proven as R. First we will prove that Ris a �lter. Obviously R is nonempty.

Let A; B 2R. Then A=X0uZ ::: uZXm, B=Y0uZ ::: uZYm where Xi; Yi2F i.

AuZB=(X0uZY0)uZ ::: uZ (XmuZYm);consequently AuZB 2R.

Let �lter C wA2RC =AtZC =(X0tZC)uZ ::: uZ (XmtZC)2R:

So R is a �lter.Let Pi2F i. Then Pi2R because Pi=(PitZP0)uZ ::: uZ (PitZPm). So F i�R; F iwR.Let now B 2F and 8i2f0; :::; mg:F iwB. Then 8i2f0; :::; mg:F i�B.Li2B for every Li2F i. L0uZ ::: uZLm2B. So B �R; B vR.So F0uF ::: uFFm=R. �

4.3.10 Separability of Core for Primary Filtrators

Theorem 4.112. A primary �ltrator with least element, whose core is a distributive lattice, iswith separable core. [TODO: Is distributivity necessary? I suspect it for every meet-semilattice.]

Proof. Let A�FB where A;B 2F.

AuFB= fAuZB j A2A; B 2Bg:So

02AuFB ,9A2A; B 2B:AuZB=0 ,

9A2A; B 2B: "AuP "B=0F ,9A2A; B 2B: "AuF "B=0F ,

9A2upA; B 2 upB:AuFB=0F

(used proposition 4.97). �

4.3.11 Distributivity of the Lattice of Filters

Theorem 4.113. If Z is a distributive lattice with greatest element, S 2 PF and A 2 F then[TODO: Can it be generalized for meet-semilattices (use generalized in�nite meet formula inrewrite-plan.pdf)? Also corollaries.]

AtFlF

S=lFhAtF iS:


Proof. Taking into account the previous section, we have:

AtFlF

S =

A\lF

S =

A\�K0uZ ::: uZKn j Ki2


=�

K0uZ ::: uZKn j K0uZ ::: uZKn2A;Ki2[

S where i=0; :::; n for n2N

=�K0uZ ::: uZKn j Ki2A^Ki2


=�

K0uZ ::: uZKn j Ki2A\[

S where i=0; :::; n for n2N

=�K0uZ ::: uZKn j Ki2

[hA\ iS where i=0; :::; n for n2N

=�

K0uZ ::: uZKn j Ki2[fA\X j X 2Sg where i=0; :::; n for n2N

=�

K0uZ ::: uZKn j Ki2[fAtFX j X 2Sg where i=0; :::; n for n2N

=

lFfAtFX j X 2Sg =

lFhAtF iS:

�

Corollary 4.114. If Z is a distributive lattice with greatest element, then F is also a distributivelattice.

Corollary 4.115. If Z is a distributive lattice with greatest element, then F is a co-brouwerianlattice.

4.3.12 Filters over Boolean Lattices

Theorem 4.116. If Z is a boolean lattice then a nFB= auFB for every a2F, B 2P (where thecomplement is taken on P).

Proof. F is a distributive lattice by corollary 4.114. Our �ltrator is �nitely meet-closed by thetheorem 4.44 and with join-closed core by the theorem 4.25. It is also up and down aligned.

So we can apply the proposition 4.74. �

4.3.12.1 Distributivity for an Element of Boolean Core

Lemma 4.117. Let F be the poset of �lters over a boolean lattice Z.Then AuF is a lower adjoint of AtF for every A2P.

Proof. Lemma 4.75. �

Theorem 4.118. Let F be the poset of �lters over a boolean lattice Z. Then A uFFF

S =FF hAuF iS for every A2P and every set S 2PF.

Proof. Direct consequence of the lemma. �

4.3.13 Generalized Filter Base

De�nition 4.119. Generalized �lter base is a �lter base on the set F.

De�nition 4.120. If S is a generalized �lter base and A =dF S, then we call S a generalized

�lter base of a �lter A.

Theorem 4.121. If Z is a distributive lattice [TODO: Can be generalized for any meet-semilat-tice?] and S is a generalized �lter base of a �lter F then for any K 2Z

K 2F,9L2S:K 2L:


Proof.

(. Because F =dF S.

). LetK 2F . Then (taken into account distributivity of Z and that S is nonempty) there existX1; :::; Xn2

SS such that X1uZ :::uZXn=K that is "X1uP :::uP"Xn="K. Consequently

(by theorem 4.44) "X1uF :::uF"Xn="K. Replacing every "Xi with suchX i2S thatXi2X i

(this is obviously possible to do), we get a �nite set T0 � S such thatdF T0 v "K. From

this there exists C 2S such that C vdF T0v"K and so K 2C. �

Corollary 4.122. If Z is a distributive lattice with least element and S is a generalized �lter baseof a �lter F then 0F2S,F =0F.

Proof. Substitute 0F as K. �

Theorem 4.123. Let Z be a distributive lattice with least element and S is a nonempty set of�lters on Z such that F0 uF ::: uF Fn =/ 0

F for every F0; :::; Fn 2 S. ThendF S =/ 0F. [TODO:

Generalize for arbitrary meet-semilattices?]

Proof. Consider the set

S 0= fF0uF ::: uFFn j F0; :::;Fn2Sg:

Obviously S 0 is nonempty and �nitely meet-closed. So S 0 is a generalized �lter base. Obviously0F 2/ S. So by properties of generalized �lter bases

dF S 0=/ 0F. But obviouslydF S =

dF S 0. SodF S=/ 0F. �

Corollary 4.124. Let Z be a distributive lattice with least element and let S 2 PZ such thatS=/ ; and A0uZ ::: uZAn=/ 0Z for every A0; :::; An2S. Then

dF h"iS=/ 0F.

Proof. Because (F;Z) is �nitely meet-closed (by the theorem 4.44). �

4.3.14 Stars for �lters

Theorem 4.125. Let Z be a bounded distributive lattice with greatest element. Then @a is a freestar for each a2F. [TODO: Generalize for arbitrary meet-semilattices?]

Proof. F is a distributive lattice by the corollary 4.114. The �ltrator (F;P) is �nitely join-closedby corollary 4.96. So we can apply the theorem 4.47. �

4.3.14.1 Stars of Filters on Boolean Lattices

In this section we will consider the set of �lters F on a boolean lattice Z.Note that P is also a boolean lattice. We will take complements on P without specifying that

the complement is taken on P.

Theorem 4.126. If Z is a boolean lattice, X 2 upA,X 2/ @A (where complement is taken onthe boolean lattice P) for every X 2P, A2F.

Proof. X 2upA,X wA,X�FA,X 2/ @A for any X 2P (taking into account theorems 4.44,4.112, 4.43). �

Corollary 4.127. If Z is a boolean lattice and A2F then

1. @A= fX j X 2P n upAg;

2. upA= fX j X 2P n @Ag

(where complement is taken on the boolean lattice P).

Corollary 4.128. If Z is a boolean lattice, @ is an injection.


For boolean lattices free stars bijectively correspond to �lters:

Theorem 4.129. If Z is a boolean lattice, then for any set S 2PP there exists a �lter A suchthat @A=S i� S is a free star.

Proof.

). That 0P2/ S is obvious. For every A; B 2A

AtPB 2S ,(AtPB)uFA=/ 0F ,(AtFB)uFA=/ 0F ,

(AuFA)tF (B uFA)=/ 0F ,AuFA=/ 0F_B uFA=/ 0F ,

A2S _B 2S

(taken into account corollary 4.114 and theorem 4.25).

(. 0P2/ S and 8A; B 2 S: (AtPB 2 S,A2 S _B 2 S). Let T = fX j X 2P n Sg. We willprove that T is a �lter.

1P2T because 0P2/ S; so T is nonempty. To prove that T is a �lter it is enough to show8X;Y 2P: (X;Y 2T,X uPY 2T ). In fact,

X;Y 2T ,X;Y 2/ S ,

:(X 2S _Y 2S) ,X tPY 2/ S ,X tPY 2T ,X uPY 2T :

So T is a �lter. To �nish the proof we will show that @T =S. In fact, for every X 2P

X 2 @T,X 2/ upT,X 2/ T,X 2S: �

Proposition 4.130. If Z is a boolean lattice then AvB, @A� @B for every A;B 2F.

Proof.

@A� @B ,fX j X 2P nAg�fX j X 2P nBg ,

P nA�P nB ,A�B ,AvB:

�

Corollary 4.131. @ is a straight monotone map if Z is a boolean lattice.

Theorem 4.132. If Z is a boolean lattice then @FF S=

Sh@ iS for every S 2PF.

Proof. For boolean lattices @ is an order embedding from the poset F to the complete lattice PP.So accordingly the lemma 4.104 it is enough to prove that there exists F 2F such that @F=

Sh@ iS.

To prove this it is enough to show that 0P2/Sh@ iS and

8A; B 2S:¡AtPB 2

[h@ iS,A2

[h@ iS _B 2

[h@ iS

�:

0P2/Sh@ iS is obvious.


Let AtPB 2Sh@ iS. Then there exists Q2h@ iS such that AtPB 2Q. Then A2Q_B 2Q,

consequently A 2Sh@ iS _B 2

Sh@ iS. Let now A 2

Sh@ iS. Then there exists Q 2 h@ iS such

as A2Q, consequently AtPB 2Q and AtPB 2Sh@ iS. �

4.3.15 More about the Lattice of Filters

De�nition 4.133. Atoms of F (for any poset Z) are called ultra�lters.

De�nition 4.134. Principal ultra�lters are also called trivial ultra�lters .

Theorem 4.135. If Z is a bounded distributive lattice [TODO: Generalize for meet-semilattices?]with least element then F is an atomic lattice.

Proof. Let F 2 F. Let choose (by Kuratowski's lemma) a maximal chain S from 0F to F . LetS 0=S nf0Fg. a=

dF S 0=/ 0F by properties of generalized �lter bases (the corollary 4.122 which usesthe fact that Z is a distributive lattice with least element). If a2/ S then the chain S can be extendedadding there element a because 0F@ avX for any X 2 S 0 what contradicts to maximality of thechain. So a2S and consequently a2S 0. Obviously a is the minimal element of S 0. Consequently(taking into account maximality of the chain) there is no Y 2 F such that 0F@ Y @ a. So a is anatomic �lter. Obviously avF . �

Obvious 4.136. If Z is a boolean lattice then F is separable.

Theorem 4.137. If Z is a boolean lattice then F is an atomistic lattice.

Proof. Because (used the theorem 3.20) F is atomic (theorem 4.135) and separable. �

Corollary 4.138. If Z is a boolean lattice then F is atomically separable.

Proof. By theorem 3.19. �

Theorem 4.139. When Z is a boolean lattice, the �ltrator (F;P) is central.

Proof. We can conclude that F is atomically separable (the corollary 4.138), with separable core(the theorem 4.112), and with join-closed core (corollary 4.96).

We need to prove Z(F)=P.Let X 2Z(F). Then there exists Y 2Z(F) such that X uFY=0F and X tFY=1F. Consequently

there is X 2 up X such that X uF Y = 0F; we also have X tF Y = 1F. Suppose X A X . Thenthere exists a2 atomsFX such that a2/ atomsFX . We can conclude also a2/ atomsF Y (otherwiseX uF Y =/ 0F). Thus a 2/ atomsF(X tF Y) and consequently X tF Y =/ 1F what is a contradiction.We have X =X 2P.

Let now X 2 P. Let Y = X . We have X uP Y = 0F and X tP Y = 1F. Thus X uF Y =dP fX uPY g=0F; X tFY =X tPY =1F. We have shown that X 2Z(F). �

4.3.16 Atomic Filters

Proposition 4.140. If Z is a meet-semilattice with least element, then a is an atom of P i� a2Pand a is an atom of F.

Proof. It is semi�ltered by the corollary 4.95, �nitely meet-closed by proposition 4.97. So we canapply the theorem 4.49. �

Proposition 4.141. If Z is a meet-semilattice with least element then, a 2 F is an atom of F i�up a=@a.

Proof. It is semi�ltered by the corollary 4.95, F is a meet-semilattice by the corollary 4.107. Sowe can apply theorem 4.50. �


Proposition 4.142. If Z is bounded distributive lattice, then atomic elements of the �ltrator(F;P) are prime. [TODO: Generalize for meet-semilattices?]

Proof. (F;P) is with �nitely join-closed core by the theorem 4.96, F is a distributive lattice bytheorem 4.114. So we can apply proposition 4.52. �

The following theorem is essentially borrowed from [18]:

Theorem 4.143. Let Z be a boolean lattice. Let a be a �lter. Then the following are equivalent:

1. a is prime.

2. For every A2Z exactly one of fA; Ag is in a.

3. a is an atom of F.

Proof.

(1))(2). Let a be prime. Then A tZ A = 1A 2 a. Therefore A 2 a _ A 2 a. But sinceAuZA=0Z2/ a it is impossible A2 aÂ2 a.

(2))(3). Obviously a=/ 0F. Let a �lter b@ a. So b� a. Let X 2 b n a. Then X 2/ a and thusX 2 a and consequently X 2 b. So 0Z=X uZX 2 b and thus b=0F. So a is atomic.

(3))(1). By the previous proposition. �

4.3.17 Some Criteria

Proposition 4.144. Let Z be an atomic complete boolean lattice. Then the following conditionsare equivalent for any F 2F:

1. F 2P;

2. 8S 2PF:¡F uF

FF S=/ 0)9K2S:F uFK=/ 0�;

3. 8S 2PP:¡F uF

FF S=/ 0)9K 2S:F uFK=/ 0�.

Proof. The �ltrator (F; P) is semi�ltered by the corollary 4.95, star separable by 4.128, with�nitely meet-closed core by proposition 4.97, with separable core by theorem 4.112. P is atomisticbecause every atomic complete boolean lattice is atomistic. F is atomistic by theorem 4.137.

So we can apply the theorem 4.53. �

Theorem 4.145. If Z is a complete boolean lattice then for each F 2 F[TODO: Too similar tothe previous theorem (proposition 4.144)! Also it seems that this theorem can be generalized.]

F 2P,8S 2PP:

GPS 2 @F)S \ @F =/ ;

!:

Proof.

8S 2PP:

GPS 2 @F)S \ @F =/ ;

!,

8S 2PP:

GPS 2/ @F(S \ @F = ;

!,

8S 2PP:

GPS 2 upF(h:iS �F

!,

8S 2PP:

lP

S 2 upF(S � upF

!;


but

F 2P )

8S 2PP:

lP

S 2upF(S �upF

!)

lPupF 2upF )

F 2P:

�

Theorem 4.146. Let Z be a boolean lattice. For any S 2PF the condition 9F 2 F: S = ?F isequivalent to conjunction of the following items:

1. S is a free star on F;

2. S is �lter closed.

Proof.

).

1. That 0F2/ ?F is obvious. For every a; b2F

atF b2 ?F ,(atF b)uFF =/ 0F ,

(auFF)tF (buFF) =/ 0F ,auFF =/ 0F_ buFF =/ 0F ,

a2 ?F _ b2 ?F

(taken into account the corollary 4.114). So ?F is a free star on F.

2. We have T � S and need to prove thatdF T uF F =/ 0F. Because hF uF iT is a

generalized �lter base, 0F2 hF uF iT,dF hF uF iT =0F,

dFT uFF =/ 0F. So it is

left to prove 0F2/ hF uF iT what follows from T �S.(. Let S be a free star on F. Then for every A; B 2P

A; B 2S \P ,A; B 2S ,

AtFB 2S ,AtZB 2S ,

AtZB 2S \P

(taken into account the theorem 4.25). So S \P is a free star on P.Thus there exists F 2F such that @F =S \P. We have upX �S,X 2S (because S is

�lter closed) for every X 2F; then (taking into account properties of generalized �lter bases)

X 2S ,upX �S ,

upX � @F ,8X 2upX :X uFF =/ 0F ,

0F2/ hF uF iupX ,lFhF uF iupX =/ 0F ,

F uFlF

upX =/ 0F ,F uFX =/ 0F ,

X 2 ?F :�


4.3.18 Filters and a Special Sublattice

Theorem 4.147. Let (F;Z) be a primary �ltrator where Z is a boolean lattice. Let A2F. Thenfor each X 2F

X 2Z(DA),9X 2P:X =X uFA:

Proof.

(. Let X = X uF A where X 2 P. Let also Y = X uF A. Then X uF Y = X uF X uF A =(X uPX)uFA=0F (used theorem 4.44) and X tF Y = (X tFX)uFA= (X tPX)uFA=

1FuFA=A (used the theorems 4.25 and corollary 4.114). So X 2Z(DA).

). Let X 2 Z(DA). Then there exists Y 2Z(DA) such that X uF Y = 0F and X tF Y =A.Then (used theorem 4.112) there exists X 2upX such that X uFY =0F. We have

X =X tF (X uFY) =X uF (X tFY)=X uFA: �

4.3.19 Core Part and Atomic Elements

Proposition 4.148. Let Z be an atomistic lattice. Then for every a 2 F such that Cor0 a existswe have

Cor0 a=GZfx j x is an atom of Z; xv ag:

Proof. (F;P) is with join-closed core by corollary 4.96. So we can apply theorem 4.67. �

4.3.20 Complements and Core Parts

Proposition 4.149. Let Z be a complete boolean lattice. Then a�= a+=Cor a for every a2F.

Proof. The �ltrator (F;P) is �ltered by the corollary 4.95. F is a complete lattice by corollary4.107. (F;P) is with co-separable core by theorem 4.73. Thus we can apply the theorem 4.60.

(F;P) is �ltered by corollary 4.95, �nitely meet-closed by proposition 4.97, with separable coreby theorem 4.112. F is a complete lattice by corollary 4.107. So we can apply the theorem 4.62. �

Proposition 4.150. Let Z be a complete lattice. Then a�2P.

Proof. F is a complete lattice by 4.107. (F;P) is a �ltrator with join-closed core by corollary 4.96.(F;P) is a �ltrator with separable core by theorem 4.112. So we can apply theorem 4.64. �

Proposition 4.151. If Z is a complete boolean lattice, then a+ is dual pseudocomplement of a,that is

a+=min fc2A j ctF a=1Fgfor every a2F.

Proof. (F;P) is �ltered by the corollary 4.95. It is with co-separable core by theorem 4.73. F isa complete lattice by corollary 4.107. So we can apply theorem 4.65. �

Proposition 4.152. For a primary �ltrator over a complete boolean lattice both edge part anddual edge part are always de�ned.

Proof. Core part and dual core part are de�ned because the core is a complete lattice. Using thetheorem 4.116. �

Proposition 4.153. If Z is a complete lattice, then for every a; b2F

Cor(auF b)=Cor auPCor b:


Proof. (F;P) is with join-closed core by corollary 4.96. F is a meet-semilattice by corollary 4.107.So we can apply theorem 4.68. Then apply proposition 4.100. �

Proposition 4.154. If Z is a complete lattice, then for every S 2PF

CorlF

S=lPhCoriS:


Corollary 4.155. If Z is a complete lattice, then for every S 2PP

CorlF

S=lP

S:

Proposition 4.156. Let Z be a complete atomistic lattice. Then for every a; b2F

Cor(atF b)=Cor atPCor b:

Proof. (F; P) is semi�ltered by corollary 4.95. It is with �nitely meet-close core by 4.97. F isstarrish by corollary 4.114. F is complete by corollary 4.107. So we can apply theorem 4.71. Thenapply proposition 4.100. �

Theorem 4.157. Let Z be a complete boolean lattice. Then (auF b)�=a�tP b� for every a; b2F.

Proof. (F;P) is a �ltered (corollary 4.95) up-aligned complete lattice �ltrator with �nitely join-closed (theorem 4.25) co-separable core (theorem 4.73) which is a complete boolean lattice. Thusby the theorem 4.60

(auF b)�=(auF b)+=Cor(auF b)=Cor auPCor b=Cor atPCor b= a+tP b+= a�tP b�

(used propositions 4.149, 4.153). �

Theorem 4.158. Let Z be a complete atomistic boolean lattice. Then (a tF b)� = a� uP b� forevery a; b2F.

Proof. (F;P) is a �ltered (corollary 4.95), distributive (corollary 4.114) complete lattice �ltrator(corollary 4.107), with �nitely meet-closed core (proposition 4.97), with separable core (theorem4.112). So we can apply the theorem 4.72. �

4.3.21 Complementive Filters and Factoring by a Filter

De�nition 4.159. Let A be a meet-semilattice and A2A. The relation � on A is de�ned by theformula

8X;Y 2A: (X�Y ,X uAA=Y uAA):

Proposition 4.160. The relation � is an equivalence relation.

Proof.

Re�exivity. Obvious.

Symmetry. Obvious.

Transitivity. Obvious. �

De�nition 4.161. When X;Y 2Z and A2F we de�ne X�Y ,"X�"Y .

Theorem 4.162. Let Z be a distributive lattice [TODO: Generalize for meet-semilattices?], A2F.Then for every X;Y 2Z

X�Y ,9A2A:X uZA=Y uZA:


Proof. 9A 2 A:X uZA= Y uZA,9A 2 A: "X uF "A= "Y uF "A)9A 2 A: "X uF "A uFA="Y uF "AuFA,9A2A: "X uFA= "Y uFA,"X uFA= "Y uFA,"X�"Y ,X�Y .

On the other hand, "X uFA= "Y uFA, fX uZA0 j A0 2 Ag= fY uZA1 j A1 2 Ag) 9A0;A1 2 A: X uZ A0 = Y uZ A1 ) 9A0; A1 2 A: X uZ A0 uZ A1 = Y uZ A0 uZ A1 ) 9A 2 A:X uZA=Y uZA. �

Proposition 4.163. The relation � is a congruence4.1 for each of the following:

1. a meet-semilattice A;

2. a distributive lattice A.

Proof. Let a0; a1; b0; b12A and a0� a1 and b0� b1.

1. a0 u b0 � a1 u b1 because (a0 u b0) u A = a0 u (b0 u A) = a0 u (b1 u A) = b1 u (a0 u A) =b1u (a1uA)= (a1u b1)uA.

2. Taking the above into account, we need to prove only a0t b0� a1t b1. We have

(a0t b0)uA=(a0uA)t (b0uA) = (a1uA)t (b1uA)= (a1t b1)uA: �

De�nition 4.164. We will denote A/ (�) = A/ ((�) \ A � A) for a set A and an equivalencerelation � on a set B�A. I will call � a congruence on A when (�)\A�A is a congruence on A.

Theorem 4.165. Let F be the set of �lters over a boolean lattice Z and A 2 F. Consider thefunction :Z(DA)!Z/� de�ned by the formula (for every p2Z(DA))

p= fX 2Z j "X uFA= pg:Then:

1. is a lattice isomorphism.

2. 8Q2 q: ¡1 q= "QuFA for every q 2Z/�.

Proof. 8p2Z(DA): p=/ ; because of theorem 4.147. Thus it is easy to see that p 2 Z/� andthat is an injection.

Let's prove that is a lattice homomorphism: (p0uF p1)= fX 2Z j "X uFA= p0uF p1g;

p0uZ/� p1 =

fX02Z j "X0uFA= p0guZ/� fX12Z j "X1uFA= p1g =

f"X0uF "X1 j X0; X12Z; "X0uFA= p0^"X1uFA= p1g �fX 02Z j "X 0uFA= p0uF p1g =

(p0uF p1):

Because p0 uZ/� p1 and (p0 uF p1) are equivalence classes, thus follows p0 uZ/� p1 = (p0uF p1).

To �nish the proof it is enough to show that 8Q2 q: q= ("QuFA) for every q 2Z/�. (Fromthis it follows that is surjective because q is not empty and thus 9Q2 q: q= ("QuFA).) Really,

("QuFA) = fX 2Z j "X uFA= "QuFAg= [Q] = q: �

This isomorphism is useful in both directions to reveal properties of both lattices Z(DA) andZ/�.

Corollary 4.166. If Z is a boolean lattice then Z/� is a boolean lattice.

Proof. Because Z(DA) is a boolean lattice (theorem 2.79). �

4.1. See Wikipedia for a de�nition of congruence.


4.3.22 Pseudodi�erence of �lters

Proposition 4.167. For a lattice F of �lters over a boolean lattice and a; b 2 F the followingexpressions are always equal:

1. a n� b=dfz 2F j av bt zg (quasidi�erence of a and b);

2. a#b=Ffz 2F j z v a^ z u b=0g (second quasidi�erence of a and b);


Proof. Theorem 3.43, taking into account corollary 4.115 theorem 4.137. �

4.4 Filters on a Set

In this section we will consider �lters on the poset Z=PU (where U is some �xed set) with theorder AvB,A�B (for A; B 2PU).

In fact, it is a complete atomistic boolean lattice withdS =

TS,FS =

SS, A= U nA for

every S 2PPU and A2PU, atoms being one-element sets.

De�nition 4.168. I will call a �lter on the lattice of all subsets of a given set U as a �lter on set .

De�nition 4.169. I will denote the set on which a �lter F is de�ned as Base(F).

Obvious 4.170. Base(F) =SF .

De�nition 4.171. I will call the primary �ltrator for Z = PU (with order on Z de�ned asAvB,A�B) for some set U as powerset �ltrator .

Proposition 4.172. The following are equivalent for a non-empty set F 2PPU:

1. F is a �lter.

2. 8X;Y 2F :X \Y 2F and F is an upper set.

3. 8X;Y 2PU: (X;Y 2F,X \Y 2F ).


Obvious 4.173. The minimal �lter on PU is PU.

Obvious 4.174. The maximal �lter on PU is fUg.

I will denote "A="UA="PUA. (The distinction between con�icting notations "UA and "PUAwill be clear from the context.)

Proposition 4.175. The powerset �ltrator is both up-aligned and down-aligned.


Proposition 4.176. Every powerset �ltrator is �ltered.

Proof. By corollary 4.95. �

Proposition 4.177. Every powerset �ltrator is with join-closed core.


Proposition 4.178. Every powerset �ltrator is with �nitely meet-closed core.

Proof. By proposition 4.97. �


Proposition 4.179. Every powerset �ltrator is with separable core.


Proposition 4.180. Every powerset �ltrator is with co-separable core.


Proposition 4.181. Cor0 a=Cor a= "Base(a)T

a for every �lter a on a set.


Proposition 4.182. Cor av a for every �lter a on a set.


Proposition 4.183. Cor a=maxdowna for every �lter a on a set.


Proposition 4.184. For the lattice F of �lters on a set U, A2F, B 2P we have:

1. B�FA,B wA;2. B�FA,B vA.


Proposition 4.185.FF S=

TS for a set S of �lters on a powerset.


Corollary 4.186. A set of �lters on a powerset is always a complete lattice.

Corollary 4.187. AtB=A\B for �lters A and B on a powerset.

Proposition 4.188. For S 2PF n f;g where F are �lters on a powerset

lFS=

�K0\ ::: \Kn j Ki2


:


Proposition 4.189. For every F0; :::;Fm2F (m2N ) where F are �lters on a powerset

F0uF ::: uFFm= fK0\ ::: \Km j Ki2F i where i=0; :::; mg:


Proposition 4.190. If A2F and S 2PF where F are �lters on a powerset then

AtFlF

S=lFhAtF iS:


Corollary 4.191. The poset of �lters on a powerset is a distributive lattice.

Corollary 4.192. The poset of �lters on a powerset is a co-brouwerian lattice.

Proposition 4.193. a nFB= auFB for every a2F, B 2P (where F is �lters on a powerset andthe complement is taken on P).

4.4 Filters on a Set 77


Proposition 4.194. Let F be the poset of �lters on a powerset. A uFFF

S =FF hA uF iS for

every A2P and every set S 2PF.


Proposition 4.195. If S is a generalized �lter base of a �lter F on a set U then for any K 2PU

K 2F,9L2S:K 2L:


Proposition 4.196. If S is a generalized �lter base of a �lter F on a set U then

0F2S,F =0F:


Proposition 4.197. Let S be a nonempty set of �lters on a set such that F0uF ::: uFFn=/ 0F for

every F0; :::;Fn2S. ThendF

S=/ 0F.


Proposition 4.198. Let S 2PU n f;g where U is a set and A0 \ ::: \ An =/ ; for every A0; :::;An2S. Then

dF h"iS=/ 0F.


Proposition 4.199. @a is a free star for each �lter a on a set.


Proposition 4.200. For a �lter A on a set: X 2upA,X 2/ @A for every X 2P, A2F.


Proposition 4.201. For a �lter A on a set:

1. @A= fX j X 2P n upAg;

2. upA= fX j X 2P n @Ag(where complement is taken on the boolean lattice P).


Proposition 4.202. @ is an injection for �lters on sets.


Proposition 4.203. For �lters on a set: for any set S 2 PP there exists a �lter A such that@A=S i� S is a free star.


Proposition 4.204. AvB, @A� @B for every �lters A, B on a set.


Proposition 4.205. @ is a straight monotone map for �lters on a set.



Proposition 4.206. @FF S=

Sh@ iS for every S 2PF where F are �lters on a set.


Proposition 4.207. The poset of �lters on a set is atomic.


Proposition 4.208. The poset of �lters on a set is separable.

Proof. By obvious 4.136. �

Proposition 4.209. The poset of �lters on a set is atomistic.


Proposition 4.210. The poset of �lters on a set is atomically separable.


Proposition 4.211. The �ltrator on a powerset is central.


Proposition 4.212. a is an atom of P i� a2P and a is an atom of F for �lters on a set.


Proposition 4.213. a2F is an atom of F i� up a= @a for �lters on a set.


Theorem 4.214. Let a be a �lter on a set. Then the following are equivalent:

1. a is prime.

2. For every A2Z exactly one of fA; Ag is in a.

3. a is an atom of F.


Proposition 4.215. The following conditions are equivalent for every �lter F on a set:

1. F 2P;

2. 8S 2PF:¡F uF

FF S=/ 0)9K2S:F uFK=/ 0�;

3. 8S 2PP:¡F uF

FF S=/ 0)9K 2S:F uFK=/ 0�.


Proposition 4.216. For every �lter F on a set

F 2P,8S 2PP:

GPS 2 @F)S \ @F =/ ;

!:


Theorem 4.217. For any S 2PF, where F are �lters on a set, the condition 9F 2 F: S = ?F isequivalent to conjunction of the following items:

1. S is a free star on F;


2. S is �lter closed.


Proposition 4.218. Let F be �lters on a set. Let A2F. Then for each X 2F

X 2Z(DA),9X 2P:X =X uFA:


Proposition 4.219. Cor a= "fp 2U j "fpg v ag andT

a= fp2 U j "fpgv ag for every �lter aon a set.

Proof. By propositions 4.148 and 4.181. �

Proposition 4.220. For every �lter a on a set a�= a+=Cor a=Cor0 a.


Corollary 4.221. For every �lter a on a set a�= a+2P.

Proposition 4.222. If a is a �lter on a set, then a+ is dual pseudocomplement of a, that is

a+=min fc2F j ctF a=1Fg:


Proposition 4.223. If a, b are �lters on a set, then

1.T(auF b)=

Ta\T

b;

2.T(atF b)=

Ta[T

b.


Proposition 4.224.T dF S=

ThTiS.


Proposition 4.225. If a, b are �lters on a set, then

1. (auF b)�= a�tP b�;

2. (atF b)�= a�uP b�.


Proposition 4.226. For every X;Y 2PU and �lter F on U we have:

"X�"Y ,9A2A:X \A=Y \A:


Proposition 4.227. Let F be the set of �lters on a set U and A 2 F. Consider the function :Z(DA)! (PU)/� de�ned by the formula (for every p2Z(DA))

p= fX 2Z j "X uFA= pg:Then:

1. is a lattice isomorphism.

2. 8Q2 q: ¡1 q= "QuFA for every q 2 (PU)/�.



Proposition 4.228. (PU)/� is a boolean lattice.


Proposition 4.229. For a lattice F of �lters on a set and a; b 2 F the following expressions arealways equal:

1. a n� b=dfz 2F j av bt zg (quasidi�erence of a and b);

2. a#b=Ffz 2F j z v a^ z u b=0g (second quasidi�erence of a and b);



Conjecture 4.230. a n� b = a#b for arbitrary �lters a, b on powersets is not provable in ZF(without axiom of choice).

4.4.1 Fréchet FilterThe consideration below is about �lters on a set U, but this can be generalized for �lters on completeatomic boolean algebras due complete atomic boolean algebras are isomorphic to algebras of setson some set U.

De�nition 4.231. = fU nX j X is a �nite subset of Ug is called either Fréchet �lter or co�nite�lter .

It is trivial that Fréchet �lter is a �lter.

Proposition 4.232. Cor=0P;T= ;.

Proof. This can be deduced from the formula 8�2U9X 2:�2/X . �

Theorem 4.233. max fX 2F j CorX =0Pg=max fX 2F jTX = ;g=.

Proof. Due the last proposition, it is enough to show that CorX =0P)X v for every �lter X .Let CorX =0P for some �lter X . Let X 2. We need to prove that X 2X .X =U n f�0; :::; �ng. U n f�ig2X because otherwise �i2"¡1CorX . So X 2X . �

Theorem 4.234. =FF fx j x is a non-trivial ultra�lterg.

Proof. It follows from the facts that Cor x = 0P for every non-trivial ultra�lter x, that F is anatomistic lattice, and the previous theorem. �

Theorem 4.235. Cor is the lower adjoint of tF¡.

Proof. Because both Cor and tF ¡ are monotone, it is enough (theorem 2.98) to prove (forevery �lters X and Y)

X vtFCorX and Cor(tFY)vY:

Cor(tFY)=CortPCorY =0PtPCorY =CorY vY .tFCorX wEdgX tFCorX =X . �

Corollary 4.236. CorX =X n� for every �lter on a set.


Corollary 4.237. CorFF

S=FF hCoriS for any set S of �lters on a powerset.


4.4.2 Number of Filters on a Set

De�nition 4.238. A collection Y of sets has �nite intersection property i� intersection of any�nite subcollection of Y is non-empty.

The following was borrowed from [7]. Thanks to Andreas Blass for email support about hisproof.

Lemma 4.239. (by Hausdor�) For an in�nite set X there is a family F of 2cardX many subsets ofX such that given any disjoint �nite subfamiliesA, B, the intersection of sets inA and complementsof sets in B is nonempty.

Proof. Let

X 0= f(P ;Q) j P 2PX is �nite; Q2PPP g:

It's easy to show that cardX 0= cardX. So it is enough to show this for X 0 instead of X . Let

F = ff(P ;Q)2X 0 j Y \P 2Qg j Y 2PXg:

To �nish the proof we show that for every disjoint �nite Y+2PPX and �nite Y¡2PPX thereexist (P ;Q)2X 0 such that

8Y 2Y+: (P ;Q)2f(P ;Q)2X 0 j Y \P 2Qg and 8Y 2Y¡: (P ;Q)2/ f(P ;Q)2X 0 j Y \P 2Qg

what is equivalent to existence (P ;Q)2X 0 such that

8Y 2Y+:Y \P 2Q and 8Y 2Y¡:Y \P 2/ Q:

For existence of this (P ;Q), it is enough existence of P such that intersections Y \P are di�erentfor di�erent Y 2Y+[Y¡.

Really, for each pair of distinct Y0; Y12Y+[Y¡ choose a point which lies in one of the sets Y0,Y1 and not in an other, and call the set of such points P . Then Y \ P are di�erent for di�erentY 2Y+[Y¡. �

Corollary 4.240. For an in�nite set X there is a family F of 2cardX many subsets of X suchthat for arbitrary disjoint subfamilies A and B the set A[fX nA j A2Bg has �nite intersectionproperty.

Theorem 4.241. Let X be a set. The number of ultra�lters on X is 22cardX

if X is in�nite andcardX if X is �nite.

Proof. The �nite case follows from the fact that every ultra�lter on a �nite set is trivial. Let Xbe in�nite. From the lemma, there exists a family F of 2cardX many subsets of X such that forevery G 2PF we have �(F ; G)=

dh"iG u

dh"ifX nA j A2F n Gg=/ 0F(X).

This �lter contains all sets from G and does not contain any sets from F n G. So for everysuitable pairs (F0;G0) and (F1;G1) there is A2�(F0;G0) such that A2�(F1;G1). Consequently all�lters �(F ;G) are disjoint. So for every pair (F ;G) where G 2PF there exist a distinct ultra�lterunder �(F ; G), but the number of such pairs (F ; G) is 22cardX. Obviously the number of all �ltersis not above 22

cardX. �

Corollary 4.242. The number of �lters on U is 22cardU

if U is in�nite and 2cardU if U is �nite.

Proof. The �nite case is obvious. The in�nite case follows from the theorem and the fact that�lters are collections of sets and there cannot be more than 22

cardUcollections of sets on U. �

4.5 Some Counter-Examples

Example 4.243. There exist a bounded distributive lattice which is not lattice with separablecenter.


Proof. The lattice with the following Hasse diagram4.2 is bounded and distributive because itdoes not contain �diamond lattice� nor �pentagon lattice� as a sublattice [40].

a

0

1

yx

Figure 4.1.

It's center is f0; 1g. xu y=0 despite up x= fx; a; 1g but y u 1=/ 0 consequently the lattice isnot with separable center. �

For further examples we will use the �lter � de�ned by the formula

�=lFf"(¡"; ") j "2R; " > 0g

and more general

�+ a=lFf"(a¡ "; a+ ") j "2R; " > 0g:

Example 4.244. There exists A2PU such thatdF h"iA=/ "

TA.

Proof. "Tf(¡"; ") j "2R; " > 0g= "f0g=/ �. �

Example 4.245. There exists a set U and a �lter a and a set S of �lters on the set U such thatauF

FF S=/FF hauF iS.

Proof. Let a = � and S = h"if("; +1) j " > 0g. Then a uFFF S = � uF "(0; +1) =/ 0F whileFF hauF iS=

FF f0Fg=0F. �

Example 4.246. There are tornings which are not weak partitions.

Proof. f�+ a j a2Rg is a torning but not weak partition of the real line. �

Lemma 4.247. Let F be the set of �lters on a set U . Then "X uFv"Y uF i� X nY is a �niteset, for every sets X;Y 2PU .

Proof. "X uF v "Y uF , fX \KX j KX 2 g � fY \KY j KY 2 g , 8KY 2 9KX 2 :Y \ KY = X \ KX, 8LY 2M9LX 2M : Y n LY = X n LX, 8LY 2M : X n (Y n LY ) 2M ,X nY 2M , where M is the set of �nite subsets of U . �

Example 4.248. There exists a �lter A on a set U such that (PU) /� and Z(D A) are notcomplete lattices.

Proof. Due to the isomorphism it is enough to prove for (PU)/�.Let take U =N and A= be the Fréchet �lter on N .Partition N into in�nitely many in�nite sets A0;A1; :::. To withhold our example we will prove

that the set {[A0]; [A1]; :::} has no supremum in (PU)/�.Let [X] be an upper bound of [A0]; [A1]; ::: that is 8i2N : "X uFw"AiuF that is Ai nX is

�nite. Consequently X is in�nite. So X \Ai=/ ;.

4.2. See Wikipedia for a de�nition of Hasse diagrams.

4.5 Some Counter-Examples 83

Choose for every i 2 N some zi 2 X \ Ai. Then fz0; z1; :::g is an in�nite subset of X (takeinto account that zi =/ zj for i =/ j). Let Y =X n fz0; z1; :::g. Then "Y uF w "Ai uF becauseAi n Y =Ai n (X n fzig) = (Ai nX) [ fzig which is �nite because Ai nX is �nite. Thus [Y ] is anupper bound for {[A0]; [A1]; :::}.

Suppose "Y uF="X uF. Then Y nX is �nite what is not true. So "Y uF@"X uF thatis [Y ] is below [X]. �

4.5.1 Weak and Strong Partition

De�nition 4.249. A family S of subsets of a countable set is independent i� the intersection ofany �nitely many members of S and the complements of any other �nitely many members of S isin�nite.

Lemma 4.250. The �in�nite� at the end of the de�nition could be equivalently replaced with �non-empty� if we assume that S is in�nite.

Proof. Suppose that some sets from the above de�nition has a �nite intersection J of cardinalityn. Then (thanks S is in�nite) get one more set X 2S and we have J \X=/ ; and J \ (N nX)=/ ;.So card(J \X) < n. Repeating this, we prove that for some �nite family of sets we have emptyintersection what is a contradiction. �

Lemma 4.251. There exists an independent family on N of cardinality c.

Proof. Let C be the set of �nite subsets of Q. Since card C = card N , it su�ces to �nd cindependent subsets of C. For each r2R let

Er= fF 2C j card(F \ (¡1; r)) is eveng:

All Er1 and Er2 are distinct for distinct r1; r2 2 R since we may consider F = fr 0g 2 C where arational number r 0 is between r1 and r2 and thus F is a member of exactly one of the sets Er1 andEr2. Thus cardfEr j r 2Rg= c.

We will show that fEr j r 2Rg is independent. Let r1; :::; rk; s1; :::; sk be distinct reals. It isenough to show that these have a nonempty intersection, that is existence of some F such that Fbelongs to all the Er and none of Es.

But this can be easily accomplished taking F having zero or one element in each of intervalsto which r1; :::; rk; s1; :::; sk split the real line. �

Example 4.252. There exists a weak partition of a �lter on a set which is not a strong partition.

Proof. (suggested by Andreas Blass) Let fXr j r2Rg be an independent family of subsets of N .We can assume a=/ b)Xa=/ Xb due the above lemma.

Let Fa be a �lter generated byXa and the complementsN nXb for all b2R, b=/ a. Independenceimplies that Fa=/ 0

F (by properties of �lter bases).Let S= fFr j r 2Rg. We will prove that S is a weak partition but not a strong partition.Let a 2 R. Then Xa 2 Fa while 8b 2 R n fag: N n Xa 2 Fb and therefore N n Xa 2FF fFb j R3 b=/ ag. Therefore FauF

FF fFb j R3 b=/ ag=0F. Thus S is a weak partition.Suppose S is a strong partition. Then for each set Z 2PR

GFfFb j b2ZguF

GFfFb j b2R nZg=0F

what is equivalent to existence of M(Z)2PN such that

M(Z)2GFfFb j b2Zg and N nM(Z)2

GFfFb j b2R nZg

that is

8b2Z:M(Z)2Fb and 8b2R nZ:N nM(Z)2Fb:


Suppose Z=/ Z 02PN . Without loss of generality we may assume that some b2Z but b2/ Z 0. ThenM(Z)2Fb and N nM(Z 0)2Fb. If M(Z)=M(Z 0) then F b=0F what contradicts to the above.

So M is an injective function from PR to PN what is impossible due cardinality issues. �

Lemma 4.253. (by Niels Diepeveen, with help of Karl Kronenfeld) Let K be a collection of freeultra�lters. We have

FK= i� 9G 2K:A2 G for every in�nite set A.

Proof.

). SupposeFK = and let A be a set such that @G 2K:A2 G. Let's prove A is �nite.

Really, 8G 2K:U nA2 G; U nA2; A is �nite.

(. Let 9G 2K:A2 G. Suppose A is a set inFK.

To �nish the proof it's enough to show that U nA is �nite.Suppose U n A is in�nite. Then 9G 2 K: U n A 2 G; 9G 2 K: A 2/ G; A 2/

FK,

contradiction. �

Lemma 4.254. (by Niels Diepeveen) If K is a non-empty set of ultra�lters such thatFK =,

then for every G 2K we haveF(K n fGg)=.

Proof. 9F 2K:A2F for every in�nite set A.The set A can be partitioned into two in�nite sets A1, A2.Take F1;F22K such that A12F1, A22F2.F1=/ F2 because otherwise A1 and A2 are not disjoint.Obviously A2F1 and A2F2.So there exist two di�erent F 2K such that A2F . Consequently9F 2K n fGg:A2F that is

F(K n fGg)=. �

Example 4.255. There exists a �lter on a set which cannot be weakly partitioned into ultra�lters.

Proof. Consider co�nite �lter on any in�nite set.Suppose K is its weak partition into ultra�lters. Then x�

F(K nfxg) for some ultra�lter x2K.

We haveF(K n fxg) @ F K (otherwise x v

F(K n fxg)) what is impossible due the last

lemma. �

Corollary 4.256. There exists a �lter on a set which cannot be strongly partitioned into ultra-�lters.

4.6 Open problems about �lters

In this section, I will formulate some conjectures about lattices of �lters on a set. If a conjecturecomes true, it may be generalized for more general lattices (such as, for example, lattices of �lterson arbitrary lattices). I deem that the main challenge is to prove the special case about lattices of�lters on a set, and generalizing the conjectures is expected to be an easy task.

4.6.1 PartitioningConsider the complete lattice [S] generated by the set S where S is a strong partition of someelement a.

Conjecture 4.257. [S] =�FF

X j X 2PS, where [S] is the complete lattice generated by a

strong partition S of �lter on a set.

Consider also the similar conjecture with weak partition instead strong partition.

Proposition 4.258. Provided that the last conjecture is true, we have that [S] is a completeatomic boolean lattice with the set of its atoms being S.

4.6 Open problems about filters 85

Remark 4.259. Consequently [S] is atomistic, completely distributive and isomorphic to a powerset algebra (see [39]).

Proof. Completeness of [S] is obvious. Let A2 [S]. Then there existsX 2PS such that A=FF X.

Let B=FF (S nX). Then B 2 [S] and AuFB=0F. AtFB=

FF S is the greatest element of [S].So we have proved that [S] is a boolean lattice.

Now let prove that [S] is atomic with the set of atoms being S. Let z 2 S and A 2 [S]. IfA=/ z then either A=0F or x2X where A=

FFX, X 2PS and x=/ z. Because S is a partition,FF

(X n fzg)uF z=0F andFF

(X n fzg)=/ 0F. So A=FF

X =FF

(X n fzg)tF zvz.Finally we will prove that elements of [S] n S are not atoms. Let A2 [S] n S and A=/ 0. Then

AwxtF y where x; y 2S and x=/ y. If A is an atom then A=x= y what is impossible. �

Proposition 4.260. The conjecture about the value of [S] is equivalent to closedness of�FF X j X 2PSunder arbitrary meets and joins.

Proof. If�FF

X j X 2 PS= [S] then trivially

�FFX j X 2 PS

is closed under arbitrary

meets and joins.If�FF

X j X 2PSis closed under arbitrary meets and joins, then it is the complete lattice

generated by the set S because it cannot be smaller than the set of all suprema of subsets of S. �

That�FF

X j X 2PSis closed under arbitrary joins is trivial. I have not succeeded to prove

that it is closed under arbitrary meets, but have proved a weaker statement that is is closed under�nite meets:

Proposition 4.261.�FF X j X 2PS

is closed under �nite meets.

Proof. Let R=�FF X j X 2PS

. Then for every X;Y 2PSGF

X uFGF

Y =GF((X \Y )[ (X nY ))uF

GFY = GF

(X \Y )tFGF

(X nY )

!uFGF

Y = GF(X \Y )uF

GFY

!tF GF

(X nY )uFGF

Y

!= GF

(X \Y )uFGF

Y

!tF 0F =

GF(X \Y )uF

GFY :

Applying the formulaFF X uF

FF Y =FF (X \Y )uF

FF Y twice we getGFX uF

GFY =GF

(X \Y )uFGF

(Y \ (X \Y )) =GF(X \Y )uF

GF(X \ Y ) =GF(X \Y ):

But for any A; B 2 R there exist X; Y 2 PS such that A =FF X, B =

FF Y . So A uF B =FFX uF

FFY =

FF(X \Y )2R. �


4.6.2 Quasidi�erence

Conjecture 4.262. a n� b=Ffau"(U nB) j B 2 bg for all a; b2F for each lattice F of �lters on

a set U .

4.6.3 Non-Formal ProblemsFind a common generalization of two theorems:

1. If Z is a meet-semilattice with greatest element then for any A;B 2F

AtFB=A\B:

2. If Z is a join-semilattice then F is a join-semilattice then and for any A;B 2F

AtFB=A\B:

Under which conditions a n� b and a#b are complementive to a?Generalize straight maps for arbitrary posets.

4.6 Open problems about filters 87

Chapter 5Common knowledge, part 2 (topology)

In this chapter I describe basics of the theory known as general topology . Starting with the nextchapter after this one I will describe generalizations of customary objects of general topologydescribed in this chapter.

The reason why I've written this chapter is to show to the reader kinds of objects which Igeneralize below in this book. For example, funcoids and a generalization of proximity spaces, andfuncoids are a generalization of pretopologies. To understand the intuitive meaning of funcoids oneneeds �rst know what are proximities and what are pretopologies.

Having said that, customary topology is not used in my de�nitions and proofs below. It is justto feed your intuition.

5.1 Metric spacesThe theory of topological spaces started immediately with the de�nition would be completelynon-intuitive for the reader. It is the reason why I �rst describe metric spaces and show thatmetric spaces give rise for a topology (see below). Topological spaces are understandable as ageneralization of topologies induced by metric spaces.

Metric spaces is a formal way to express the notion of distance . For example, there are distancejx¡ y j between real numbers x and y, distance between points of a plane, etc.

De�nition 5.1. A metric space is a set U together with a function d:U �U!R (distance) suchthat for every x; y; z 2U :

1. d(x; y)> 0;2. d(x; y)= 0,x= y;

3. d(x; y)= d(y; x) (symmetry);

4. d(x; z)6 d(x; y) + d(y; z) (triangle inequality).

Exercise 5.1. Show that the Euclid space Rn (with the standard distance) is a metric space for every n2N .

De�nition 5.2. Open ball of radius r > 0 centered at point a2U is the set

Br(a)= fx2U j d(a; x)<rg:

De�nition 5.3. Closed ball of radius r > 0 centered at point a2U is the set

Br[a] = fx2U j d(a; x)6 rg:

5.1.1 Open and closed sets

De�nition 5.4. A set A in a metric space is called open when 8a2A9r > 0:Br(a)�A.

De�nition 5.5. A set A in a metric space is closed when its complement U nA is open.

De�nition 5.6. Closure cl(A) of a set A in a metric space is the set of points y such that

8"> 09a2A: d(y; a)<":

89

Proposition 5.7. cl(A)�A.

Proof. It follows from d(a; a)= 0<". �

Exercise 5.2. Prove cl(A[B)= cl(A)[ cl(B) for every subsets A and B of a metric space.

5.1.2 Continuity

De�nition 5.8. A function f from a metric space A to a metric space B is called continuous atpoint a2A when

8"> 09� > 08x2A: (d(a; x)<�) d(f(a); f(x))<"):

De�nition 5.9. A function f is called continuous when it is continuous at every point of itsdomain.

5.2 Pretopological spaces

Pretopological space can be de�ned in two equivalent ways: a neighborhood system or a preclosureoperator . To be more clear I will call pretopological space only the �rst (neighborhood system) andthe second call a preclosure space.

De�nition 5.10. Pretopological space is a set U together with a �lter �(x) on U for every x2U ,such that "Ufxgv�(x). �(x) is called a pretopology on U .

De�nition 5.11. Preclosure on a set U is an unary operation cl on PU such that for every A;B 2PU :

1. cl(;) = ;;

2. cl(A)�A;

3. cl(A[B)= cl(A)[ cl(B).

I call a preclosure together with a set U as preclosure space.

Theorem 5.12. Small pretopological spaces and small preclosure spaces bijectively correspondto each other by the formulas:

cl(A)= fx2U j A2 @�(x)g; (5.1)

�(x)= fA2PU j x2/ cl(U nA)g: (5.2)

Proof. First let's prove that cl de�ned by formula (5.1) is really a preclosure.cl(;) = ; is obvious. If x 2A then A 2 @�(x) and so cl(A)�A. cl(A[B) = fx 2U j A[B 2

@�(x)g= fx2U j A2 @�(x)_B 2 @�(x)g= cl(A)[ cl(B). So, it is really a preclosure.Next let's prove that � de�ned by formula (5.2) is a pretopology. That �(x) is an upper

set is obvious. Let A; B 2�(x). Then x 2/ cl(U nA) ^ x 2/ cl(U n B); x 2/ cl(U nA) [ cl(U nB) =cl((U nA)[ (U nB))= cl(U n (A\B)); A\B 2�(x). We have proved that �(x) is a �lter.

Let's prove "Ufxg v�(x). If A2�(x) then x2/ cl(U nA) and consequently x2/ U nA; x2A;A2"Ufxg. So "Ufxgv�(x) and thus � is a pretopology.

It is left to prove that the functions de�ned by the above formulas are mutually inverse.Let cl0 be a preclosure, let � is the pretopology induced by cl0 by the formula (5.2), let cl1

is the preclosure induced by � by the formula (5.1). Let's prove cl1 = cl0. Really, x 2 cl1(A),�(x)�/ "UA, 8X 2�(x):X \ A =/ ; , 8X 2PU : (x 2/ cl0(U nX))X \ A =/ ;), 8X 0 2PU :(x 2/ cl0(X 0) ) A n X 0 =/ ;) , 8X 0 2 PU : (A n X 0 = ; ) x 2 cl0(X 0)) , 8X 0 2 PU :(A�X 0) x2 cl0(X 0)),x2 cl0(A). So cl1(A)= cl0(A).

90 Common knowledge, part 2 (topology)

Let now �0 be a pretopology, let cl is the closure induced by �0 by the formula (5.1), let�1 is the pretopology induced by cl by the formula (5.2). Really A 2 �1(x), x 2/ cl(U n A),�0(x)�"U(U nA),"UAw�0(x),A2�0(x) (used proposition 4.184). So �1(x)=�0(x).

That these functions are mutually inverse, is now proved. �

5.2.1 Pretopology induced by a metricEvery metric space induces a pretopology by the formula:

�(x)=lf"UBr(x) j r2R; r > 0g:

Exercise 5.3. Show that it is a pretopology.

Proposition 5.13. The preclosure corresponding to this pretopology is the same as the preclosureof the metric space.

Proof. I denote the preclosure of the metric space as clM and the preclosure corresponding to ourpretopology as clP . We need to show clP = clM.

Really: clP(A) = fx 2 U j A 2 @�(x)g= fx 2 U j 8" > 0:B"(x)�/ Ag= fy 2 U j 8" > 09a 2A:d(y; a)<"g= clM(A) for every set A2PU . �

5.3 Topological spaces

Proposition 5.14. For the set of open sets of a metric space (U ; d) it holds:

1. Union of any (possibly in�nite) number of open sets is an open set.

2. Intersection of a �nite number of open sets is an open set.

3. U is an open set.

Proof. Let S be a set of open sets. Let a2SS. Then there exists A2S such that a2A. Because

A is open we have Br(a)�A for some r > 0. Consequently Br(a)�S

S that isS

S is open.Let A0; :::; An be open sets. Let a2A0\ ::: \An for some n2N . Then there exist ri such that

Bri(a)�Ai. So Br(a)�A0\ ::: \An for r=min fr0; :::; rng that is A0\ ::: \An is open.That U is an open set is obvious. �

The above proposition suggests the following de�nition:

De�nition 5.15. A topology on a set U is a collection O (called the set of open sets) of subsetsof U such that:

1. Union of any (possibly in�nite) number of open sets is an open set.

2. Intersection of a �nite number of open sets is an open set.

3. U is an open set.

The pair (U ;O) is called a topological space .

Remark 5.16. From the above it is clear that every metric induces a topology.

Proposition 5.17. Empty set is always open.

Proof. Empty set is union of an empty set. �

De�nition 5.18. A closed set is a complement of an open set.

Topology can be equivalently expresses in terms of closed sets:A topology on a set U is a collection (called the set of closed sets) of subsets of U such that:

1. Intersection of any (possibly in�nite) number of closed sets is a closed set.

5.3 Topological spaces 91

2. Union of a �nite number of closed sets is a closed set.

3. ; is a closed set.

Exercise 5.4. Show that the de�nitions using open and closed sets are equivalent.

5.3.1 Relationships between pretopologies and topologies

5.3.1.1 Topological space induced by preclosure space

Having a preclosure space (U ; cl) we de�ne a topological space whose closed sets are such setsA2PU that cl(A)=A.

Proposition 5.19. This really de�nes a topology.

Proof. Let S be a set of closed sets. First, we need to prove thatT

S is a closed set. We havecl(T

S)�A for every A2S. Thus cl(T

S)�T

S and consequently cl(T

S) =T

S. SoT

S is aclosed set.

Let now A0; :::; An be closed sets, then

cl(A0[ ::: [An)= cl(A0)[ ::: [ cl(An) =A0[ ::: [An

that is A0[ ::: [An is a closed set.That ; is a closed set is obvious. �

Having a pretopological space (U ; �) we de�ne a topological space whose open sets are

fX 2PU j 8x2X:X 2�(x)g:

Proposition 5.20. This really de�nes a topology.

Proof. Let set S �fX 2PU j 8x2X :X 2�(x)g. Then 8X 2S8x2X :X 2�(x). Thus

8x2[

S9X 2S:X 2�(x)

and so 8x2S

S:S

S 2�(x). SoS

S is an open set.Let now A0; :::; An2fX 2PU j 8x2X :X 2�(x)g for n2N . Then 8x2Ai:Ai2�(x) and so

8x2A0\ ::: \An:Ai2�(x);

thus 8x2A0\ ::: \An:A0\ ::: \An2�(x). So A0\ ::: \An2fX 2PU j 8x2X :X 2�(x)g.That U is an open set is obvious. �

Proposition 5.21. Topology � de�ned by a pretopology and topology � de�ned by the corre-sponding preclosure, are the same.

Proof. Let A2PU .A is �-closed,cl(A) =A, cl(A)�A,8x2U : (A2 @�(x))x2A);A is � -open,8x 2 A: A 2 �(x) , 8x 2 U : (x 2 A) A 2 �(x)) , 8x 2 U : (x 2/ U n A)

U nA2/ @�(x)).So �-closed and � -open are negations of each other. It follows �= � . �

5.3.1.2 Preclosure space induced by topological space

We de�ne a preclosure and a pretopology induced by a topology and then show these two areequivalent.

Having a topological space we de�ne a preclosure space by the formula

cl(A)=\fX 2PU j X is a closed set; X �Ag:

Proposition 5.22. It is really a preclosure.


Proof. cl(;)= ; because ; is a closed set. cl(A)�A is obvious. cl(A[B)=TfX 2PU j X is a

closed set; X �A[Bg=TfX1[X2 j X1; X22PU are closed sets; X1�A; X2�Bg=

TfX12

PU j X1 is closed a set;X1�Ag[TfX22PU j X2 is closed a set;X2�Bg= cl(A)[ cl(B). Thus

cl is a preclosure. �

Or: �(x)=df"UX j X 2O; x2Xg.

It is trivially a pretopology (used the fact that U 2O).

Proposition 5.23. The preclosure and the pretopology de�ned in this section above correspondto each other (by the formulas from theorem 5.12).

Proof. We need to prove cl(A) = fx2U j �(x)�/ "UAg, that is\fX 2PU j X is a closed set; X �Ag=

�x2U j

lf"UX j X 2O; x2Xg�/ "UA

:

Equivalently transforming it, we get:TfX 2PU j X is a closed set; X �Ag= fx2U j 8X 2O: (x2X)"UX �/ "UA)g;TfX 2PU j X is a closed set; X �Ag= fx2U j 8X 2O: (x2X)X �/ A)g:

x2TfX 2PU j X is a closed set;X �Ag,8X 2PU : (X is a closed set^X �A)x2X),

8X 02O: (U nX 0�A)x2U nX 0),8X 02O: (X 0�A)x2/X 0),8X 2O: (x2X)X �/ A). Soour equivalence holds. �

Proposition 5.24. If � is the topology induced by pretopology �, in turn induced by topology�, then � = �.

Proof. The set of closed sets of � is fA 2 PU j cl�(A) = Ag = fA 2 PU jTfX 2 PU j X

is a closed set in �; X � Ag = Ag = fA 2PU j A is a closed set in �g (taken into account thatintersecting closed sets is a closed set). �

De�nition 5.25. Idempotent closures are called Kuratowski closures .

Theorem 5.26. The above de�ned correspondences between topologies and pretopologies,restricted to Kuratowski closures, is a bijection.

Proof. Taking into account the above proposition, it's enough to prove that:If � is the pretopology induced by topology �, in turn induced by a Kuratowski closure �, then

� = �.cl�(A) =

TfX 2PU j X is a closed set in �;X �Ag=

TfX 2PU j cl�(X) =X;X �Ag=T

fcl�(X) j X 2PU ;cl�(X)=X;X�cl�(A)g=Tfcl�(cl�(X)) j X=Ag=cl�(cl�(A))=cl�(A). �

5.3.1.3 Topology induced by a metric

De�nition 5.27. Every metric space induces a topology in this way: A set X is open i�

8x2X9"> 0:Br(x)�X:

Exercise 5.5. Prove it is really a topology and this topology is the same as the topology, induced by thepretopology, in turn induced by our metric space.

5.4 Proximity spaces

Let (U ; d) be metric space. We will de�ne distance between sets A; B 2PU by the formula

d(A; B)= inf fd(a; b) j a2A; b2Bg:

(Here �inf � denotes in�mum on the real line.)

De�nition 5.28. Sets A; B 2PU are near (denoted A�B) i� d(A; B)= 0.

5.4 Proximity spaces 93

� de�ned in this way (for a metric space) is an example of proximity as de�ned below.

De�nition 5.29. A proximity space is a set (U ; �) conforming to the following axioms (for everyA; B;C 2PU):

1. A\B=/ ;)A�B;

2. if A�B then A=/ ; and B=/ ;;3. A�B)B �A (symmetry);

4. (A[B) �C,A�C _B �C;

5. C � (A[B),C �A_C �B;

6. A��B implies existence of P ; Q2PU with A��P , B ��Q and P [Q=U .

Exercise 5.6. Show that proximity generated by a metric space is really a proximity (conforms to the aboveaxioms).

De�nition 5.30. Quasi-proximity is de�ned as the above but without the symmetry axiom.

De�nition 5.31. Closure is generated by a proximity by the following formula:

cl(A)= fa2U j fag �Ag:

Proposition 5.32. Every closure generated by a proximity is a Kuratowski closure.

Proof. First prove it is a preclosure. cl(;) = ; is obvious. cl(A)�A is obvious. cl(A[B) = fa 2U j fag�A[Bg=fa2U j fag�A_fag�Bg=fa2U j fag�Ag[fa2U j fag�Bg=cl(A)[cl(B).

It is remained to prove that cl is idempotent, that is cl(cl(A)) = cl(A). It is enough to showcl(cl(A))� cl(A), that is if x2/ cl(A) then x2/ cl(cl(A)).

If x2/ cl(A) then fxg ��A. So there are P ; Q2PU such that fxg ��P , A��Q, P [Q=U . ThenU n Q � P , so fxg ��U n Q and hence x 2 Q. Hence U n cl(A) � Q, and so cl(A) � U n Q � P .Consequently fxg ��cl(A) and hence x2/ cl(cl(A)). �


Chapter 6

Funcoids

In this chapter (and several following chapters) the word �lter will refer to a �lter on a set (ratherthan a �lter on an arbitrary poset).

6.1 Informal introduction into funcoids

Funcoids are a generalization of proximity spaces and a generalization of pretopological spaces.Also funcoids are a generalization of binary relations.

That funcoids are a common generalization of �spaces� (proximity spaces, (pre)topologicalspaces) and binary relations (including monovalued functions) makes them smart for describingproperties of functions in regard of spaces. For example the statement �f is a continuous functionfrom a space � to a space �� can be described in terms of funcoids as the formula f � �v� � f (seebelow for details).

Most naturally funcoids appear as a generalization of proximity spaces.Let � be a proximity that is certain binary relation so that A � B is de�ned for every sets A

and B. We will extend it from sets to �lters by the formula:

A � 0B,8A2A; B 2B:A�B:

Then (as it will be proved below) there exist two functions �; � 2FF such that

A � 0B,Bu�A=/ 0F,Au �B=/ 0F:

The pair (�; �) is called funcoid when Bu�A=/ 0F,Au �B=/ 0F. So funcoids are a generalizationof proximity spaces.

Funcoids consist of two components the �rst � and the second �. The �rst component of afuncoid f is denoted as hf i and the second component is denoted as hf¡1i. (The similarity of thisnotation with the notation for the image of a set under a function is not a coincidence, we will seethat in the case of principal funcoids (see below) these coincide.)

One of the most important properties of a funcoid is that it is uniquely determined by just oneof its components. That is a funcoid f is uniquely determined by the function hf i. Moreover afuncoid f is uniquely determined by values of hf i on principal �lters.

Next we will consider some examples of funcoids determined by speci�ed values of the �rstcomponent on sets.

Funcoids as a generalization of pretopological spaces: Let � be a pretopological space that

is a map � 2 Ff for some set f. Then we de�ne �0X=defF f�x j x 2Xg for every set X 2Pf.

We will prove that there exists a unique funcoid f such that �0 = hf ijP�" where P is the set ofprincipal �lters on f. So funcoids are a generalization of pretopological spaces. Funcoids are also ageneralization of preclosure operators: For every preclosure operator p on a set f it exists a uniquefuncoid f such that hf ijP�"= " � p.

For every binary relation p on a set f there exists unique funcoid f such that

8X 2Pf: hf i"X = "hpiX

(where hpi is de�ned in the introduction), recall that a funcoid is uniquely determined by the valuesof its �rst component on sets. I will call such funcoids principal . So funcoids are a generalizationof binary relations.

95

Composition of binary relations (i.e. of principal funcoids) complies with the formulas:

hg � f i= hgi � hf i and h(g � f)¡1i= hf¡1i � hg¡1i:

By the same formulas we can de�ne composition of every two funcoids. Funcoids with this com-position form a category (the category of funcoids).

Also funcoids can be reversed (like reversal of X and Y in a binary relation) by the formula(�; �)¡1= (�; �). In the particular case if � is a proximity we have �¡1= � because proximitiesare symmetric.

Funcoids behave similarly to (multivalued) functions but acting on �lters instead of acting onsets. Below these will be de�ned domain and image of a funcoid (the domain and the image of afuncoid are �lters).

6.2 Basic de�nitionsDe�nition 6.1. Let us call a funcoid from a set A to a set B a quadruple (A; B; �; �) where�2F(B)F(A), � 2F(A)F(B) such that

8X 2F(A);Y 2F(B): (Y �/ �X ,X �/ �Y):

Further we will assume that all funcoids in consideration are small without mentioning itexplicitly.

De�nition 6.2. Source and destination of every funcoid (A;B;�; �) are de�ned as:

Src(A;B;�; �) =A and Dst(A;B;�; �)=B:

I will denote FCD(A;B) the set of funcoids from A to B.I will denote FCD the set of all funcoids (for small sets).

De�nition 6.3. h(A;B;�; �)i=def� for a funcoid (A;B;�; �).

De�nition 6.4. The reverse funcoid (A;B;�; �)¡1=(B;A; �;�) for a funcoid (A;B;�; �).

Note 6.5. The reverse funcoid is not an inverse in the sense of group theory or category theory.

Proposition 6.6. If f is a funcoid then f¡1 is also a funcoid.

Proof. It follows from symmetry in the de�nition of funcoid. �

Obvious 6.7. (f¡1)¡1= f for a funcoid f .

De�nition 6.8. The relation [f ]2P(F(Src f) � F(Dst f)) is de�ned (for every funcoid f andX 2F(Src f), Y 2F(Dst f)) by the formula X [f ]Y,Y �/ hf iX .

Obvious 6.9. X [f ] Y , Y �/ hf iX , X �/ hf¡1iY for every funcoid f and X 2 F(Src f),Y 2F(Dst f).

Obvious 6.10. [f¡1]=[f ]¡1 for a funcoid f .

Theorem 6.11. Let A, B be sets.

1. For given value of hf i 2F(B)F(A) there exists no more than one funcoid f 2 FCD(A;B).

2. For given value of [f ]2P(F(A)�F(B)) there exists no more than one funcoid f 2 FCD(A;B).

Proof. Let f ; g 2FCD(A;B).Obviously, hf i = hgi ) [f ]=[g] and hf¡1i = hg¡1i ) [f ]=[g]. So it's enough to prove that

[f ]=[g])hf i= hgi.

96 Funcoids

Provided that [f ]=[g] we have Y �/ hf iX , X [f ] Y ,X [g] Y , Y �/ hgiX and consequentlyhf iX = hgiX for every X 2F(A), Y 2F(B) because a set of �lters is separable, thus hf i= hgi. �

Proposition 6.12. hf i0F(Src f)=0F(Dst f) for every funcoid f .

Proof. Y �/ hf i0F(Src f), 0F(Src f)�/ hf¡1iY , 0,Y �/ 0F(Dst f). Thus hf i0F(Src f)= 0F(Dst f) byseparability of �lters. �

Proposition 6.13. hf i(I t J )= hf iI t hf iJ for every funcoid f and I ;J 2F(Src f).

Proof.

?hf i(I tJ ) =

fY 2F j Y �/ hf i(I t J )g =

fY 2F j I t J �/ hf¡1iYg =

fY 2F j I �/ hf¡1iY _ J �/ hf¡1iYg =

fY 2F j Y �/ hf iI _ Y �/ hf iJ g =

fY 2F j Y �/ hf iI t hf iJ g =

?(hf iI t hf iJ ):

Thus hf i(I tJ )= hf iI t hf iJ because F(Dst f) is separable. �

Proposition 6.14. For every f 2FCD(A;B) for every sets A and B we have:

1. K [f ] I t J ,K [f ] I _K [f ]J for every I ;J 2F(B), K2F(A).

2. I tJ [f ]K,I [f ]K_J [f ]K for every I ;J 2F(A), K2F(B).

Proof.

1. K [f ] I t J , (I t J ) u hf iK=/ 0F(B),I u hf iK=/ 0F(B) _ J u hf iK=/ 0F(B),K [f ] I _K [f ]J .

2. Similar. �

6.2.1 Composition of funcoids

De�nition 6.15. Funcoids f and g are composable when Dst f = Src g.

De�nition 6.16. Composition of composable funcoids is de�ned by the formula

(B;C;�2; �2) � (A;B;�1; �1)= (A;C;�2 ��1; �1 � �2):

Proposition 6.17. If f , g are composable funcoids then g � f is a funcoid.

Proof. Let f =(A;B;�1; �1), g=(B;C;�2; �2). For every X 2F(A), Y 2F(C) we have

Y �/ (�2 ��1)X ,Y �/ �2�1X ,�1X �/ �2Y,X �/ �1 �2Y,X �/ (�1 � �2)Y :

So (A;C;�2 ��1; �1 � �2) is a funcoid. �

Obvious 6.18. hg � f i= hgi � hf i for every composable funcoids f and g.

Proposition 6.19. (h � g) � f =h � (g � f) for every composable funcoids f , g, h.

Proof. h(h� g)� f i=hh� gi�hf i=(hhi�hgi)�hf i= hhi�(hgi�hf i)=hhi�hg� f i= hh� (g� f)i. �

Theorem 6.20. (g � f)¡1= f¡1 � g¡1 for every composable funcoids f and g.

Proof. h(g � f)¡1i= hf¡1i � hg¡1i= hf¡1 � g¡1i. �

6.2 Basic definitions 97

6.3 Funcoid as continuationLet f be a funcoid.

De�nition 6.21. hf i� is the function P(Src f)!F(Dst f) de�ned by the formula

hf i�X = hf i"Src fX:

De�nition 6.22. [f ]� is the relation between P(Src f) and P(Dst f) de�ned by the formula

X [f ]�Y ,"Src fX [f ] "DstY :

Obvious 6.23.

1. hf i�= hf i � "Src f;

2. [f ]�=("Dst )¡1 � [f ]�"Src f.

Obvious 6.24. hgihf i�X = hg � f i�X for every X 2P(Src f).

Theorem 6.25. For every funcoid f and X 2F(Src f), Y 2F(Dst f)

1. hf iX =dhhf i�iX ;

2. X [f ]Y,8X 2X ; Y 2Y :X [f ]�Y .

Proof.2. X [f ]Y,Y uhf iX =/ 0F(Dst f),8Y 2Y: "Dst fY u hf iX =/ 0F(Dst f),8Y 2Y :X [f ] "Dst fY .Analogously X [f ]Y,8X 2X : "Src fX [f ]Y . Combining these two equivalences we get

X [f ]Y,8X 2X ; Y 2Y: "Src fX [f ] "Dst fY ,8X 2X ; Y 2Y :X [f ]�Y :

1. Y u hf iX =/ 0F(Dst f),X [f ]Y,8X 2X : "Src fX [f ]Y,8X 2X :Y u hf i�X =/ 0F(Dst f).Let's denote W = fY u hf i�X j X 2X g. We will prove that W is a generalized �lter base. To

prove this it is enough to show that V = fhf i�X j X 2X g is a generalized �lter base.Let P ; Q 2 V . Then P = hf i�A, Q= hf i�B where A; B 2 X ; A \ B 2 X and R v P u Q for

R= hf i�(A\B)2V . So V is a generalized �lter base and thus W is a generalized �lter base.0F(Dst f)2/W,

dW =/ 0F(Dst f) by properties of generalized �lter bases. That is

8X 2X : Y u hf i�X =/ 0F(Dst f),Y ulhhf i�iX =/ 0F(Dst f):

Comparing with the above, Y uhf iX =/ 0F(Dst f),Yudhhf i�iX =/ 0F(Dst f). So hf iX =

dhhf i�iX

because the lattice of �lters is separable. �

Corollary 6.26. Let f be a funcoid.

1. The value of f can be restored knowing hf i�.2. The value of f can be restored knowing [f ]�.

Proposition 6.27. For every f 2FCD(A;B) we have (for every I ; J 2PA)

hf i�;=0F(B); hf i�(I [J) = hf i�I t hf i�Jand

:(I [f ]� ;); I [ J [f ]�K, I [f ]�K _J [f ]�K (for every I ; J 2PA, K 2PB);:(; [f ]� I); K [f ]� I [J,K [f ]� I _K [f ]� J (for every I ; J 2PB, K 2PA):

Proof. hf i�; = hf i"A; = hf i0F(A) = 0F(B); hf i�(I [ J) = hf i"A(I [ J) = hf i"AI t hf i"AJ =hf i�I t hf i�J .

I [f ]� ; , 0F(B) �/ hf i"AI , 0; I [ J [f ]� K , "A(I [ J) [f ] "BK , "BK �/ hf i"A(I [ J),"BK �/ hf i�(I [ J),"BK�/ hf i�I t hf i�J,"BK�/ hf i�I _"BK �/ hf i�J, I [f ]�K _J [f ]�K.

The rest follows from symmetry. �

98 Funcoids

Theorem 6.28. Fix sets A and B. Let LF =�f 2FCD(A;B): hf i� and LR=�f 2FCD(A;B): [f ]�.1. LF is a bijection from the set FCD(A;B) to the set of functions �2F(B)PA that obey the

conditions (for every I ; J 2PA)

�;=0F(B); �(I [J) =�I t�J: (6.1)

For such � it holds (for every X 2F(A))

hLF¡1�iX =lh�iX : (6.2)

2. LR is a bijection from the set FCD(A;B) to the set of binary relations � 2P(PA�PB)that obey the conditions

:(I � ;); I [ J �K, I �K _J �K (for every I ; J 2PA, K 2PB);:(; � I); K � I [J,K �I _K �J (for every I ; J 2PB, K 2PA).

(6.3)

For such � it holds (for every X 2F(A), Y 2F(B))

X [LR¡1 �]Y,8X 2X ; Y 2Y :X �Y : (6.4)

Proof. Injectivity of LF and LR, formulas (6.2) (for �2 imLF) and (6.4) (for �2 imLR), formulas(6.1) and (6.3) follow from two previous theorems. The only thing remained to prove is that forevery � and � that obey the above conditions a corresponding funcoid f exists.

2. Let de�ne �2F(B)PA by the formula @(�X)= fY 2PB j X �Y g for every X 2PA. (It isobvious that fY 2PB j X �Y g is a free star.) Analogously it can be de�ned � 2F(A)PB by theformula @(�Y ) = fX 2PA j X � Y g. Let's continue � and � to �02 F(B)F(A) and � 02 F(A)F(B)

by the formulas

�0X =lh�iX and � 0 Y =

lh�iY

and � to � 02P(F(A)�F(B)) by the formula

X � 0Y,8X 2X ; Y 2Y:X �Y :

Y u�0X =/ 0F(B),Y udh�iX =/ 0F(B),

dhY u ih�iX =/ 0F(B). Let's prove that

W = hY u ih�iX

is a generalized �lter base: To prove it is enough to show that h�iX is a generalized �lter base. IfA;B 2 h�iX then exist X1; X22X such that A=�X1, B=�X2.

Then �(X1\X2)2h�iX . So h�iX is a generalized �lter base and thusW is a generalized �lterbase.

By properties of generalized �lter bases,dhY u ih�iX =/ 0F(B) is equivalent to

8X 2X :Y u�X =/ 0F(B);

what is equivalent to 8X 2X ; Y 2 Y: "BY u�X =/ 0F(B),8X 2X ; Y 2 Y: Y 2 @(�X),8X 2X ;Y 2 Y : X � Y . Combining the equivalencies we get Y u �0 X =/ 0F(B) , X � 0 Y . AnalogouslyX u � 0Y=/ 0F(A),X � 0Y . So Y u�0X =/ 0F(B),X u � 0Y=/ 0F(A), that is (A;B;�0; � 0) is a funcoid.From the formula Y u�0X =/ 0F(B),X � 0Y it follows that

X [(A;B;�0; � 0)]�Y ,"BY u�0 "AX =/ 0F(B),"AX � 0 "BY ,X �Y :

1. Let de�ne the relation � 2P(PA�PB) by the formula X �Y ,"BY u�X =/ 0F(B).That :(I � ;) and :(; � I) is obvious. We have I [ J � K , "BK u �(I [ J) =/ 0F(B) ,

"BK u (�I t�J)=/ 0F(B),"BK u�I =/ 0F(B)_"BK u�J =/ 0F(B), I �K _J �K andK � I [ J , "B(I [ J) u �K =/ 0F(B), ("BI t "BJ) u �K =/ 0F(B), "BI u �K =/ 0F(B) _

"BJ u�K =/ 0F(B),K � I _K �J .That is the formulas (6.3) are true.Accordingly to the above there exists a funcoid f such that

X [f ]Y,8X 2X ; Y 2Y:X �Y :

6.3 Funcoid as continuation 99

8X 2 PA; Y 2 PB:¡"BY u hf i"AX =/ 0F(B), "AX [f ] "BY , X � Y , "BY u �X =/ 0F(B)

�,

consequently 8X 2PA:�X = hf i"AX = hf i�X . �

Note that by the last theorem to every proximity � corresponds a unique funcoid. So funcoidsare a generalization of (quasi-)proximity structures. Reverse funcoids can be considered as a gen-eralization of conjugate quasi-proximity.

De�nition 6.29. Any (multivalued) function F : A! B corresponds to a funcoid "FCD(A;B)F 2FCD(A;B), where by de�nition

"FCD(A;B)F

�X =

dh"BihhF iiX for every X 2F(A).

Using the last theorem it is easy to show that this de�nition is monovalued and does notcontradict to former stu�. (Take �= "B � hF i.)

De�nition 6.30. "FCDf =def¡Src f ;Dst f ; "FCD(Src f ;Dst f)GR f

�for every Rel-morphism f .

De�nition 6.31. Funcoids corresponding to a binary relation (= multivalued function) are calledprincipal funcoids .

We may equate principal funcoids with corresponding binary relations by the method ofappendix B in [29]. This is useful for describing relationships of funcoids and binary relations,such as for the formulas of continuous functions and continuous funcoids (see below).

Theorem 6.32. If S is a generalized �lter base on Src f then hf idS=

dhhf iiS for every funcoid

f .

Proof. hf idS vhf iX for every X 2S and thus hf i

dS v

dhhf iiS.

By properties of generalized �lter bases:hf idS=

dhhf i�i

dS=d

hhf i�ifX j 9P 2S:X 2Pg=dfhf i�X j 9P 2S:X 2Pgw

dfhf iP j P 2Sg=

dhhf iiS: �

6.4 Lattices of funcoids

De�nition 6.33. f v g=def

[f ]�[g] for f ; g 2 FCD.

Thus every FCD(A;B) is a poset. (It's taken into account that [f ]=/ [g] when f =/ g.)We will consider �ltrators (�ltrators of funcoids ) whose base is FCD(A;B) and whose core are

principal funcoids from A to B.

Lemma 6.34. hf i�X =dfhF i�X j F 2up f g for every funcoid f and set X 2P(Src f).

Proof. Obviously hf i�X vdfhF i�X j F 2up f g.

Let B 2 hf i�X . Let FB=X �B [ ((Src f) nX)�Dst f .hFBiX =B.Let P 2 P(Src f). We have ; =/ P � X ) "Dst f hFBiP = "Dst fB w hf i�P and ; =/

P * X ) "Dst f hFBiP = "Dst fDst f w hf i�P . Thus "Dst f hFBiP w hf i�P for every P andso "FCD(Src f ;Dst f)FB w f that is FB 2up f .

Thus 8B 2 hf i�X :B 2dfhF i�X j F 2 up f g because B 2

"FCD(Src f ;Dst f)FB

��X .

SodfhF i�X j F 2up f gv hf i�X . �

Theorem 6.35. hf iX =dfhF iX j F 2up f g for every funcoid f and X 2F(Src f).

Proof.dfhF iX j F 2up f g=

dfdhhF i�iX j F 2up f g=

dfdfhF i�X j X 2X g j F 2up f g=d

fdfhF i�X j F 2up f g j X 2X g=

dfhf i�X j X 2X g= hf iX (the lemma used). �

Conjecture 6.36. Every �ltrator of funcoids is: [TODO: Solved. See rewrite-plan.pdf]

1. with separable core;

100 Funcoids

2. with co-separable core.

Below it is shown that FCD(A;B) are complete lattices for every sets A and B. We will applylattice operations to subsets of such sets without explicitly mentioning FCD(A;B).

Theorem 6.37. FCD(A;B) is a complete lattice (for every sets A and B). For every R2PFCD(A;B) and X 2PA, Y 2PB

1. X [FR]�Y ,9f 2R:X [f ]�Y ;

2. hFRi�X =

Ffhf i�X j f 2Rg.

Proof. Accordingly [26] to prove that it is a complete lattice it's enough to prove existence of alljoins.

2. �X=defF

fhf i�X j f 2Rg. We have �;=0F(Dst f);

�(I [ J) =Gfhf i�(I [J) j f 2Rg

=Gfhf i�I t hf i�J j f 2Rg

=Gfhf i�I j f 2Rgt

Gfhf i�J j f 2Rg

= �I t�J:

So hhi�=� for some funcoid h. Obviously

8f 2R:hw f: (6.5)

And h is the least funcoid for which holds the condition (6.5). So h=FR.

1. X [FR]� Y , "Dst fY u h

FRi�X =/ 0F(Dst f), "Dst fY u

Ffhf i�X j f 2Rg=/ 0F(Dst f),

9f 2R: "Dst fY u hf i�X =/ 0F(Dst f),9f 2R:X [f ]�Y (used proposition 4.194). �

In the next theorem, compared to the previous one, the class of in�nite joins is replaced withlesser class of �nite joins and simultaneously class of sets is changed to more wide class of �lters.

Theorem 6.38. For every f ; g 2FCD(A;B) and X 2F(A) (for every sets A, B)

1. hf t giX = hf iX t hgiX ;

2. [f t g]=[f ][[g].

Proof. 1. Let �X =defhf iX t hgiX ; �Y=defhf¡1iY t hg¡1iY for every X 2F(A), Y 2F(B). Then

Y u�X =/ 0F(B) , Y uhf iX =/ 0F(B)_Y u hgiX =/ 0F(B)

, X uhf¡1iY =/ 0F(A)_X u hg¡1iY =/ 0F(A)

, X u �Y =/ 0F(A):

So h= (A;B;�; � ) is a funcoid. Obviously hw f and hw g. If pw f and pw g for some funcoidp then hpiX w hf iX t hgiX = hhiX that is pwh. So f t g=h.

2. X [f t g] Y , Y u hf t giX =/ 0F(B), Y u (hf iX t hgiX ) =/ 0F(B), Y u hf iX =/ 0F(B) _Y u hgiX =/ 0F(B),X [f ]Y _X [g]Y for every X 2F(A), Y 2F(B). �

De�nition 6.39. GR f =def�

F 2P(Src f �Dst f) j "FCD(Src f ;Dst f)F w f.

De�nition 6.40. xyGR f =deff(Src f ;Dst f ;F ) j F 2GR f g.

Remark 6.41. xyGR f is a set of Rel-morphisms.

6.5 More on composition of funcoids

Proposition 6.42. [g � f ]=[g]�hf i= hg¡1i¡1 � [f ] for every composable funcoids f and g.

6.5 More on composition of funcoids 101

Proof. X [g � f ]Y,Yuhg� f iX =/ 0F(Dst g),Yuhgihf iX =/ 0F(Dst g),hf iX [g]Y,X ([g]�hf i)Yfor every X 2 F(Src f), Y 2 F(Dst g). [g � f ]=[(f¡1 � g¡1)¡1],[f¡1 � g¡1]¡1=([f¡1]�hg¡1i)¡1 =hg¡1i¡1 � [f ]. �

Corollary 6.43. [f ]=[idDst f]�hf i for every funcoid f .

The following theorem is a variant for funcoids of the statement (which de�nes compositionsof relations) that x (g � f)z,9y: (xf y^ y gz) for every x and z and every binary relations f and g.

Theorem 6.44. For every sets A, B, C and f 2FCD(A;B), g2FCD(B;C) and X 2F(A), Z 2F(C)

X [g � f ]Z,9y 2 atomsF(B): (X [f ] y ^ y [g]Z):

Proof.

9y 2 atomsF(B): (X [f ] y^ y [g]Z) ,9y 2 atomsF(B):

¡Z u hgiy=/ 0F(C)^ y u hf iX =/ 0F(B)

�,

9y 2 atomsF(B):¡Z uhgiy=/ 0F(C)^ y vhf iX

�)

Z uhgihf iX =/ 0F(C) ,X [g � f ]Z:

Reversely, if X [g � f ]Z then hf iX [g]Z, consequently there exists y2atoms hf iX such that y [g]Z;we have X [f ] y. �

Theorem 6.45. For every sets A, B, C

1. f � (gt h)= f � g t f �h for g; h2FCD(A;B) and f 2FCD(B;C);

2. (gt h) � f = g � f th � f for g; h2FCD(B;C) and f 2FCD(A;B).

Proof. I will prove only the �rst equality because the other is analogous.For every X 2F(A), Z 2F(C)

X [f � (g th)]Z , 9y 2 atomsF(B): (X [g th] y ^ y [f ]Z), 9y 2 atomsF(B): ((X [g] y _X [h] y)^ y [f ]Z), 9y 2 atomsF(B): ((X [g] y ^ y [f ]Z)_ (X [h] y ^ y [f ]Z)), 9y 2 atomsF(B): (X [g] y ^ y [f ]Z)_9y 2 atomsF(B): (X [h] y ^ y [f ]Z), X [f � g]Z _X [f �h]Z, X [f � g t f �h]Z:

�

Remark 6.46. The above theorem can be proved without atomic �lters by the formula hf � (gth)iX = hf ihg t hiX = hf i(hgiX t hhiX ) = hf ihgiX t hf ihhiX = hf � giX t hf � hiX =hf � gt f �hiX . [TODO: This may be useful for a pointfree generalization. Preserve old proof forhistory.]

6.6 Domain and range of a funcoid

De�nition 6.47. Let A be a set. The identity funcoid idFCD(A)=(A;A; idF(A); idF(A)).

Obvious 6.48. The identity funcoid is a funcoid.

De�nition 6.49. Let A be a set, A2F(A). The restricted identity funcoid

idAFCD=(A;A;Au;Au ):

102 Funcoids

Proposition 6.50. The restricted identity funcoid is a funcoid.

Proof. We need to prove that (AuX )uY =/ 0F(A), (AuY)uX =/ 0F(A) what is obvious. �

Obvious 6.51.

1.¡idFCD(A)

�¡1= idFCD(A);

2. (idAFCD)¡1= idAFCD.

Obvious 6.52. For every X ;Y 2F(A)

1. X�idFCD(A)

�Y,X uY =/ 0F(A);

2. X [idAFCD]Y,AuX uY =/ 0F(A).

De�nition 6.53. I will de�ne restricting of a funcoid f to a �lter A2F(Src f) by the formula

f jA=f � idAFCD:

De�nition 6.54. Image of a funcoid f will be de�ned by the formula im f = hf i1F(Src f).Domain of a funcoid f is de�ned by the formula dom f = im f¡1.

Obvious 6.55. For every binary relation f between sets A and B

1. im "FCD(A;B) f = "Bim f ;

2. dom "FCD(A;B) f = "Adom f .

Proposition 6.56. hf iX = hf i(X u dom f) for every funcoid f , X 2F(Src f).

Proof. For every Y 2 F(Dst f) we have Y u hf i(X u dom f) =/ 0F(Dst f),X u dom f u hf¡1iY =/0F(Src f),X u im f¡1 u hf¡1iY =/ 0F(Src f),X u hf¡1iY =/ 0F(Src f),Y u hf iX =/ 0F(dst f). Thushf i(X udom f) = hf iX because the lattice of �lters is separable. �

Proposition 6.57. hf iX = im(f jX) for every funcoid f , X 2F(Src f).

Proof. im(f jX) = hf � idXFCDi1F(Src f)= hf ihidXFCDi1F(Src f)= hf i¡X u 1F(Src f)

�= hf iX . �

Proposition 6.58. X udom f =/ 0F(Src f),hf iX =/ 0F(Dst f) for every funcoid f and X 2F(Src f).

Proof. X u dom f =/ 0F(Src f), X u hf¡1i1F(Dst f) =/ 0F(Src f), 1F(Dst f) u hf iX =/ 0F(Dst f),hf iX =/ 0F(Dst f). �

Corollary 6.59. dom f =F �

a2 atomsF(Src f) j hf ia=/ 0F(Dst f).Proof. This follows from the fact that F(Src f) is an atomistic lattice. �

Proposition 6.60. dom(f jA)=Audom f for every funcoid f and A2F(Src f).

Proof. dom(f jA) = im(idAFCD� f¡1) = hidAFCDihf¡1i1F(Dst f)=Auhf¡1i1F(Dst f)=Au dom f . �

Theorem 6.61. im f =dhimiup f and dom f =

dhdomiup f for every funcoid f .

Proof. im f = hf i1F(Src f)=d �

hF i1F(Src f) j F 2 up f=dfimF j F 2 up f g=

dhimiup f .

The second formula follows from symmetry. �

Proposition 6.62. For every composable funcoids f , g:

1. If im f w dom g then im(g � f)= im g.

2. If im f v dom g then dom(g � f) =dom f .

6.6 Domain and range of a funcoid 103

Proof.

1. im(g � f) = hg � f i1F(Src f) = hgihf i1F(Src f) = hgiim f = hgi(im f u dom g) = hgidom g =

hgi1F(Src g)= im g.

2. dom(g � f)= im(f¡1 � g¡1) what by proved above is equal to im f¡1 that is dom f . �

Lemma 6.63. �B 2 F(B): 1F�FCDB is an upper adjoint of �f 2 FCD(A;B): im f (for every setsA, B).

Proof. We need to prove im f vB, f v 1F�FCDB what is obvious. �

Corollary 6.64. Image and domain of funcoids preserve joins.

Proof. By properties of Galois connections and duality. �

6.7 Categories of funcoidsI will de�ne two categories, the category of funcoids and the category of funcoid triples .

The category of funcoids is de�ned as follows:


� The set of morphisms from a set A to a set B is FCD(A;B).

� The composition is the composition of funcoids.

� Identity morphism for a set is the identity funcoid for that set.

To show it is really a category is trivial.The category of funcoid triples is de�ned as follows:

� Objects are �lters on small sets.

� The morphisms from a �lter A to a �lter B are triples (A; B; f) where f 2 FCD(Base(A);Base(B)) and dom f vA^ im f vB.

� The composition is de�ned by the formula (B; C; g) � (A;B; f)= (A; C; g � f).� Identity morphism for a �lter A is idAFCD.

To prove that it is really a category is trivial.

6.8 Specifying funcoids by functions or relations on atomic�ltersTheorem 6.65. For every funcoid f and X 2F(Src f), Y 2F(Dst f)

1. hf iX =Fhhf iiatomsX ;

2. X [f ]Y,9x2 atomsX ; y 2 atomsY :x [f ] y.

Proof. 1.

Y u hf iX =/ 0F(Dst f) , X u hf¡1iY =/ 0F(Src f)

, 9x2 atomsX :xu hf¡1iY =/ 0F(Src f)

, 9x2 atomsX :Y u hf ix=/ 0F(Dst f):

@ hf iX =Fh@ ihhf iiatomsX =@

Fhhf iiatomsX . So hf iX =

Fhhf iiatomsX by proposition 4.202.

2. If X [f ] Y , then Y u hf iX =/ 0F(Dst f), consequently there exists y 2 atoms Y such thatyu hf iX =/ 0F(Dst f), X [f ] y. Repeating this second time we get that there exists x2atomsX suchthat x [f ] y. From this it follows

9x2 atomsX ; y 2 atomsY :x [f ] y:

104 Funcoids

The reverse is obvious. �

Corollary 6.66. Let f be a funcoid.

� The value of f can be restored knowing hf ijatomsF(Src f).

� The value of f can be restored knowing [f ]jatomsF(Src f)�atomsF(Dst f).

Theorem 6.67. Let A and B be sets.

1. A function �2F(B)atomsF(A) such that (for every a2 atomsF(A))

�avl G

�h�i � atoms � "A�a (6.6)

can be continued to the function hf i for a unique f 2 FCD(A;B);

hf iX =Gh�iatomsX (6.7)

for every X 2F(A).

2. A relation �2P¡atomsF(A)�atomsF(B)

�such that (for every a2atomsF(A), b2atomsF(B))

8X 2 a; Y 2 b9x2 atoms "AX; y 2 atoms "BY :x � y) a � b (6.8)

can be continued to the relation [f ] for a unique f 2FCD(A;B);

X [f ]Y,9x2 atomsX ; y 2 atomsY :x � y (6.9)

for every X 2F(A), Y 2F(B).

Proof. Existence of no more than one such funcoids and formulas (6.7) and (6.9) follow from theprevious theorem.

1. Consider the function �02F(B)PA de�ned by the formula (for every X 2PA)

�0X =Gh�iatoms "AX:

Obviously �0 ;=0F(B). For every I ; J 2PA

�0(I [ J) =Gh�iatoms "A(I [ J)

=Gh�i(atoms "AI [ atoms "AJ)

=G

(h�iatoms "AI [ h�iatoms "AJ)=Gh�iatoms "AI t

Gh�iatoms "AJ

= �0 I t�0J:

Let continue �0 till a funcoid f (by the theorem 6.28): hf iX =dh�0iX .

Let's prove the reverse of (6.6):l G

�h�i � atoms � "A�a =

l G�h�i

�hatomsih"Aia

vl G

�h�i�ffagg

=l �¡G

�h�i�fag

=l �G

h�ifag

=l �G

f�ag=lf�ag=�a:

Finally,

�a=l G

�h�i � atoms � "A�a=

lh�0ia= hf ia;

so hf i is a continuation of �.2. Consider the relation � 02P(PA�PB) de�ned by the formula (for everyX 2PA, Y 2PB)

X � 0Y ,9x2 atoms "AX; y 2 atoms "BY :x � y:Obviously :(X � 0 ;) and :(; � 0Y ).

6.8 Specifying funcoids by functions or relations on atomic filters 105

For suitable I and J we have:

I [J � 0Y , 9x2 atoms "A(I [J); y 2 atoms "BY : x � y, 9x2 atoms "AI [ atoms "BJ ; y2 atoms "BY :x � y, 9x2 atoms "AI ; y 2 atoms "BY :x � y_9x2 atoms "AJ ; y2 atoms "BY :x � y, I � 0Y _J � 0Y ;

similarly X � 0 I [J,X � 0 I _X � 0J for suitable I and J . Let's continue � 0 till a funcoid f (by thetheorem 6.28):

X [f ]Y,8X 2X ; Y 2Y:X � 0Y :

The reverse of (6.8) implication is trivial, so

8X 2 a; Y 2 b9x2 atoms "AX; y 2 atoms "BY :x � y, a � b:

8X 2 a; Y 2 b9x2 atoms "AX; y 2 atoms "BY :x � y,8X 2 a; Y 2 b:X � 0Y , a [f ] b.So a � b, a [f ] b, that is [f ] is a continuation of �. �

One of uses of the previous theorem is the proof of the following theorem:

Theorem 6.68. If A and B are sets, R2PFCD(A;B), x2 atomsF(A), y 2 atomsF(B), then

1. hdRix=

dfhf ix j f 2Rg;

2. x [dR] y,8f 2R:x [f ] y.

Proof. 2. Let denote x � y,8f 2R:x [f ] y. For every a2 atomsF(A), b2 atomsF(B)

8X 2 a; Y 2 b9x2 atoms "AX; y 2 atoms "BY :x � y)8f 2R;X 2 a; Y 2 b9x2 atoms "AX; y 2 atoms "BY :x [f ] y)

8f 2R;X 2 a; Y 2 b:X [f ]�Y )8f 2R: a [f ] b,

a � b:

So by theorem 6.67, � can be continued till [p] for some funcoid p2FCD(A;B).For every funcoid q 2 FCD(A; B) such that 8f 2 R: q v f we have x [q] y ) 8f 2 R:

x [f ] y,x � y, x [p] y, so q v p. Consequently p=dR.

From this x [dR] y,8f 2R:x [f ] y.

1. From the former y 2 atoms hdRix, y u h

dRix =/ 0F(B), 8f 2 R: y u hf ix =/ 0F(B),

y 2dhatomsifhf ix j f 2Rg, y 2 atoms

dfhf ix j f 2Rg for every y 2 atomsF(A). From this it

follows hdRix=

dfhf ix j f 2Rg. �

Theorem 6.69. g � f =dfG�F j F 2 up f ;G2 up gg for every composable funcoids f and g.

Proof. Let x2 atomsF(Src f). Then

hg � f ix =

hgihf ix = (theorem 6.35)lfhGihf ix j G2 up gg = (theorem 6.35)l �

hGilfhF ix j F 2up f g j G2 up g

= (theorem 6.32)l �l

fhGihF ix j F 2up f g j G2 up g

=lfhGihF ix j F 2 up f ;G2 up gg =lfhG �F ix j F 2 up f ;G2 up gg = (theorem 6.68)lfG�F j F 2up f ;G2up gg

�x:

Thus g � f =dfG�F j F 2up f ;G2up gg. �

Theorem 6.70. Let A, B, C be sets, f 2FCD(A;B), g 2FCD(B;C), h2 FCD(A;C). Then

g � f �/ h, g�/ h � f¡1:

106 Funcoids

Proof.

g � f �/ h ,9a2 atomsF(A); c2 atomsF(C): a [(g � f)uh] c ,

9a2 atomsF(A); c2 atomsF(C): (a [g � f ] c^ a [h] c) ,9a2 atomsF(A); b2 atomsF(B); c2 atomsF(C): (a [f ] b^ b [g] c^ a [h] c) ,

9b2 atomsF(B); c2 atomsF(C): (b [g] c^ b [h � f¡1] c) ,9b2 atomsF(B); c2 atomsF(C): b [g u (h � f¡1)] c ,

g�/ h � f¡1:�

6.9 Direct product of �lters

A generalization of Cartesian product of two sets is funcoidal product of two �lters:

De�nition 6.71. Funcoidal product of �lters A and B is such a funcoid A�FCDB2FCD(Base(A);Base(B)) that for every X 2F(Base(A)), Y 2F(Base(B))

X [A�FCDB]Y,X �/ A^Y �/ B:

Proposition 6.72. A�FCDB is really a funcoid and

hA�FCDBiX =

(B if X �/ A0F(Base(B)) if X �A:

Proof. Obvious. �

Obvious 6.73. "FCD(U ;V )(A�B)= "UA�FCD"VB for sets A�U and B �V .

Proposition 6.74. f vA�FCDB,dom f vA^ im f vB for every f 2FCD(A;B) and A2F(A),B 2F(B).

Proof. If f v A �FCD B then dom f v dom(A �FCD B) v A, im f v im(A �FCD B) v B. Ifdom f vA^ im f vB then

8X 2F(A);Y 2F(B):¡X [f ]Y)X uA=/ 0F(A)^Y uB=/ 0F(B)

�;

consequently f vA�FCDB. �

The following theorem gives a formula for calculating an important particular case of a meeton the lattice of funcoids:

Theorem 6.75. f u (A �FCD B) = idBFCD � f � idAFCD for every funcoid f and A 2 F(Src f),B 2F(Dst f).

Proof. h=def

idBFCD� f � idAFCD. For every X 2F(Src f)

hhiX = hidBFCDihf ihidAFCDiX =B u hf i(AuX ):

From this, as easy to show, h v f and h v A �FCD B. If g v f ^ g v A �FCD B for ag 2FCD(Src f ;Dst f) then dom g vA, im gvB,

hgiX =Bu hgi(AuX )vBu hf i(AuX ) = hidBFCDihf ihidAFCDiX = hhiX ;

g vh. So h= f u (A�FCDB). �

Corollary 6.76. f jA=f u¡A�FCD1F(Dst f)� for every funcoid f and A2F(Src f).

6.9 Direct product of filters 107

Proof. f u¡A�FCD1F(Dst f)�= id1F(Dst f)

FCD � f � idAFCD= f � idAFCD= f jA. �

Corollary 6.77. f �/ A�FCDB,A [f ]B for every funcoid f , A2F(Src f), B 2 (Dst f).

Proof. f �/ A �FCD B , hf u (A �FCD B)i�(Src f) =/ 0F(Dst f) , hidBFCD � f � idAFCDi�(Src f) =/0F(Dst f),hidBFCDihf ihidAFCDi1F(Src f)=/ 0F(Dst f),Bu hf i

¡Au 1F(Src f)

�=/ 0F(Dst f),Bu hf iA=/

0F(Dst f),A [f ]B. �

Corollary 6.78. Every �ltrator of funcoids is star-separable.

Proof. The set of funcoidal products of principal �lters is a separation subset of the lattice offuncoids. �

Theorem 6.79. Let A, B be sets. If S 2P(F(A)�F(B)) thenlfA�FCDB j (A;B)2Sg=

ldomS �FCD

limS:

Proof. If x2 atomsF(A) then by theorem 6.68lfA�FCDB j (A;B)2Sg

�x=

lfhA�FCDBix j (A;B)2Sg:

If x�/d

domS then

8(A;B)2S:¡xuA=/ 0F(A)^ hA�FCDBix=B

�;

fhA�FCDBix j (A;B)2Sg= imS;

if x�d

domS then

9(A;B)2S:¡xuA=0F(A)^ hA�FCDBix=0F(B)

�;

fhA�FCDBix j (A;B)2Sg3 0F(B):

So lfA�FCDB j (A;B)2Sg

�x=

( dimS if x�/

ddomS

0F(B) if x�d

domS:

From this the statement of the theorem follows. �

Corollary 6.80. For every A0;A12F(A), B0;B12F(B) (for every sets A, B)

(A0�FCDB0)u (A1�FCDB1)= (A0uA1)�FCD (B0uB1):

Proof. (A0�FCDB0) u (A1�FCDB1) =dfA0�FCDB0;A1�FCDB1g what is by the last theorem

equal to (A0uA1)�FCD (B0uB1). �

Theorem 6.81. If A, B are sets and A2 F(A) then A�FCD is a complete homomorphism fromthe lattice F(B) to the lattice FCD(A;B), if also A=/ 0F(A) then it is an order embedding.

Proof. Let S 2PF(B), X 2PA, x2 atomsF(A).GhA�FCD iS

��X =

GfhA�FCDBi�X j B 2Sg

=

( FS ifX 2 @A

0F(B) ifX 2/ @A

=A�FCD

GS��X;l

hA�FCD iS�x =

lfhA�FCDBix j B 2Sg

=

( dS if x�/ A

0F(B) if x�A=A�FCD

lS�x:

108 Funcoids

ThusFhA�FCD iS=A�FCDF S and

dhA�FCD iS=A�FCDd S.

If A=/ 0F(A) then obviously the function A�FCD is injective. �

The following proposition states that cutting a rectangle of atomic width from a funcoid alwaysproduces a rectangular (representable as a funcoidal product of �lters) funcoid (of atomic width).

Proposition 6.82. If f is a funcoid and a is an atomic �lter on Src f then

f ja=a�FCD hf ia:

Proof. Let X 2F(Src f).

X �/ a)hf jaiX = hf ia; X � a)hf jaiX =0F(Dst f): �

6.10 Atomic funcoidsTheorem 6.83. An f 2 FCD(A;B) is an atom of the lattice FCD(A;B) (for some sets A, B) i�it is a funcoidal product of two ultra�lters.

Proof.

). Let f 2FCD(A;B) be an atom of the lattice FCD(A;B). Let's get elements a2atomsdom fand b2 atoms hf ia. Then for every X 2F(A)

X � a)ha�FCD biX =0F(B)vhf iX ; X �/ a)ha�FCD biX = bvhf iX :

So a�FCD bv f ; because f is atomic we have f = a�FCD b.

(. Let a2 atomsF(A), b2 atomsF(B), f 2FCD(A;B). If b�hf ia then :(a [f ] b), f � a�FCD b;if bvhf ia then 8X 2F(A): (X �/ a)hf iX w b), f w a�FCD b. Consequently f � a�FCD b_f w a�FCD b; that is a�FCD b is an atom. �

Theorem 6.84. The lattice FCD(A;B) is atomic (for every sets A, B).

Proof. Let f be a non-empty funcoid from A to B. Then dom f =/ 0F(A), thus by the theorem4.207 there exists a2atomsdom f . So hf ia=/ 0F(B) thus it exists b2atoms hf ia. Finally the atomicfuncoid a�FCD bv f . �

Theorem 6.85. The lattice FCD(A;B) is separable (for every sets A, B).

Proof. Let f ; g 2 FCD(A;B), f @ g. Then there exists a 2 atomsF(A) such that hf ia@ hgia. Sobecause the lattice F(B) is atomically separable, there exists b 2 atomsF(B) such that hf ia u b=0F(B) and bvhgia. For every x2 atomsF(A)

hf iau ha�FCD bia= hf iau b=0F(B);

x=/ a)hf ixu ha�FCD bix= hf ixu 0F(B)=0F(B):

Thus hf ixu ha�FCD bix=0F(B) and consequently f � a�FCD b.

ha�FCD bia= bvhgia;x=/ a)ha�FCD bix=0F(B)vhgix:

Thus ha�FCD bixvhgix and consequently a�FCD bv g.So the lattice FCD(A;B) is separable by the theorem 3.14. �

Corollary 6.86. The lattice FCD(A;B) is:

1. separable;

2. atomically separable;

3. conforming to Wallman's disjunction property.

6.10 Atomic funcoids 109


Remark 6.87. For more ways to characterize (atomic) separability of the lattice of funcoids seesubsections �Separation subsets and full stars� and �Atomically separable lattices�.

Corollary 6.88. The lattice FCD(A;B) is an atomistic lattice.

Proof. Let f 2 FCD(A;B). Suppose contrary to the statement to be proved thatF

atoms f @ f .Then there exists a2 atoms f such that au

Fatoms f =0FCD(A;B) what is impossible. �

Proposition 6.89. atoms(f t g) = atoms f [ atoms g for every funcoids f ; g 2 FCD(A; B) (forevery sets A, B).

Proof. a�FCDb�/ f t g,a [f t g] b,a [f ] b_a [g] b,a�FCDb�/ f _a�FCDb�/ g for every atomic�lters a and b. �

Theorem 6.90. For every f ; g; h2FCD(A;B), R2PFCD(A;B) (for every sets A and B)

1. f u (gt h)= (f u g)t (f uh);2. f t

dR=

dhf t iR.

Proof. We will take into account that the lattice of funcoids is an atomistic lattice.1. atoms(f u (g t h)) = atoms f \ atoms(g t h) = atoms f \ (atoms g [ atoms h) =

(atoms f \atoms g)[ (atoms f \atomsh)=atoms(f u g)[atoms(f uh)=atoms((f u g)t (f uh)).2. atoms(f t

dR) = atoms f [ atoms

dR = atoms f [

ThatomsiR =

Th(atoms f) [

ihatoms iR=Thatomsihf t iR= atoms

dhf t iR. (Used the following equality.)

h(atoms f)[ ihatoms iR =

f(atoms f)[A j A2 hatoms iRg =

f(atoms f)[A j 9C 2R:A= atoms Cg =

f(atoms f)[ (atomsC) j C 2Rg =

fatoms(f tC) j C 2Rg =

fatomsB j 9C 2R:B= f tCg =

fatomsB j B 2 hf t iRg =

hatomsihf t iR:�

Note that distributivity of the lattice of funcoids is proved through using atoms of this lattice.I have never seen such method of proving distributivity.

Corollary 6.91. The lattice FCD(A;B) is co-brouwerian (for every sets A, B).

Conjecture 6.92. Distributivity of the lattice FCD(A; B) of funcoids (for arbitrary sets A andB) is not provable in ZF (without axiom of choice).

The next proposition is one more (among the theorem 6.44) generalization for funcoids ofcomposition of relations.

Proposition 6.93. For every composable funcoids f , g

atoms(g � f) =(x�FCD z j x2 atomsF(Src f); z 2 atomsF(Dst g);

9y 2 atomsF(Dst f): (x�FCD y 2 atoms f ^ y�FCD z 2 atoms g)

):

Proof. x�FCDz�/ g � f,x [g � f ]z,9y2atomsF(Dst f): (x�FCD y�/ f ^ y�FCDz�/ g) (it was usedthe theorem 6.44). �

Corollary 6.94. g � f =FfG�F j F 2atoms f ;G2atoms gg for every composable funcoids f , g.

110 Funcoids

Theorem 6.95. Let f be a funcoid.

1. X [f ]Y,9F 2 atoms f :X [F ]Y for every X 2F(Src f), Y 2F(Dst f);

2. hf iX =FF2atoms f hF iX for every X 2F(Src f).

Proof. 1. 9F 2atoms f :X [F ]Y,9a2atomsF(Src f); b2atomsF(Dst f): (a�FCDb�/ f ^X [a�FCD b]

Y),9a 2 atomsF(Src f); b 2 atomsF(Dst f): (a�FCD b�/ f ^ a �FCD b�/ X �FCD Y),9F 2 atoms f :(F �/ f ^F �/ X �FCDY), f �/ X �FCDY,X [f ]Y .

2. Let Y 2F(Dst f). Suppose Y�/ hf iX . Then X [f ]Y ; 9F 2atoms f :X [F ]Y; 9F 2atoms f :Y�/hF iX ; Y�/

FF 2atoms f hF iX . So hf iX v

FF2atoms f hF iX . The contrary hf iX w

FF2atoms f hF iX

is obvious. �

Problem 6.96. Let A and B be in�nite sets. Characterize the set of all coatoms of the latticeFCD(A; B) of funcoids from A to B. Particularly, is this set empty? Is FCD(A; B) a coatomiclattice? coatomistic lattice?

6.11 Complete funcoidsDe�nition 6.97. I will call co-complete such a funcoid f that hf i�X is a principal �lter for everyX 2P(Src f).

Obvious 6.98. Funcoid f is co-complete i� hf iX 2P for every X 2P.

De�nition 6.99. I will call generalized closure such a function �2 (PB)PA (for some sets A, B)that

1. �;= ;;2. 8I ; J 2PA:�(I [ J)=�I [�J .

Obvious 6.100. A funcoid f is co-complete i� hf i�= "Dst f �� for a generalized closure �.

Remark 6.101. Thus funcoids can be considered as a generalization of generalized closures. Atopological space in Kuratowski sense is the same as re�exive and transitive generalized closure.So topological spaces can be considered as a special case of funcoids.

De�nition 6.102. I will call a complete funcoid a funcoid whose reverse is co-complete.

Theorem 6.103. The following conditions are equivalent for every funcoid f :

1. funcoid f is complete;

2. 8S 2PF(Src f); J 2P(Dst f): (FS [f ] "Dst fJ,9I 2S: I [f ] "Dst fJ);

3. 8S 2PP(Src f); J 2P(Dst f): (S

S [f ]� J,9I 2S: I [f ]� J);4. 8S 2PF(Src f): hf i

FS=

Fhhf iiS;

5. 8S 2PP(Src f): hf i�S

S=Fhhf i�iS;

6. 8A2P(Src f): hf i�A=Ffhf i�fag j a2Ag.

Proof.

(3))(1). For every S 2PP(Src f), J 2P(Dst f)

"Src f[

S u hf¡1i�J =/ 0F(Src f),9I 2S: "Src fI u hf¡1i�J =/ 0F(Src f);

consequently by proposition 4.215 we have that hf¡1i�J is a principal �lter.

(1))(2). For every S 2 PF(Src f), J 2 P(Dst f) we have hf¡1i�J is a principal �lter,consequently G

S u hf¡1i�J =/ 0F(Src f),9I 2S: I u hf¡1i�J =/ 0F(Src f):

6.11 Complete funcoids 111

From this follows (2).

(6))(5). hf i�S

S =Ffhf i�fag j a 2

SSg =

F Sffhf i�fag j a 2 Ag j A 2 Sg =F

fFfhf i�fag j a2Ag j A2Sg=

Ffhf i�A j A2Sg=

Fhhf i�iS.

(2))(4). "Dst fJ �/ hf iFS,

FS [f ]"Dst fJ,9I 2S:I [f ]"Dst fJ,9I 2S: "Dst fJ �/ hf iI,

"Dst fJ �/Fhhf iiS (used theorem 4.215).

(2))(3), (4))(5), (5))(3), (5))(6). Obvious. �

The following proposition shows that complete funcoids are a direct generalization of pretopo-logical spaces.

Proposition 6.104. To specify a complete funcoid f it is enough to specify hf i� on one-elementsets, values of hf i� on one element sets can be speci�ed arbitrarily.

Proof. From the above theorem is clear that knowing hf i� on one-element sets hf i� can be foundon every set and then the value of hf i can be inferred for every �lter.

Choosing arbitrarily the values of hf i� on one-element sets we can de�ne a complete funcoidthe following way: hf i�X =

Ffhf i�f�g j �2Xg for every X 2P(Src f). Obviously it is really a

complete funcoid. �

Theorem 6.105. A funcoid is principal i� it is both complete and co-complete.

Proof.

). Obvious.

(. Let f be both a complete and co-complete funcoid. Consider the relation g de�ned bythat "Dst f hgif�g= hf i�f�g (g is correctly de�ned because f corresponds to a generalizedclosure). Because f is a complete funcoid f is the funcoid corresponding to g. �

Theorem 6.106. If R2PFCD(A;B) is a set of (co-)complete funcoids thenFR is a (co-)complete

funcoid (for every sets A and B).

Proof. It is enough to prove for co-complete funcoids. LetR2PFCD(A;B) be a set of co-completefuncoids. Then for every X 2P(Src f)G

R��X =

Gfhf i�X j f 2Rg

is a principal �lter (used theorem 6.37). �

Corollary 6.107. If R is a set of binary relations between sets A and B thenF

"FCD(A;B)�R=

"FCD(A;B)S

R.

Proof. From two last theorems. �

Theorem 6.108. Filtrators of funcoids are �ltered.

Proof. It's enough to prove that every funcoid is representable as an (in�nite) meet (on the latticeFCD(A;B)) of some set of principal funcoids.

Let f 2FCD(A;B), X 2PA, Y 2hf iX , g(X ;Y )=def"AX �FCD"BY t"AX �FCD1F(B). For every

K 2PA

hg(X;Y )i�K = h"AX �FCD"BY i�K t"AX �FCD1F(B)

��K=

0BBB@8><>:0F(B) ifK= ;"BY if ;=/ K �X1F(B) ifK *X

1CCCAwhf i�K;so g(X;Y )w f . For every X 2PA

lfhg(X ;Y )i�X j Y 2 hf i�Xg=

lf"BY j Y 2 hf i�Xg= hf i�X ;

112 Funcoids

consequently lfg(X ;Y ) j X 2PA; Y 2 hf i�Xg

��X vhf i�X

that is lfg(X ;Y ) j X 2PA; Y 2 hf i�Xgv f

and �nally

f =lfg(X ;Y ) j X 2PA; Y 2 hf i�Xg: �

Theorem 6.109.

1. g is metacomplete if g is a complete funcoid.

2. g is co-metacomplete if g is a co-complete funcoid.

Proof.

1. Let R be funcoids from a set A to a set B and g from B to some C. Thenhg �

FRi�X = hgih

FRi�X = hgi

Ffhf i�X j f 2 Rg =

Ffhgihf i�X j f 2 Rg =F

fhg � f i�X j f 2Rg= hFfg � f j f 2Rgi�X = h

Fhg � iRi�X for every set X �A. So

g � (FR)=

Fhg � iR.

2. By duality. �

Conjecture 6.110. g is complete if g is a metacomplete funcoid.

I will denote ComplFCD and CoComplFCD the sets of small complete and co-complete fun-coids correspondingly. ComplFCD(A; B) are complete funcoids from A to B and likewise withCoComplFCD(A;B).

Obvious 6.111. ComplFCD and CoComplFCD are closed regarding composition of funcoids.

Proposition 6.112. ComplFCD and CoComplFCD (with induced order) are complete lattices.

Proof. It follows from the theorem 6.106. �

Theorem 6.113. Atoms of the lattice ComplFCD(A; B) are exactly funcoidal products of theform "Af�g�FCD b where �2A and b is an ultra�lter on B.

Proof. First, it's easy to see that "Af�g�FCD b are elements of ComplFCD(A;B). Also 0FCD(A;B)

is an element of ComplFCD(A;B)."Af�g�FCD b are atoms of ComplFCD(A;B) because they are atoms of FCD(A;B).It remains to prove that if f is an atom of ComplFCD(A;B) then f = "Af�g�FCD b for some

�2A and an ultra�lter b on B.Suppose f 2 FCD(A; B) is a non-empty complete funcoid. Then there exists � 2 A such

that hf i�f�g=/ 0F(B). Thus "Af�g �FCD b v f for some ultra�lter b on B. If f is an atom thenf = "Af�g�FCD b. �

Theorem 6.114.

1. A funcoid f is complete i� there exists a function G: Src f!F(Dst f) such that

f =Gf"Src ff�g�FCDG(�) j �2Src f g: (6.10)

2. A funcoid f is co-complete i� there exists a function G:Dst f!F(Src f) such that

f =GfG(�)�FCD"Dst ff�g j �2Dst f g:

Proof. We will prove only the �rst as the second is symmetric.

). Let f be complete. Then take

G(�) =G �

b2 atomsF(Dst f) j "Src ff�g�FCD bv f

6.11 Complete funcoids 113

and we have (6.10) obviously.

(. Let (6.10) hold. Then G(�)=F

atomsG(�) and thus

f =Gf"Src ff�g�FCD b j �2Src f ; b2 atomsG(�)g

and so f is complete. �

Theorem 6.115.

1. For a complete funcoid f there exists exactly one function F 2F(Dst f)Src f such that

f =Gf"Src ff�g�FCDF (�) j �2Src f g:

2. For a co-complete funcoid f there exists exactly one function F 2F(Src f)Dst f such that

f =GfF (�)�FCD"Dst ff�g j �2Dst f g:

Proof. We will prove only the �rst as the second is similar. Let

f =Gf"Src ff�g�FCDF (�) j �2Src f g=

Gf"Src f f�g�FCDG(�) j �2 Src f g

for some F ;G2F(Dst f)Src f. We need to prove F =G. Let � 2Src f .

hf i�f�g=Gfh"Src ff�g�FCDF (�)i�f�g j �2Src f g=F (�):

Similarly hf i�f�g=G(�). So F (�)=G(�). �

6.12 Funcoids corresponding to pretopologies

Let � be a pretopology on a set U and cl the preclosure corresponding to it (see theorem 5.12).Both induce a funcoid, I will show that these two funcoids are reverse of each other:

Theorem 6.116. Let f be a complete funcoid de�ned by the formula hf i�fxg=�(x) for everyx2U , let g be a co-complete funcoid de�ned by the formula hgi�X = "Ucl(X) for every X 2PU .Then g= f¡1.

Remark 6.117. It is obvious that funcoids f and g exist.

Proof. X [g]�Y ,"UY �/ hgi"UX,Y �/ cl(X),9y2Y :�(y)�/ "UX,9y 2Y : hf i�fyg�/ "UX,(proposition 4.194 and properties of complete funcoids),hf i�Y �/ "UX,Y [f ]�X.

So g= f¡1. �

6.13 Completion of funcoids

Theorem 6.118. Cor f =Cor0 f for an element f of a �ltrator of funcoids.

Proof. By theorems 4.34 and 6.108. �

De�nition 6.119. Completion of a funcoid f 2 FCD(A; B) is the complete funcoid Compl f 2FCD(A;B) de�ned by the formula hCompl f i�f�g= hf i�f�g for �2Src f .

De�nition 6.120. Co-completion of a funcoid f is de�ned by the formula

CoCompl f =(Compl f¡1)¡1:

Obvious 6.121. Compl f v f and CoCompl f v f .

114 Funcoids

Proposition 6.122. The �ltrator (FCD(A;B);ComplFCD(A;B)) is �ltered.

Proof. Because the �ltrator of funcoids is �ltered. �

Theorem 6.123. Compl f = Cor(FCD(A;B);ComplFCD(A;B))f = Cor0(FCD(A;B);ComplFCD(A;B))f for

every funcoid f 2FCD(A;B).

Proof. Cor(FCD(A;B);ComplFCD(A;B))f =Cor0(FCD(A;B);ComplFCD(A;B))f using theorem 4.34 since the

�ltrator (FCD(A;B);ComplFCD(A;B)) is �ltered.Let g2up(FCD(A;B);ComplFCD(A;B)) f . Then g2ComplFCD(A;B) and gw f . Thus g=Compl gw

Compl f .Thus 8g 2up(FCD(A;B);ComplFCD(A;B)) f : gwCompl f .Let 8g 2 up(FCD(A;B);ComplFCD(A;B)) f :hv g for some h2ComplFCD(A;B).Then hv

dup(FCD(A;B);ComplFCD(A;B)) f = f and consequently h=ComplhvCompl f .

Thus

Compl f =lComplFCD(A;B)

up(FCD(A;B);ComplFCD(A;B)) f =Cor(FCD(A;B);ComplFCD(A;B))f: �

Theorem 6.124. hCoCompl f i�X =Cor hf i�X for every funcoid f and set X 2P(Src f).

Proof. CoCompl f v f thus hCoCompl f i�X v hf i�X but hCoCompl f i�X is a principal �lterthus hCoCompl f i�X vCor hf i�X .

Let �X =Cor hf i�X. Then �;=0F(Dst f) and

�(X [Y )=Cor hf i�(X [Y )=Cor(hf i�X t hf i�Y )=Cor hf i�X tCor hf i�Y =�X t�Y

(used theorem 4.223). Thus � can be continued till hgi for some funcoid g. This funcoid is co-complete.

Evidently g is the greatest co-complete element of FCD(Src f ;Dst f) which is lower than f .Thus g=CoCompl f and Cor hf i�X =�X = hgi�X = hCoCompl f i�X . �

Theorem 6.125. ComplFCD(A;B) is an atomistic lattice.

Proof. Let f 2ComplFCD(A;B). hf i�X=Ffhf i�fxg j x2Xg=

F �f j"Src ffxg

��fxg j x2X=F �f j"Src ffxg

��X j x 2X

, thus f =

F �f j"Src ffxg j x 2X

. It is trivial that every f j"Src ffxg

is a join of atoms of ComplFCD(A;B). �

Theorem 6.126. A funcoid is complete i� it is a join (on the lattice FCD(A;B)) of atomic completefuncoids.

Proof. It follows from the theorem 6.106 and the previous theorem. �

Corollary 6.127. ComplFCD(A;B) is join-closed.

Theorem 6.128. ComplFR=

FhCompliR for every R2PFCD(A;B) (for every sets A, B).

Proof. hComplF

Ri�X =FfhF

Ri�f�g j � 2 Xg =FfFfhf i�f�g j f 2 Rg j � 2 Xg =F

fFfhf i�f�g j � 2Xg j f 2Rg=

FfhCompl f i�X j f 2Rg= h

FhCompliRi�X for every set

X . �

Corollary 6.129. Compl is a lower adjoint.

Conjecture 6.130. Compl is not an upper adjoint (in general).

Proposition 6.131. Compl f =F �

f j"Src ff�g j �2Src ffor every funcoid f .

Proof. Let denote R the right part of the equality to prove.

6.13 Completion of funcoids 115

hRi�f�g=F �

f j"Src ff�g��f�g j �2 Src f

= hf i�f�g for every � 2Src f and R is complete

as a join of complete funcoids.Thus R is the completion of f . �

Conjecture 6.132. Compl f = f n� (�FCDf) for every funcoid f .

This conjecture may be proved by considerations similar to these in the section �Fréchet �lter�.

Lemma 6.133. Co-completion of a complete funcoid is complete.

Proof. Let f be a complete funcoid.hCoCompl f i�X = Cor hf i�X = Cor

Ffhf i�fxg j x 2 Xg =

FfCor hf i�fxg j x 2 Xg =F

fhCoCompl f i�fxg j x2Xg for every set X . Thus CoCompl f is complete. �

Theorem 6.134. ComplCoCompl f =CoComplCompl f =Cor f for every funcoid f .

Proof. Compl CoCompl f is co-complete since (used the lemma) CoCompl f is co-complete.Thus Compl CoCompl f is a principal funcoid. CoCompl f is the greatest co-complete fun-coid under f and Compl CoCompl f is the greatest complete funcoid under CoCompl f . SoCompl CoCompl f is greater than any principal funcoid under CoCompl f which is greater thanany principal funcoid under f . Thus ComplCoCompl f is the greatest principal funcoid under f .Thus ComplCoCompl f =Cor f . Similarly CoComplCompl f =Cor f . �

Question 6.135. Is ComplFCD(A;B) a co-brouwerian lattice for every sets A,B? [TODO: Solved.]

6.13.1 More on completion of funcoids

Proposition 6.136. For every composable funcoids f and g

1. Compl(g � f)wCompl g �Compl f ;

2. CoCompl(g � f)wCoCompl g �CoCompl f .

Proof.

1. Compl g �Compl f =Compl(Compl g �Compl f)vCompl(g � f).

2. CoCompl g �CoCompl f =CoCompl(CoCompl g �CoCompl f)vCoCompl(g � f). �

Proposition 6.137. For every composable funcoids f and g [TODO: Errors in this theorem,corrected in the LyX version of this text.]

1. CoCompl(g � f)= (CoCompl g) � f if f is a co-complete funcoid.

2. Compl(f � g)= f �Compl g if f is a complete funcoid.

Proof.

1. hCoCompl(g � f)i�X = Cor hg � f i�X = Cor hgihf i�X = hCoCompl gihf i�X =h(CoCompl g) � f i�X for every X .

2. (CoCompl(g � f))¡1 = f¡1 � (CoCompl g)¡1; Compl(g � f)¡1 = f¡1 � Compl g¡1;Compl(f¡1 � g¡1) = f¡1 � Compl g¡1. After variable replacement we get Compl(f � g) =f �Compl g (after the replacement f is a complete funcoid). �

Corollary 6.138. CoCompl((Compl g)� f)=Compl(g� (CoCompl f))=(Compl g)�(CoCompl f).

Proof. By the theorem:Compl(g � (CoCompl f)) = (Compl g) � (CoCompl f);CoCompl((Compl f) � g) = (Compl f) � (CoCompl g).After variable replacement CoCompl((Compl g) � f) = (Compl g) � (CoCompl f). �

116 Funcoids

Proposition 6.139. For every composable funcoids f and g

1. Compl(g � (Compl f)) =Compl(g � f);2. CoCompl((CoCompl g) � f) =CoCompl(g � f).

Proof.

1. hg � (Compl f)i�fxg= hgihCompl f i�fxg= hgihf i�fxg= hg � f i�fxg.Thus Compl(g � (Compl f))=Compl(g � f).

2. (Compl(g � (Compl f)))¡1= (Compl(g � f))¡1; CoCompl(g � (Compl f))¡1=CoCompl(g �f)¡1; CoCompl((Compl f)¡1 � g¡1) = CoCompl(f¡1 � g¡1); CoCompl((CoCompl f¡1) �g¡1) = CoCompl(f¡1 � g¡1). After variable replacement CoCompl((CoCompl g) � f) =CoCompl(g � f). �

6.13.1.1 Open maps

De�nition 6.140. An open map from a topological space to a topological space is a functionwhich maps open sets into open sets.

An obvious generalization of this is open map f from an endofuncoid � to an endofuncoid �,which is by de�nition a function (or rather a principal, entirely de�ned, monovalued funcoid) fromOb � to Ob � such that

8x2Ob �; V 2 h�i�fxg: hf i�V wh� ihf i�fxg:

This formula is equivalent (exercise!) to

8x2Ob �: hf ih�i�fxgw h� ihf i�fxg:

It can be abstracted/simpli�ed further (now for an arbitrary funcoid f from Ob � to Ob �):

Compl(f � �)wCompl(� � f):

De�nition 6.141. An open funcoid from an endofuncoid � to an endofuncoid � is a funcoid ffrom Ob � to Ob � such that Compl(f � �)wCompl(� � f).

Theorem 6.142. Let �, �, � be endofuncoids. Let f be a co-complete open funcoid from Ob � toOb� and g is an open funcoid from Ob� to Ob�. Then g� f is an open funcoid from Ob� to Ob�.

Proof. Let Compl(f � �)wCompl(� � f) and Compl(g � �)wCompl(� � g).Compl(g � f � �)wCompl(g �Compl(f � �))wCompl(g �Compl(� � f))=Compl(g �Compl(�)�

f)=Compl(g �Compl(�)) � f =Compl(g � �) � f wCompl(� � g) � f =Compl(� � g � f): �

Obvious 6.143. A funcoid f is open i� f � �wCompl(� � f).

Corollary 6.144. A co-complete funcoid f is open i� f � �w (Compl �) � f . Thus f is open i� itis a continuous morphism from � to Compl � with the reverse order of funcoids . (See a de�nitionof a continuous morphism below.)

6.14 Monovalued and injective funcoids

Following the idea of de�nition of monovalued morphism let's call monovalued such a funcoid fthat f � f¡1v idim f

FCD.Similarly, I will call a funcoid injective when f¡1 � f v iddom f

FCD .

Obvious 6.145. A funcoid f is:

� monovalued i� f � f¡1v idFCD(Dst f);

� injective i� f¡1 � f v idFCD(Src f).

6.14 Monovalued and injective funcoids 117

In other words, a funcoid is monovalued (injective) when it is a monovalued (injective) mor-phism of the category of funcoids. Monovaluedness is dual of injectivity.

Obvious 6.146.

1. A morphism (A; B; f) of the category of funcoid triples is monovalued i� the funcoid f ismonovalued.

2. A morphism (A; B; f) of the category of funcoid triples is injective i� the funcoid f isinjective.

Theorem 6.147. The following statements are equivalent for a funcoid f :

1. f is monovalued.

2. 8a2 atomsF(Src f): hf ia2 atomsF(Dst f)[�0F(Dst f).

3. 8I ;J 2F(Dst f): hf¡1i(I uJ ) = hf¡1iI u hf¡1iJ .

4. 8I ; J 2P(Dst f): hf¡1i�(I \ J)= hf¡1i�I u hf¡1i�J .

Proof.

(2))(3). Let a2 atomsF(Src f), hf ia= b. Then because b2 atomsF(Dst f)[�0F(Dst f)

(I uJ )u b=/ 0F(Dst f),I u b=/ 0F(Dst f)^J u b=/ 0F(Dst f);

a [f ] I uJ , a [f ] I ^ a [f ]J ;I u J [f¡1] a,I [f¡1] a^J [f¡1] a;

au hf¡1i(I u J ) =/ 0F(Src f), au hf¡1iI =/ 0F(Src f)^ au hf¡1iJ =/ 0F(Src f);

hf¡1i(I uJ )= hf¡1iI u hf¡1iJ :

(3))(1). hf¡1ia u hf¡1ib = hf¡1i(a u b) = hf¡1i0F(Dst f) = 0F(Src f) for every two distinctultra�lters a and b on Dst f . This is equivalent to :(hf¡1ia [f ] b); b � hf ihf¡1ia; b �hf � f¡1ia; :(a [f � f¡1] b). So a [f � f¡1] b) a = b for every ultra�lters a and b. This ispossible only when f � f¡1v idFCD(Dst f).

(4))(3). hf¡1i(I u J ) =dhhf¡1i�i(I u J ) =

dhhf¡1i�ifI \ J j I 2 I ; J 2 J g =d

fhf¡1i�(I \J) j I 2I ; J 2J g=dfhf¡1i�I uhf¡1i�J j I 2I ; J 2J g=

dfhf¡1i�I j I 2

Igudfhf¡1i�J j J 2J g= hf¡1iI u hf¡1iJ .

(3))(4). Obvious.

:(2)):(1). Suppose hf ia 2/ atomsF(Dst f) [�0F(Dst f) for some a 2 atomsF(Src f). Then

there exist two atomic �lters p and q on Dst f such that p =/ q and hf ia w p ^ hf ia w q.Consequently p �/ hf ia; a �/ hf¡1ip; a v hf¡1ip; hf � f¡1ip = hf ihf¡1ip w hf ia w q;hf � f¡1ipvp and hf � f¡1ip=/ 0F(Dst f). So it cannot be f � f¡1v idFCD(Dst f). �

Corollary 6.148. A binary relation corresponds to a monovalued funcoid i� it is a function.

Proof. Because 8I ; J 2 P(im f): hf¡1i�(I \ J) = hf¡1i�I u hf¡1i�J is true for a funcoid fcorresponding to a binary relation if and only if it is a function. �

Remark 6.149. This corollary can be reformulated as follows: For binary relations (principalfuncoids) the classic concept of monovaluedness and monovaluedness in the above de�ned sense ofmonovaluedness of a funcoid are the same.

Proposition 6.150. Every monovalued funcoid is metamonovalued.

Proof. h(dG) � f ix= h

dGihf ix=

dg2G hgihf ix=

dg2G hg � f ix=

dg2G (g � f)

�x for every

ultra�lter x2 atomsF(Src f). Thus (dG) � f =

dg2G (g � f). �

Corollary 6.151. Every injective funcoid is metainjective.

118 Funcoids

Conjecture 6.152. Every metamonovalued funcoid is monovalued.

6.15 T0-, T1-, T2-, and T3-separable funcoidsFor funcoids it can be generalized T0-, T1-, T2-, and T3- separability. Worthwhile note that T0 andT2 separability is de�ned through T1 separability.

De�nition 6.153. Let call T1-separable such endofuncoid f that for every �; � 2Ob f is true

�=/ �):(f�g [f ]� f�g):

Proposition 6.154. An endofuncoid f is T1-separable i� Cor f v idFCD(Ob f).

Proof. 8x; y 2 Ob f : (fxg [f ]� fyg ) x = y) , 8x; y 2 Ob f : (fxg [Cor f ]� fyg ) x = y) ,Cor f v idFCD(Ob f). �

De�nition 6.155. Let call T0-separable such funcoid f 2FCD(A;A) that f u f¡1 is T1-separable.

De�nition 6.156. Let call T2-separable such funcoid f that f¡1 � f is T1-separable.

For symmetric transitive funcoids T1- and T2-separability are the same (see theorem 3.51).

Obvious 6.157. A funcoid f is T2-separable i� �=/ �)hf i�fag�hf i�f�g for every �; � 2Src f .

De�nition 6.158. Regular funcoid is an endofuncoid f such that hf ihf¡1i�C�hf i�fpg( p2/ Cfor every p2Ob f and C 2POb f .

Obvious 6.159. Funcoid f is regular i�:

1. hf � f¡1i�C�hf i�fpg( p2/ C;

2. hf¡1 � f � f¡1i�C�"Ob ffpg( p2/ C;

3. hf¡1 � f � f¡1i�C v"Ob fC;

4. f¡1 � f � f¡1v idFCD(Ob f).

De�nition 6.160. An endofuncoid is T3- i� it is both T2- and regular.

6.16 Filters closed regarding a funcoidDe�nition 6.161. Let's call closed regarding a funcoid f 2 FCD(A; A) such �lter A 2 F(Src f)that hf iA vA.

This is a generalization of closedness of a set regarding an unary operation.

Proposition 6.162. If I and J are closed (regarding some funcoid f), S is a set of closed �lterson Src f , then

1. I tJ is a closed �lter;

2.dS is a closed �lter.

Proof. Let denote the given funcoid as f . hf i(I tJ )= hf iI thf iJ vItJ , hf idSv

dhhf iiSvd

S. Consequently the �lters I tJ anddS are closed. �

Proposition 6.163. If S is a set of �lters closed regarding a complete funcoid, then the �lterFS

is also closed regarding our funcoid.

Proof. hf iFS=

Fhhf iiS v

FS where f is the given funcoid. �

6.16 Filters closed regarding a funcoid 119

Chapter 7Reloids

7.1 Basic de�nitions

De�nition 7.1. I call a reloid from a set A to a set B a triple (A;B;F ) where F 2F(A�B).

De�nition 7.2. Source and destination of every reloid (A;B;F ) are de�ned as

Src(A;B;F ) =A and Dst(A;B;F )=B:

I will denote RLD(A;B) the set of reloids from A to B.I will denote RLD the set of all reloids (for small sets).

De�nition 7.3. GR(A;B;F )=defF , xyGR(A;B;F )=

deff(A;B;K) j K2F g for every reloid (A;B;F ).

Note that xyGR (A;B;F ) is a set of morphisms of the category Rel.

De�nition 7.4.

� "RLD(A;B)f=def(A;B; "A�Bf) for every relation f 2P(A�B).

� "RLDf =(Src f ;Dst f ; "Src f�Dst fGR f) for every Rel-morphism f .

De�nition 7.5. I call members of a set h"RLDiRel(A;B) as principal reloids.

Reloids are a generalization of uniform spaces. Also reloids are generalization of binary rela-tions.

De�nition 7.6. The reverse reloid of a reloid is de�ned by the formula

(A;B;F )¡1=(B;A; fK¡1 j K 2F g):

Note 7.7. The reverse reloid is not an inverse in the sense of group theory or category theory.

Reverse reloid is a generalization of conjugate quasi-uniformity.

De�nition 7.8. Every set RLD(A;B) is a poset by the formula f v g,GR f vGR g. We willapply lattice operations to subsets of RLD(A;B) without explicitly mentioning RLD(A;B).

Obvious 7.9. The poset RLD(A;B) is isomorphic to the poset F(A�B) for every sets A, B.

7.2 Composition of reloids

De�nition 7.10. Reloids f and g are composable when Dst f =Src g.

De�nition 7.11. Composition of (composable) reloids is de�ned by the formula

g � f =lf"RLD(G�F ) j F 2 xyGR f ;G2 xyGR gg:

Obvious 7.12. Composition of reloids is a reloid.

121

Theorem 7.13. (h � g) � f =h � (g � f) for every composable reloids f , g, h.

Proof. For two nonempty collections A and B of sets I will denote

A�B,8K 2A9L2B:L�K ^8K 2B9L2A:L�K:

It is easy to see that � is a transitive relation.I will denote B �A= fL �K j K 2A; L2Bg.Let �rst prove that for every nonempty collections of relations A, B, C

A�B)A�C�B �C:Suppose A�B and P 2A �C that is K 2A and M 2C such that P =K �M . 9K 0 2B:K 0�Kbecause A� B. We have P 0=K 0 �M 2B � C. Obviously P 0� P . So for every P 2 A � C thereexists P 02B �C such that P 0�P ; the vice versa is analogous. So A�C�B �C.

GR((h � g) � f) � GR(h � g) � GR f , GR(h � g) � (GR h) � (GR g). By proven aboveGR((h� g) � f)� (GRh) � (GR g) � (GR f).

Analogously GR(h � (g � f))� (GRh) � (GR g) � (GR f).So GR(h � (g � f))�GR((h � g) � f) what is possible only if GR(h � (g � f)) =GR((h � g) � f).

Thus (h� g) � f =h � (g � f). �

Theorem 7.14. For every reloid f :

1. f � f =df"RLD(F �F ) j F 2 xyGR f g if Src f =Dst f ;

2. f¡1 � f =df"RLD(F¡1 �F ) j F 2 xyGR f g;

3. f � f¡1=df"RLD(F �F¡1) j F 2 xyGR f g.

Proof. I will prove only (1) and (2) because (3) is analogous to (2).

1. It's enough to show that 8F ; G 2 xyGR f9H 2 xyGR f :H �H v G � F . To prove it takeH =F uG.

2. It's enough to show that 8F ;G2xyGR f9H 2xyGR f :H¡1�H vG¡1�F . To prove it takeH =F uG. Then H¡1 �H =(F uG)¡1 � (F uG)vG¡1 �F . �

Theorem 7.15. For every sets A, B, C if g; h2RLD(A;B) then

1. f � (gt h)= f � g t f �h for every f 2RLD(B;C);

2. (gt h) � f = g � f th � f for every f 2RLD(C;A).

Proof. We'll prove only the �rst as the second is dual.By the in�nite distributivity law for �lters we have

f � g t f �h =lf"RLD(F �G) j F 2 xyGR f ;G2 xyGR ggtlf"RLD(F �H) j F 2 xyGR f ;H 2 xyGRhg

=lf"RLD(F1 �G)t"RLD(F2 �H) j F1; F22 xyGR f ;G2 xyGR g;H 2 xyGRhg

=lf"RLD((F1 �G)t (F2 �H)) j F1; F22 xyGR f ;G2 xyGR g;H 2 xyGRhg:

Obviouslylf"RLD((F1 �G)t (F2 �H)) j F1; F22 xyGR f ;G2 xyGR g;H 2 xyGRhg wl

f"RLD(((F1uF2) �G)t ((F1uF2) �H)) j F1; F22 xyGR f ;G2 xyGR g;H 2 xyGRhg =lf"RLD((F �G)t (F �H)) j F 2 xyGR f ;G2 xyGR g;H 2 xyGRhg =l

f"RLD(F � (GtH)) j F 2 xyGR f ;G2 xyGR g;H 2 xyGRhg:

Because G2 xyGR g ^H 2 xyGRh)GtH 2 xyGR(g th) we havelf"RLD(F � (GtH)) j F 2 xyGR f ;G2 xyGR g;H 2 xyGRhg wl

f"RLD(F �K) j F 2 xyGR f ;K 2 xyGR (gt h)g =

f � (g th):

122 Reloids

Thus we have proved f � gt f �hw f � (gth). But obviously f � (gth)w f � g and f � (gth)w f �hand so f � (g th)w f � g t f �h. Thus f � (g th)= f � g t f �h. �

Theorem 7.16. Let A, B, C be sets, f 2RLD(A;B), g 2RLD(B;C), h2RLD(A;C). Then

g � f �/ h, g�/ h � f¡1:

Proof. g � f �/ h,df"RLD(G �F ) j F 2 xyGR f ; G2 xyGR gg u

dh"RLDixyGRh=/ 0RLD(A;C),d

f"RLD((G � F ) u H) j F 2 xyGR f ; G 2 xyGR g; H 2 xyGR hg =/ 0RLD(A;C), 8F 2 xyGR f ;

G2xyGR g;H 2xyGRh:"RLD((G�F )uH)=/ 0RLD(A;C),8F 2xyGR f ;G2xyGR g;H 2xyGRh:G�F �/ H (used properties of generalized �lter bases).

Similarly g�/ h � f¡1,8F 2 xyGR f ;G2 xyGR g;H 2 xyGRh:G�/ H �F¡1.Thus g � f �/ h, g�/ h � f¡1 because G�F �/ H,G�/ H �F¡1 by proposition 3.70. �

Theorem 7.17. For every composable reloids f and g

1. g � f =Ffg �F j F 2 atoms f g.

2. g � f =FfG � f j G2 atoms gg.

Proof. We will prove only the �rst as the second is dual.Ffg � F j F 2 atoms f g = g � f , 8x 2 RLD(Src f ; Dst g): (x �/ g � f , x �/

Ffg � F j F 2

atoms f g)(8x2RLD(Src f ;Dst g): (x�/ g � f,9F 2atoms f :x�/ g �F ),8x2RLD(Src f ;Dst g):(g¡1 � x�/ f,9F 2 atoms f : g¡1 � x�/ F ) what is obviously true. �

Corollary 7.18. If f and g are composable reloids, then

g � f =GfG �F j F 2 atoms f ;G2 atoms gg:

Proof. g � f =Ffg � F j F 2 atoms f g =

FfFfG � F j G 2 atoms gg j F 2 atoms f g =F

fG�F j F 2 atoms f ;G2 atoms gg. �

7.3 Direct product of �lters

De�nition 7.19. Reloidal product of �lters A and B is de�ned by the formula

A�RLDB=defl �

"RLD(Base(A);Base(B))(A�B) j A2A; B 2B:

Obvious 7.20. "UA�RLD "VB= "RLD(U ;V )(A�B) for every sets A�U , B �V .

Theorem 7.21. A�RLDB=Ffa�RLD b j a2 atomsA; b2 atomsBg for every �lters A, B.

Proof. Obviously

A�RLDB wGfa�RLD b j a2 atomsA; b2 atomsBg:

Reversely, let

K 2GRGfa�RLD b j a2 atomsA; b2 atomsBg:

Then K 2GR(a�RLD b) for every a2 atomsA, b2 atomsB; K �Xa� Yb for some Xa 2 a, Yb 2 b;K�

SfXa�Yb j a2atomsA; b2atomsBg=

SfXa j a2atomsAg�

SfYb j b2atomsBg�A�B

where A2A, B 2B; K 2GR(A�RLDB). �

Theorem 7.22. If A0;A12F(A), B0;B12F(B) for some sets A, B then

(A0�RLDB0)u (A1�RLDB1)= (A0uA1)�RLD (B0uB1):

7.3 Direct product of filters 123

Proof.

(A0�RLDB0)u (A1�RLDB1)= f"RLD(P uQ) j P 2 xyGR(A0�RLDB0); Q2 xyGR(A1�RLDB1)g=�"RLD(A;B)((A0�B0)\ (A1�B1)) j A02A0; B02B0; A12A1; B12B1

=�"RLD(A;B)((A0\A1)� (B0\B1)) j A02A0; B02B0; A12A1; B12B1

=�"RLD(A;B)(K �L) j K 2A0uA1; L2B0uB1

= (A0uA1)�RLD (B0uB1):

�

Theorem 7.23. If S 2P(F(A)�F(B)) for some sets A, B thenlfA�RLDB j (A;B)2Sg=

ldomS �RLD

limS:

Proof. Let P =d

domS, Q=d

imS; l=dfA�RLDB j (A;B)2Sg.

P �RLDQv l is obvious.Let F 2GR(P �RLDQ). Then there exist P 2P and Q2Q such that F �P �Q.P =P1\ ::: \Pn where Pi2 domS and Q=Q1\ ::: \Qm where Qj 2 imS.P �Q=

Ti; j (Pi�Qj).

Pi�Qj 2GR(A�RLDB) for some (A;B)2S. P �Q=Ti; j (Pi�Qj)2GR l. So F 2GR l. �

Corollary 7.24.dhA�RLD iT =A�RLDd T if A is a �lter and T is a set of �lters with common

base.

Proof. Take S= fAg�T where T is a set of �lters.Then

dfA�RLDB j B 2T g=A�RLDd T that is

dhA�RLD iT =A�RLDd T . �

De�nition 7.25. I will call a reloid convex i� it is a join of direct products.

7.4 Restricting reloid to a �lter. Domain and image

De�nition 7.26. Identity reloid for a set A is de�ned by the formula idRLD(A)= "RLD(A;A)idA.

Obvious 7.27.¡idRLD(A)

�¡1= idRLD(A).

De�nition 7.28. I de�ne restricting a reloid f to a �lter A as f jA=f u¡A�RLD 1F(Dst f)�.

De�nition 7.29. Domain and image of a reloid f are de�ned as follows:

dom f =lh"Src f ihdomiGR f ; im f =

lh"Dst f ihimiGR f:

Proposition 7.30. f vA�RLDB,dom f vA^ im f vB for every reloid f and �ltersA2F(Src f),B 2F(Dst f).

Proof.

). It follows from dom(A�RLDB)vA^ im(A�RLDB)vB.(. dom f vA,8A2A9F 2GR f : domF �A. Analogously

im f vB,8B 2B9G2GR f : imG�B.Let dom f v A ^ im f v B, A 2 A, B 2 B. Then there exist F ; G 2 GR f such that

domF �A^ imG�B. Consequently F \G2GR f , dom(F \G)�A, im(F \G)�B thatis F \G�A�B. So there exists H 2GR f such that H �A�B for every A2A, B 2B.So f vA�RLDB. �

De�nition 7.31. I call restricted identity reloid for a �lter A the reloid

idARLD=def¡

idRLD(Base(A))�jA:

124 Reloids

Theorem 7.32. idARLD=d �

"RLD(Base(A);Base(A))idA j A2Afor every �lter A.

Proof. Let K 2 GRd �

"RLD(Base(A);Base(A))idA j A 2 A, then there exists A 2 A such that

K � idA. ThenidARLD v "RLD(Base(A);Base(A))idBase(A) u

¡A �RLD 1F(Base(A))

�v "RLD(Base(A);Base(A))idBase(A) u¡

"Base(A)A�RLD1F(Base(A))�="RLD(Base(A);Base(A))idBase(A)u"RLD(Base(A);Base(A))(A�Base(A))=

"RLD(Base(A);Base(A))(idBase(A)\ (A�Base(A)))="RLD(Base(A);Base(A))idAv"RLD(Base(A);Base(A))K.Thus K 2GR idARLD.Reversely letK 2GRidARLD=GR

¡idRLD(Base(A))u

¡A�RLD1F(Base(A))

��, then there exists A2A

such that K 2GR "RLD(Base(A);Base(A))(idBase(A)\ (A�Base(A))) =GR "RLD(Base(A);Base(A))idAwGR

d �"RLD(Base(A);Base(A))idA j A2A

. �

Corollary 7.33. (idARLD)¡1= idARLD.

Theorem 7.34. f jA=f � idARLD for every reloid f and A2F(Src f).

Proof. We need to prove that f u¡A�RLD 1F(Dst f)�= f �

d �"RLD(Src f ;Src f)idA j A2A

.

We have f �d �

"RLD(Src f ;Src f)idA j A2A=d �

"RLD(Src f ;Dst f)(F � idA) j F 2GR f ;A2A=d �

"RLD(Src f ;Dst f)(F jA) j F 2GR f ; A 2A=d �

"RLD(Src f ;Dst f)(F \ (A �Dst f)) j F 2GR f ;

A 2 A=d �

"RLD(Src f ;Dst f)F j F 2 GR fud �

"RLD(Src f ;Dst f)(A � Dst f) j A 2 A=

f u¡A�RLD 1F(Dst f)�. �

Theorem 7.35. (g � f)jA=g � (f jA) for every composable reloids f and g and A2F(Src f).

Proof. (g � f)jA=(g � f) � idARLD= g � (f � idARLD)= g � (f jA). �

Theorem 7.36. f u (A�RLDB)= idBRLD� f � idARLD for every reloid f andA2F(Src f), B2F(Dst f).

Proof. f u (A�RLD B) = f u¡A �RLD 1F(Dst f)� u ¡1F(Src f)�RLD B

�= f jAu

¡1F(Src f)�RLD B

�=

(f � idARLD) u¡1F(Src f) �RLD B

�=¡(f � idARLD)¡1 u

¡1F(Src f) �RLD B

�¡1�¡1 = ¡(idARLD � f¡1) u¡B �RLD 1F(Src f)

��¡1=(idARLD � f¡1 � idBRLD)¡1= idBRLD � f � idARLD. �

Theorem 7.37. f j"Srcf�g="Srcf�g�RLD im(f j"Srcf�g) for every reloid f and �2 Src f .

Proof. First,

im(f j"Srcf�g) =lh"Dst f ihimiGR(f j"Srcf�g) =l

h"Dst f ihimiGR¡f u¡"Src f f�g�RLD 1F(Dst f)�� =l

f"Dst fim(F \ (f�g�Dst f)) j F 2GR f g =lf"Dst fim(F jf�g) j F 2GR f g:

Taking this into account we have:

"Srcf�g�RLD im(f j"Srcf�g) =l �"RLD(Src f ;Dst f)(f�g�K) j K 2 im(f j"Srcf�g)

=l �

"RLD(Src f ;Dst f)(f�g� im(F jf�g)) j F 2GR f

=l �"RLD(Src f ;Dst f)(F jf�g) j F 2GR f

=l �

"RLD(Src f ;Dst f)(F \ (f�g�Dst f)) j F 2GR f

=l �"RLD(Src f ;Dst f)F j F 2GR f

u"RLD(Src f ;Dst f)(f�g�Dst f) =

f u"RLD(Src f ;Dst f)(f�g�Dst f) =

f j"Srcf�g:�

7.4 Restricting reloid to a filter. Domain and image 125

Lemma 7.38. �B 2F(B): 1F�RLDB is an upper adjoint of �f 2RLD(A;B): im f (for every setsA, B).

Proof. We need to prove im f vB, f v 1F�RLDB what is obvious. �

Corollary 7.39. Image and domain of reloids preserve joins.

Proof. By properties of Galois connections and duality. �

7.5 Categories of reloids

I will de�ne two categories, the category of reloids and the category of reloid triples .The category of reloids is de�ned as follows:


� The set of morphisms from a set A to a set B is RLD(A;B).

� The composition is the composition of reloids.

� Identity morphism for a set is the identity reloid for that set.

To show it is really a category is trivial.The category of reloid triples is de�ned as follows:

� Objects are �lters on small sets.

� The morphisms from a �lter A to a �lter B are triples (A; B; f) where f 2 RLD(Base(A);Base(B)) and dom f vA, im f vB.

� The composition is de�ned by the formula (B; C; g) � (A;B; f)= (A; C; g � f).� Identity morphism for a �lter A is idARLD.


7.6 Monovalued and injective reloids

Following the idea of de�nition of monovalued morphism let's callmonovalued such a reloid f thatf � f¡1v idim f

RLD.Similarly, I will call a reloid injective when f¡1 � f v iddom f

RLD .

Obvious 7.40. A reloid f is

� monovalued i� f � f¡1v idRLD(Dst f);

� injective i� f¡1 � f v idRLD(Src f).

In other words, a reloid is monovalued (injective) when it is a monovalued (injective) morphismof the category of reloids.

Monovaluedness is dual of injectivity.

Obvious 7.41.

1. A morphism (A; B; f) of the category of reloid triples is monovalued i� the reloid f ismonovalued.

2. A morphism (A;B; f) of the category of reloid triples is injective i� the reloid f is injective.

Theorem 7.42.

1. A reloid f is a monovalued i� there exists a function (monovalued binary relation) F 2GR f .

2. A reloid f is a injective i� there exists an injective binary relation F 2GR f .

126 Reloids

3. A reloid f is a both monovalued and injective i� there exists an injection (a monovaluedand injective binary relation = injective function) F 2GR f .

Proof. The reverse implications are obvious. Let's prove the direct implications:

1. Let f be a monovalued reloid. Then f � f¡1v idRLD(Dst f). So there exists

h2GR(f � f¡1)=GRl �

"RLD(Dst f ;Dst f)(F �F¡1) j F 2GR f

such that "RLD(Dst f ;Dst f)hv idRLD(Dst f). It's simple to show that fF �F¡1 j F 2GR f g isa �lter base. Consequently there exists F 2GR f such that F �F¡1� idDst f that is F is afunction.

2. Similar.

3. Let f be a both monovalued and injective reloid. Then by proved above there exist F ;G 2 GR f such that F is monovalued and G is injective. Thus F \ G 2 GR f is bothmonovalued and injective. �

Conjecture 7.43. A reloid f is monovalued i�

8g 2RLD(Src f ;Dst f): (gv f)9A2F (Src f): g= f jA):

7.7 Complete reloids and completion of reloids

Definition 7.44. A complete reloid is a reloid representable as a join of reloidal products"Af�g�RLD b where �2A and b is an ultra�lter on B for some sets A and B.

De�nition 7.45. A co-complete reloid is a reloid representable as a join of reloidal productsa�RLD "Bf�g where � 2B and a is an ultra�lter on A for some sets A and B.

I will denote the sets of complete and co-complete reloids correspondingly as ComplRLD andCoComplRLD.

Obvious 7.46. Complete and co-complete are dual.

Theorem 7.47.

1. A reloid f is complete i� there exists a function G: Src f!F(Dst f) such that

f =Gf"Src ff�g�RLDG(�) j �2Src f g: (7.1)

2. A reloid f is co-complete i� there exists a function G:Dst f!F(Src f) such that

f =GfG(�)�RLD "Dst ff�g j �2Dst f g:

Proof. We will prove only the �rst as the second is symmetric.

). Let f be complete. Then take

G(�) =G �

b2 atomsF(Dst f) j "Src ff�g�RLD bv f

and we have (7.1) obviously.

(. Let (7.1) hold. Then G(�)=F

atomsG(�) and thus

f =Gf"Src ff�g�RLD b j �2Src f ; b2 atomsG(�)g

and so f is complete. �

Obvious 7.48. Complete and co-complete reloids are convex.

7.7 Complete reloids and completion of reloids 127

Obvious 7.49. Principal reloids are complete and co-complete.

Obvious 7.50. Join (on the lattice of reloids) of complete reloids is complete.

Corollary 7.51. ComplRLD (with the induced order) is a complete lattice.

Theorem 7.52. A reloid which is both complete and co-complete is principal.

Proof. Let f be a complete and co-complete reloid. We have

f =Gf"Src ff�g�RLDG(�) j �2Src f g and f =

GfH(�)�RLD "Dst ff�g j � 2Dst f g

for some functions G: Src f!F(Dst f) and H:Dst f!F(Src f). For every �2Src f we have

G(�) =

im f j"Src ff�g =

im¡f u¡"Src ff�g�RLD 1F(Dst f)�� = (*)

imG �

(H(�)�RLD "Dst ff�g)u¡"Src ff�g�RLD 1F(Dst f)� j � 2Dst f

=

imGf(H(�)u"Src ff�g)�RLD "Dst ff�g j � 2Dst f g =

imG ( (

"Src ff�g�RLD "Dst ff�g ifH(�)�/ "Src ff�g0RLD(Src f ;Dst f) ifH(�)�"Src ff�g

!j � 2Dst f

)=

imGf"Src ff�g�RLD "Dst ff�g j � 2Dst f ;H(�)�/ "Src ff�gg =

imG �

"RLD(Src f ;Dst f)f(�; �)g j � 2Dst f ;H(�)�/ "Src ff�g

=Gf"Dst ff�g j � 2Dst f ;H(�)�/ "Src ff�gg

* proposition 4.194 was used.Thus G(�) is a principal �lter that is G(�) = "Dst fg(�) for some g: Src f ! Dst f ;

"Src f f�g�RLDG(�) = "RLD(Src f ;Dst f)(f�g� g(�)); f is principal as a join of principal reloids. �

Conjecture 7.53. Composition of complete reloids is complete. [TODO: Solved.]

Theorem 7.54.

1. For a complete reloid f there exists exactly one function F 2F(Dst f)Src f such that

f =Gf"Src ff�g�RLDF (�) j �2 Src f g:

2. For a co-complete reloid f there exists exactly one function F 2F(Src f)Dst f such that

f =GfF (�)�RLD "Dst f f�g j �2Dst f g:

Proof. We will prove only the �rst as the second is similar. Let

f =Gf"Src f f�g�RLDF (�) j �2Src f g=

Gf"Src f f�g�RLDG(�) j �2Src f g

for some F ;G2F(Dst f)Src f. We need to prove F =G. Let � 2Src f .

f u¡"Src ff�g�RLD 1F(Dst f)� = (proposition 4.194)G �

("Src f f�g�RLDF (�))u¡"Src ff�g�RLD 1F(Dst f)� j �2Src f

=

"Src ff�g�RLDF (�):

Similarly f u¡"Src ff�g �RLD 1F(Dst f)� = "Src ff�g �RLD G(�). Thus "Src ff�g �RLD F (�) =

"Src f f�g�RLDG(�) and so F (�)=G(�). �

De�nition 7.55. Completion and co-completion of a reloid f 2 RLD(A; B) are de�ned by theformulas:

Compl f =Cor(RLD(A;B);ComplRLD(A;B))f ; CoCompl f =Cor(RLD(A;B);CoComplRLD(A;B))f:

128 Reloids

Theorem 7.56. Atoms of the lattice ComplRLD(A;B) are exactly reloidal products of the form"Af�g�RLD b where �2A and b is an ultra�lter on B.

Proof. First, it's easy to see that "Af�g�RLD b are elements of ComplRLD(A;B). Also 0RLD(A;B)

is an element of ComplRLD(A;B)."Af�g�RLD b are atoms of ComplRLD(A;B) because they are atoms of RLD(A;B).It remains to prove that if f is an atom of ComplRLD(A;B) then f = "Af�g�RLD b for some

�2A and an ultra�lter b on B.Suppose f is a non-empty complete reloid. Then "Af�g �RLD b v f for some � 2 A and an

ultra�lter b on B. If f is an atom then f = "Af�g�RLD b. �

Obvious 7.57. ComplRLD(A;B) is an atomistic lattice.

Proposition 7.58. Compl f =F �

f j"Src ff�g j �2Src ffor every reloid f .

Proof. Let's denote R the right part of the equality to be proven.That R is a complete reloid follows from the equality

f j"Src ff�g="Src f f�g�RLD im¡f j"Src ff�g

�:

The only thing left to prove is that g vR for every complete reloid g such that g v f .Really let g be a complete reloid such that gv f . Then

g=Gf"Src ff�g�RLDG(�) j �2 Src f g

for some function G: Src f!F(Dst f).We have "Src ff�g�RLDG(�)= g j"Src ff�gvf j"Src ff�g. Thus g vR. �

Conjecture 7.59. Compl f uCompl g=Compl(f u g) for every f ; g 2RLD(A;B).

Theorem 7.60. ComplFR=

FhCompliR for every set R2PRLD(A;B) for every sets A, B.

Proof.

ComplG

R =G �¡GR�j"Af�g j �2A

= (proposition 4.194)G �G

ff j"Af�g j �2Ag j f 2R

=GhCompliR:

�

Lemma 7.61. Completion of a co-complete reloid is principal.

Proof. Let f be a co-complete reloid. Then there is a function F :Dst f!F(Src f) such that

f =GfF (�)�RLD "Dst f f�g j �2Dst f g:

So

Compl f =G �¡GfF (�)�RLD "Dst f f�g j �2Dst f g

�j"Src ff�g j � 2Src f

=G �¡G

fF (�)�RLD "Dst f f�g j �2Dst f g�u¡"Src f f�g�RLD 1F(Dst f)� j � 2Src f

= (*)G �G �

(F (�)�RLD "Dst ff�g)u¡"Src ff�g�RLD 1F(Dst f)� j �2Dst f

j � 2Src f

=G �G

f"Src ff�g�RLD "Dst ff�g j �2Dst f g j � 2 Src f ; "Src f f�gvF (�):

* proposition 4.194.Thus Compl f is principal. �

Theorem 7.62. ComplCoCompl f =CoComplCompl f =Cor f for every reloid f .

7.7 Complete reloids and completion of reloids 129

Proof. We will prove only ComplCoCompl f =Cor f . The rest follows from symmetry.From the lemma ComplCoCompl f is principal. It is obvious ComplCoCompl f v f . So to �nish

the proof we need to show only that for every principal reloid F v f we have F vComplCoCompl f .Really, obviously F vCoCompl f and thus F =ComplF vComplCoCompl f . �

Question 7.63. Is ComplRLD(A;B) a distributive lattice? Is ComplRLD(A;B) a co-brouwerianlattice?

Conjecture 7.64. If f is a complete reloid, then it is metacomplete.

Conjecture 7.65. If f is a metacomplete reloid, then it is complete.

Conjecture 7.66. Compl f = f n�¡Src f �RLD 1F(Dst f)� for every reloid f .

By analogy with similar properties of funcoids described above:

Proposition 7.67. For composable reloids f and g it holds

1. Compl(g � f)w (Compl g) � (Compl f);

2. CoCompl(g � f)w (CoCompl g) � (CoCompl f).

Proof.

1. (Compl g) � (Compl f)vCompl((Compl g) � (Compl f))vCompl(g � f).2. By duality. �

Conjecture 7.68. For composable reloids f and g it holds

1. Compl(g � f)= (Compl g) � f if f is a co-complete reloid;

2. CoCompl(f � g)= f �CoCompl g if f is a complete reloid;

3. CoCompl((Compl g) � f) =Compl(g � (CoCompl f))= (Compl g) � (CoCompl f);

4. Compl(g � (Compl f)) =Compl(g � f);5. CoCompl((CoCompl g) � f) =CoCompl(g � f).

130 Reloids

Chapter 8

Relationships between funcoids and reloids

8.1 Funcoid induced by a reloid

Every reloid f induces a funcoid (FCD)f 2FCD(Src f ;Dst f) by the following formulas (for everyX 2F(Src f), Y 2F(Dst f)):

X [(FCD)f ]Y,8F 2 xyGR f :X ["FCDF ]Y ;h(FCD)f iX =

lfh"FCDF iX j F 2 xyGR f g:

We should prove that (FCD)f is really a funcoid.

Proof. We need to prove that

X [(FCD)f ]Y,Y uh(FCD)f iX =/ 0F(Dst f),X u h(FCD)f¡1iY =/ 0F(Src f):

The above formula is equivalent to:

8F 2 xyGR f :X ["FCDF ]Y ,Y u

lfh"FCDF iX j F 2 xyGR f g=/ 0F(Dst f) ,

X ulfh"FCDF¡1iY j F 2 xyGR f g=/ 0F(Src f):

We have Y udfh"FCDF iX j F 2 xyGR f g=

dfY u h"FCDF iX j F 2 xyGR f g.

Let's denote W = fY u h"FCDF iX j F 2 xyGR f g.8F 2 xyGR f :X ["FCDF ]Y,8F 2 xyGR f :Y u h"FCDF iX =/ 0F(Dst f), 0F(Dst f)2/W .We need to prove only that 0F(Dst f)2/W,

dW =/ 0F(Dst f). (The rest follows from symmetry.)

This follows from the fact that W is a generalized �lter base.Let's prove that W is a generalized �lter base. For this it's enough to prove that V =

fh"FCDF iX j F 2 xyGR f g is a generalized �lter base. Let A; B 2 V that is A = h"FCDP iX ,B = h"FCDQiX where P ; Q 2 xyGR f . Then for C = h"FCD(P u Q)iX is true both C 2 Vand C vA;B. So V is a generalized �lter base and thus W is a generalized �lter base. �

Proposition 8.1. (FCD)"RLDf = "FCDf for every Rel-morphism f .

Proof. X [(FCD)"RLDf ]Y,8F 2xyGR"RLDf :X ["FCDF ]Y,X ["FCDf ]Y (for every X 2F(Src f),Y 2F(Dst f)). �

Theorem 8.2. X [(FCD)f ]Y,X �RLDY �/ f for every reloid f and X 2F(Src f), Y 2F(Dst f).

Proof.

X �RLDY �/ f , 8F 2GR f ; P 2X �RLDY:P �/ F, 8F 2GR f ;X 2X ; Y 2Y:X �Y �/ F, 8F 2GR f ;X 2X ; Y 2Y: "Src fX

�"FCD(Src f ;Dst f)F

�"Dst fY

, 8F 2GR f :X�"FCD(Src f ;Dst f)F

�Y

, X [(FCD)f ]Y:�

131

Theorem 8.3. (FCD)f =dh"FCDixyGR f for every reloid f .

Proof. Let a be an ultra�lter on Src f .h(FCD)f ia=

dfh"FCDF ia j F 2 xyGR f g by the de�nition of (FCD).

hdh"FCDixyGR f ia=

dfh"FCDF ia j F 2 xyGR f g by theorem 6.68.

So h(FCD)f ia= hdh"FCDixyGR f ia for every ultra�lter a. �

Lemma 8.4. For every two �lter bases S and T of morphisms Rel(U ;V ) and every set A�Ulh"RLDiS=

lh"RLDiT)

lf"V hF iA j F 2Sg=

lf"V hGiA j G2T g:

Proof. Letdh"RLDiS=

dh"RLDiT .

First let prove that fhF iA j F 2Sg is a �lter base. LetX;Y 2fhF iA j F 2Sg. ThenX=hFX iAand Y = hFY iA for some FX ; FY 2 S. Because S is a �lter base, we have S 3 FZ v FX u FY . SohFZiAvX uY and hFZiA2fhF iA j F 2Sg. So fhF iA j F 2Sg is a �lter base.

Suppose X 2df"V hF iA j F 2 Sg. Then there exists X 0 2 fhF iA j F 2 Sg where X w X 0

because fhF iA j F 2Sg is a �lter base. That is X 0= hF iA for some F 2S. There exists G2T suchthat GvF because T is a �lter base. Let Y 0= hGiA. We have Y 0vX 0vX; Y 02fhGiA j G2T g;Y 02

df"V hGiA j G2T g; X 2

df"V hGiA j G2T g. The reverse is symmetric. �

Lemma 8.5. fG�F j F 2GR f ;G2GR gg is a �lter base for every reloids f and g.

Proof. Let denote D= fG � F j F 2GR f ; G 2GR gg. Let A2D ^B 2D. Then A=GA �FA^B =GB � FB for some FA; FB 2GR f , GA; GB 2GR g. So A \ B � (GA \GB) � (FA \ FB) 2Dbecause FA\FB 2GR f and GA\GB 2GR g. �

Theorem 8.6. (FCD)(g � f) = ((FCD)g) � ((FCD)f) for every composable reloids f and g.

Proof.

h(FCD)(g � f)i�X =lf"Dst ghH iX j H 2GR(g � f)g

=l �

"Dst ghH iX j H 2GRlf"RLD(G�F ) j F 2 xyGR f ;G2 xyGR gg

:

Obviouslylf"RLD(G �F ) j F 2 xyGR f ;G2 xyGR gg =l

h"RLDixyGRlf"RLD(G �F ) j F 2 xyGR f ;G2 xyGR gg;

from this by lemma 8.4 (taking into account that

fG�F j F 2GR f ;G2GR ggand

GRlf"RLD(G�F ) j F 2 xyGR f ;G2 xyGR gg

are �lter bases)l �

"Dst ghH iX j H 2GRlf"RLD(G�F ) j F 2 xyGR f ;G2 xyGR gg

=l

f"Dst ghG�F iX j F 2GR f ;G2GR gg:

On the other side

h((FCD)g) � ((FCD)f)i�X = h(FCD)gih(FCD)f i�X= h(FCD)gi

lf"Dst ghF iX j F 2 xyGR f g

=l �

h"FCDGilf"Dst ghF iX j F 2 xyGR f g j G2 xyGR g

:

Let's prove that fhF iX j F 2 xyGR f g is a �lter base. If A; B 2 fhF iX j F 2 xyGR f g thenA= hF1iX , B = hF2iX where F1; F2 2 xyGR f . A \B � hF1 u F2iX 2 fhF iX j F 2 xyGR f g. SofhF iX j F 2 xyGR f g is really a �lter base.

132 Relationships between funcoids and reloids

By theorem 6.32 we have

h"FCDGilf"Dst ghF iX j F 2 xyGR f g=

lf"Dst ghGihF iX j F 2 xyGR f g:

So continuing the above equalities,

h((FCD)g) � ((FCD)f)i�X =l �l

f"Dst ghGihF iX j F 2 xyGR f g j G2 xyGR g

=lf"Dst ghGihF iX j F 2 xyGR f ;G2 xyGR gg

=lf"Dst ghG�F iX j F 2 xyGR f ;G2 xyGR gg:

Combining these equalities we get h(FCD)(g � f)i�X = h((FCD)g) � ((FCD)f)i�X for every setX 2P(Src f). �

Proposition 8.7. (FCD)idARLD= idAFCD for every �lter A.

Proof. Recall that idARLD=d �

"Base(A)idA j A2A. For every X ;Y 2F(Base(A)) we have:

X [(FCD)idARLD]Y,X �RLDY �/ idARLD,8A2A:X �RLDY �/ "RLD(Base(A);Base(A))idA,8A2A:X�"FCD(Base(A);Base(A))idA

�Y , 8A 2 A: X u Y �/ "Base(A)A, X u Y �/ A, X [idAFCD] Y (used

properties of generalized �lter bases). �

Proposition 8.8.

1. (FCD)f is a monovalued funcoid if f is a monovalued reloid.

2. (FCD)f is an injective funcoid if f is an injective reloid.

Proof. We will prove only the �rst as the second is dual. Let f be a monovalued reloid. Thenf � f¡1 v idRLD(Dst f); (FCD)(f � f¡1) v idFCD(Dst f); (FCD)f � ((FCD)f)¡1 v idFCD(Dst f) that is(FCD)f is a monovalued funcoid. �

Proposition 8.9. (FCD)(A�RLDB) =A�FCDB for every �lters A, B.

Proof. X [(FCD)(A�RLDB)]Y,8F 2 xyGR(A�RLDB):X ["FCDF ]Y (for every X 2F(Base(A)),Y 2F(Base(B))).

Evidently8F 2 xyGR(A�RLDB):X ["FCDF ]Y)8A2A; B 2B:X

�"FCD(Base(A);Base(B))(A�B)

�Y.

Let 8A 2A; B 2 B: X�"FCD(Base(A);Base(B))(A�B)

�Y . Then if F 2GR(A�RLD B), there are

A2A, B 2B such that F �A�B. So X ["FCDF ]Y .We have proved 8F 2 xyGR(A �RLD B): X ["FCDF ] Y , 8A 2 A; B 2 B:

X�"FCD(Base(A);Base(B))(A�B)

�Y .

Further 8A 2A; B 2B:X�"FCD(Base(A);Base(B))(A�B)

�Y,8A2A; B 2B:

¡X �/ "Base(A)A^

Y �/ "Base(B)B�,X �/ A^Y �/ B,X [A�FCDB]Y .

Thus X [(FCD)(A�RLDB)]Y,X [A�FCDB]Y. �

Proposition 8.10. dom (FCD)f = dom f and im (FCD)f = im f for every reloid f .

Proof. im (FCD)f = h(FCD)f i1F(Src f)=df"Dst f hF i(Src f) j F 2GR f g=

df"Dst fim F j F 2

GR f g=dh"Dst f ihimiGR f = im f .

dom (FCD)f = dom f is similar. �

Proposition 8.11. (FCD)(f u (A �RLD B)) = (FCD)f u (A �FCD B) for every reloid f andA2F(Src f) and B 2F(Dst f).

Proof. (FCD)(f u (A�RLDB)) = (FCD)(idBRLD � f � idARLD) = (FCD)idBRLD � (FCD)f � (FCD)idARLD=idBFCD� (FCD)f � idAFCD=(FCD)f u (A�FCDB). �

Corollary 8.12. (FCD)(f jA) = ((FCD)f)jA for every reloid f and a �lter A2F(Src f).

8.1 Funcoid induced by a reloid 133

Proposition 8.13. h(FCD)f iX = im(f jX) for every reloid f and a �lter X 2F(Src f).

Proof. im(f jX) = im(FCD)(f jX) = im(((FCD)f)jX)= h(FCD)f iX . �

Proposition 8.14. (FCD)f =F �

x�FCD y j x2 atomsF(Src f); y2 atomsF(Src f); x�RLD y�/ ffor

every reloid f .

Proof. (FCD)f =F �

x �FCD y j x 2 atomsF(Src f); y 2 atomsF(Dst f); x �FCD y �/ (FCD)f, but

x�FCD y�/ (FCD)f,x [(FCD)f ] y,x�RLD y�/ f , thus follows the theorem. �

8.2 Reloids induced by a funcoid

Every funcoid f 2 FCD(A;B) induces a reloid from A to B in two ways, intersection of outwardrelations and union of inward reloidal products of �lters:

(RLD)outf =lh"RLDixyGR f ;

(RLD)inf =GfA�RLDB j A2F(A);B 2F(B);A�FCDB v f g:

Theorem 8.15. (RLD)inf =F �

a�RLD b j a2 atomsF(A); b2 atomsF(B); a�FCD bv f.

Proof. It follows from theorem 7.21. �

Remark 8.16. It seems that (RLD)in has smoother properties and is more important than(RLD)out. (However see also the exercise below for (RLD)in not preserving identities.)

Proposition 8.17. GR "RLDf =GR "FCDf for every Rel-morphism f .

Proof. X 2GR "RLDf,X � f,X 2GR "FCDf . �

Proposition 8.18. (RLD)out"FCDf = "RLDf for every Rel-morphism f .

Proof. (RLD)out"FCDf =dh"RLDixyGR f = "RLDmin xyGR f = "RLDf taking into account the

previous proposition. �

Surprisingly, a funcoid is greater inward than outward:

Theorem 8.19. (RLD)outf v (RLD)inf for every funcoid f .

Proof. We need to prove

(RLD)outf vGfA�RLDB j A2F(Src f);B 2F(Dst f);A�FCDB v f g:

Let

K 2GfA�RLDB j A2F(Src f);B 2F(Dst f);A�FCDB v f g:

Then

K 2 "RLD(Src f ;Dst f)[fXA�YB j A2F(Src f);B 2F(Dst f);A�FCDB v f g

= (RLD)out"FCD[fXA�YB j A2F(Src f);B 2F(Dst f);A�FCDB v f g

= (RLD)outGf"FCD(XA�YB) j A2F(Src f);B 2F(Dst f);A�FCDB v f g

w (RLD)outG

atoms f= (RLD)outf

where XA2A, YB2B. So K 2 (RLD)outf . �

Theorem 8.20. (FCD)(RLD)inf = f for every funcoid f .


Proof. For every sets X 2P(Src f), Y 2P(Dst f)

X [(FCD)(RLD)inf ]�Y ,"Src fX �RLD "Dst fY �/ (RLD)inf ,

"RLD(Src f ;Dst f)(X �Y )�/G �

a�RLD b j a2 atomsF(A); b2 atomsF(B); a�FCD bv f, (*)

9a2 atomsF(A); b2 atomsF(B): (a�FCD bv f ^ av"Src fX ^ bv"Dst fY ) ,X [f ]�Y :

* proposition 4.215.Thus (FCD)(RLD)inf = f . �

Remark 8.21. The above theorem allows to represent funcoids as reloids.

Obvious 8.22. (RLD)in(A�FCDB) =A�RLDB for every �lters A, B.

Conjecture 8.23. (RLD)outidAFCD= idARLD for every �lter A.

Exercise 8.1. Prove that generally (RLD)inidAFCD =/ idARLD.

Conjecture 8.24. dom(RLD)inf = dom f and im(RLD)inf = im f for every funcoid f . [TODO:easy using products of ultra�lters?]

Proposition 8.25. dom(f jA)=Audom f for every reloid f and �lter A2F(Src f).

Proof. dom(f jA) =dom (FCD)(f jA)= dom ((FCD)f)jA=Au dom (FCD)f =Au dom f . �

Theorem 8.26. For every composable reloids f , g:


2. If im f v dom g then dom(g � f) =dom g.

Proof.

1. im(g � f)= im(FCD)(g � f)= im((FCD)g � (FCD)f)= im (FCD)g= im g.

2. Similar. �

Conjecture 8.27. (RLD)in(g � f)= (RLD)in g � (RLD)in f for every composable funcoids f and g.[TODO: Solved.]

Theorem 8.28. a �RLD b v (RLD)inf , a �FCD b v f for every funcoid f and a 2 atomsF(Src f),b2 atomsF(Dst f). [TODO: Move to �funcoidal reloids� section?]

Proof. a�FCD bv f) a�RLD bv (RLD)inf is obvious.a�RLD bv (RLD)inf) a�RLD b�/ (RLD)inf) a [(FCD)(RLD)inf ] b) a [f ] b) a�FCD bv f . �

Conjecture 8.29. If A�RLDBv(RLD)inf then A�FCDBv f for every funcoid f andA2F(Src f),B 2F(Dst f).

Theorem 8.30. GR (FCD)g �GR g for every reloid g.

Proof. Let K 2GR g. Then for every sets X 2PSrc g, Y 2PDst gX [K] Y ,X ["FCDK]�Y ,X [(FCD)"RLDK]�Y (X [(FCD)g]�Y .Thus "FCDK w (FCD)g that is K 2GR (FCD)g. �

Theorem 8.31. g � (A �RLD B) � f = h(FCD)f¡1iA �RLD h(FCD)giB for every reloids f , g and�lters A2F(Dst f), B 2F(Src g). [TODO: Similar proposition for funcoids?]

8.2 Reloids induced by a funcoid 135

Proof. g � (A �RLD B) � f =d �

"RLD(Src f ;Dst g)(G � (A � B) � F ) j F 2 GR f ; G 2 GR g;

A 2 A; B 2 B=d �

"RLD(Src f ;Dst g)(hF¡1iA � hGiB) j F 2 GR f ; G 2 GR g; A 2 A;B 2 B

=df"Src f hF¡1iA �RLD "Dst ghGiB j F 2 GR f ; G 2 GR g; A 2 A; B 2 Bg = (theorem

7.23)=df"Src f hF¡1iA j F 2 GR f ; A 2 Ag �RLD d f"Dst ghGiB j G 2 GR g; B 2 Bg =d �

"FCD(Dst f ;Src f)F¡1�"Dst fA j F 2 GR f ; A 2 A

�RLD d �

"FCD(Src g;Dst g)G�"Src gB j G 2

GR g; B 2 B=d �

"FCD(Dst f ;Src f)F¡1�A j F 2 GR f

�RLD d �

"FCD(Src g;Dst g)G�B j G 2

GR g= (by de�nition of (FCD))=h(FCD)f¡1iA�RLD h(FCD)giB. �

Corollary 8.32.

1. (A�RLDB) � f = h(FCD)f¡1iA�RLDB;

2. g � (A�RLDB) =A�RLD h(FCD)giB.

8.3 Galois connections between funcoids and reloids

Theorem 8.33. (FCD): RLD(A; B)! FCD(A; B) is the lower adjoint of (RLD)in: FCD(A; B)!RLD(A;B) for every sets A, B.

Proof. Because (FCD) and (RLD)in are trivially monotone, it's enough to prove (for every f 2RLD(A;B), g 2 FCD(A;B))

f v (RLD)in(FCD)f and (FCD)(RLD)ingv g:

The second formula follows from the fact that (FCD)(RLD)ing= g.

(RLD)in(FCD)f =G �a�RLD b j a2 atomsF(A); b2 atomsF(B); a�FCD bv (FCD)f

=G �

a�RLD b j a2 atomsF(A); b2 atomsF(B); a [(FCD)f ] b

=G �a�RLD b j a2 atomsF(A); b2 atomsF(B); a�RLD b�/ f

wG �

p2 atoms(a�RLD b) j a2 atomsF(A); b2 atomsF(B); p�/ f

=G �p2 atomsRLD(A;B) j p�/ f

=G

fp j p2 atoms f g= f:

�

Corollary 8.34.

1. (FCD)FS=

Fh(FCD)iS if S 2PRLD(A;B).

2. (RLD)indS=

dh(RLD)iniS if S 2PFCD(A;B).

Proposition 8.35. (RLD)in(f u (A�FCDB)) = ((RLD)inf)u (A�RLDB) for every funcoid f andA2F(Src f), B 2F(Dst f).

Proof. (RLD)in(f u (A�FCDB))= ((RLD)inf)u (RLD)in(A�FCDB)= ((RLD)inf)u (A�RLDB). �

Corollary 8.36. (RLD)in(f jA)= ((RLD)inf)jA.

Conjecture 8.37. (RLD)in is not a lower adjoint (in general).

Conjecture 8.38. (RLD)out is neither a lower adjoint nor an upper adjoint (in general).

Exercise 8.2. Prove that cardFCD(A;B) = 22maxfA;B g

if A or B is an in�nite set (provided that A and B arenonempty).

Lemma 8.39. "FCDf(x; y)gv (FCD)g,"RLDf(x; y)gv g for every reloid g.


Proof. "FCDf(x; y)g v (FCD)g , "FCDf(x; y)g �/ (FCD)g , fxg [(FCD)g]� fyg , "RLDf(x;y)g�/ g,"RLDf(x; y)gv g. �

Theorem 8.40. Cor (FCD)g=(FCD)Cor g for every reloid g.

Proof. Cor (FCD)g =Ff"FCDf(x; y)g j "FCDf(x; y)g v (FCD)gg =

Ff"FCDf(x; y)g j "RLDf(x;

y)gv gg=Ff(FCD)"RLDf(x; y)g j "RLDf(x; y)gv gg=(FCD)

Ff"RLDf(x; y)g j "RLDf(x; y)gv gg=

(FCD)Cor g. �

Conjecture 8.41. For every funcoid g

1. Cor (RLD)ing=(RLD)inCor g;

2. Cor (RLD)outg=(RLD)outCor g.

8.4 Funcoidal reloidsDe�nition 8.42. I call funcoidal such a reloid � that

X �RLDY �/ �)9X 02F(Base(X )) nf0g;Y 02F(Base(Y)) nf0g: (X 0vX ^Y 0vY ^X 0�RLDY 0v �)

for every X 2F(Src �), Y 2F(Dst �).

Proposition 8.43. A reloid � is funcoidal i� x�RLD y�/ �)x�RLD y v � for every ultra�lters xand y on respective sets.

Proof.

). x�RLD y�/ �)9X 02 atomsx; Y 02 atoms y:X 0�RLDY 0v �)x�RLD y v �.

(. X �RLD Y �/ �)9x 2 atoms X ; y 2 atoms Y : x �RLD y �/ �)9x 2 atoms X ; y 2 atoms Y:x �RLD y v � ) 9X 0 2 F(Base(X )) n f0g; Y 0 2 F(Base(Y)) n f0g: (X 0 v X ^ Y 0 v Y ^X 0�RLDY 0v �). �

Proposition 8.44. (RLD)in(FCD)f =F �

a�RLD b j a2 atomsF(Src �); b2 atomsF(Dst �); a�RLD b�/f.

Proof. (RLD)in(FCD)f =F �

a�RLD b j a2 atomsF(Src f); b2 atomsF(Dst f); a�FCD bv (FCD)f=F �

a�RLD b j a 2 atomsF(Src f); b 2 atomsF(Dst f); a [(FCD)f ] b=F �

a�RLD b j a 2 atomsF(Src f);b2 atomsF(Src f); a�RLD b�/ f

. �

De�nition 8.45. I call (RLD)in(FCD)f funcoidization of a reloid f .

Lemma 8.46. (RLD)in(FCD)f is funcoidal for every reloid f .

Proof. x�RLD y�/ (RLD)in(FCD) f)x�RLD yv (RLD)in(FCD)f . �

Theorem 8.47. (RLD)in is a bijection from FCD(A;B) to the set of funcoidal reloids from A to B.

Proof. Let f 2FCD(A;B). Prove that (RLD)in f is funcoidal.Really (RLD)in f = (RLD)in (FCD)(RLD)inf and thus we can use the lemma stating that it is

funcoidal.It remains to prove (RLD)in(FCD) f = f for a funcoidal reloid f . ((FCD)(RLD)in g= g for every

funcoid g is already proved above.)(RLD)in(FCD) f =

F �x �RLD y j x 2 atomsF(Src �); y 2 atomsF(Dst �); x �RLD y �/ f

=F �

p 2 atoms(x �RLD y) j x 2 atomsF(Src �); y 2 atomsF(Dst �); x �RLD y �/ f=F �

p 2atoms(x�RLD y) j x2 atomsF(Src �); y 2 atomsF(Dst �); x�RLD yv f

=F

atoms f = f . �

Corollary 8.48. Funcoidal reloids are convex.

Proof. Every (RLD)in f is obviously convex. �

8.4 Funcoidal reloids 137

Chapter 9

On distributivity of composition with a prin-cipal reloid

9.1 Decomposition of composition of binary relations

Remark 9.1. Sorry for an unfortunate choice of terminology: �composition� and �decomposition�are unrelated.

The idea of the proof below is that composition of binary relations can be decomposed into twooperations: and dom:

g f = f((x; z); y) j xfy ^ ygzg:

Composition of binary relations can be decomposed: g � f =dom(g f).It can be decomposed even further: g f =�0 f \�1 g where

�0 f = f((x; z); y) j xfy; z 2fg and �1 f = f((x; z); y) j yfz; x2fg:

(Here f is the Grothendieck universe.)Now we will do a similar trick with reloids.

9.2 Decomposition of composition of reloids

A similar thing for reloids:

g � f =l �

"RLD(Src f ;Dst g)(G�F ) j F 2GR f ;G2GR g=l �

"RLD(Src f ;Dst g)dom(GF ) j F 2

GR f ;G2GR g:

Lemma 9.2. fGF j F 2GR f ;G2GR gg is a �lter base.

Proof. Let P ; Q 2 fGF j F 2GR f ; G 2GR gg. Then P =G0F0, Q=G1F1 for some F0;F12 f , G0; G12 g. Then F0\F12GR f , G0\G12GR g and thus

P \Q� (F0\F1) (G0\G1)2fGF j F 2GR f ;G2GR gg: �

Corollary 9.3.�"RLD(Src f�Dst g;f)(GF ) j F 2GR f ;G2GR g

is a generalized �lter base.

Proposition 9.4. g � f = domd �

"RLD(Src f�Dst g;f)(GF ) j F 2GR f ;G2GR g.

Proof. "RLD(Src f ;Dst g)dom(G F )w domd �

"RLD(Src f�Dst g;f)(G F ) j F 2GR f ; G 2GR g.

Thus

g � f w doml �

"RLD(Src f�Dst g;f)(GF ) j F 2GR f ;G2GR g:

Let X 2domd �

"RLD(Src f�Dst g;f)(GF ) j F 2GR f ;G2GR g. Then there exist Y such that

X�Y 2GRd �

"RLD(Src f�Dst g;f)(GF ) j F 2GR f ;G2GR g. So because it is a generalized

�lter base X �Y �GF for some F 2GR f , G2GR g. Thus X 2dom(GF ), X 2GR(g � f). �

139

We can de�ne g f for reloids f , g:

g f = fGF j F 2GR f ;G2GR gg:Then

g � f =l

"RLD(Src f ;Dst g)�hdomi(g f)= doml

"RLD(Src f�Dst g;f)�(g f):

9.3 Lemmas for the main result

Lemma 9.5. (g f)\ (h f)= (g\ h) f for binary relations f , g, h.

Proof. (g \ h) f =�0 f \�1 (g \ h) = �0 f \ (�1 g \�1 h) = (�0 f \�1 g) \ (�0 f \�1 h) =(g f)\ (h f): �

Lemma 9.6. Let F = "RLD(SrcF ;DstF )f be a principal reloid, T is a set of reloids from DstF to aset V .

l �"RLD(Src f�V ;f)(G f) j G2GR

GT=G �l

"RLD(Src f�V ;f)�(GF ) j G2T

:

Proof.d �

"RLD(Src f�V ;f)(G f) j G2GRFTwF �d


is

obvious.Let K 2

F �d "RLD(Src f�V ;f)

�(GF ) j G2T

. Then for each G2T

K 2l

"RLD(Src f�V ;f)�(GF );

K 2d

"RLD(Src f�V ;f)�f¡ f j ¡2Gg.

Then K 2f¡ f j ¡2GRGg by properties of generalized �lter bases.K 2f(¡0\ ::: \¡n) f j n2N ;¡i2Gg= f(¡0 f)\ ::: \ (¡n f) j n2N ;¡i2Gg.8G2T :K � (¡G;0 f)\ ::: \ (¡G;n f) for some n2N ;¡G;i2G.K � (¡0 f)\ ::: \ (¡n f) where ¡i=

Sg2G ¡g;i 2GR

FT .

K 2f(¡0 f)\ ::: \ (¡n f) j n2N g.So K 2 f(¡00 f) \ ::: \ (¡n

0 f) j n 2 N ; ¡i0 2 GR

FT g = f(¡00 \ ::: \ ¡n

0 ) f j n 2 N ;

¡i02GR

FT g=

d "RLD(Src f�V ;f)

�fG f j G2GR

FT g. �

9.4 Proof of the main result

Theorem 9.7. (FT ) �F =

FfG�F j G2T g for every principal reloid F = "RLD(Src f ;Dst g)f and

a set T of reloids from DstF to some set V . (In other words principal reloids are co-metacompleteand thus also metacomplete by duality.)

Proof. ¡GT��F =

l "RLD(Src f ;V )

�hdomi

¡¡GT�F

�= dom

l "RLD(Src f�V ;f)

�¡¡GT�F

�= dom

l "RLD(Src f�V ;f)

��G f j G2GR

GT;G

fG �F j G2T g =G �l

"RLD(Src f ;V )�hdomi(GF ) j G2T

=G �

doml


= dom

G �l "RLD(Src f�V ;f)

�(GF ) j G2T

:

It's enough to provel �

"RLD(Src f�V ;f)(G f) j G2GRG

T=G �l


140 On distributivity of composition with a principal reloid

but this is the statement of the lemma. �

9.5 Embedding reloids into funcoids

De�nition 9.8. Let f be a reloid. The funcoid

�f 2FCD(P(Src f � Src f);P(Dst f �Dst f))

is de�ned by the formulas:

h�f ix= f � x and h� f¡1iy= f¡1 � y

where x are endoreloids on Src f and y are endoreloids on Dst f .

Proposition 9.9. It is really a funcoid (if we equate reloids x and y with corresponding �lters onCartesian products of sets).

Proof. y�/ h�f ix, y�/ f � x, f¡1 � y�/ x,h� f¡1iy�/ x. �

Corollary 9.10. (�f)¡1= � f¡1.

De�nition 9.11. It can be continued to arbitrary funcoids x having destination Src f by theformula h�� f ix= h�f iidSrc f � x= f � x.

Proposition 9.12. � is an injection.

Proof. Consider x= idSrc f. �

Proposition 9.13. �(g � f)= (�g) � (�f).

Proof. h�(g � f)ix = g � f � x = h�gih�f ix = (h�gi � h�f i)x. Thus h�(g � f)i = h�gi � h�f i =h(�g) � (�f)i and so �(g � f)= (�g) � (�f). �

Theorem 9.14. �FF =

Fh�iF for a set F of reloids.

Proof. It's enough to prove h�FF i�X = h

Fh�iF i�X for a set X .

Really, h�FF i�X = h�

FF i"X =

FF � "X =

Fff � "X j f 2 F g=

Ffh�f i"X j f 2 F g=

hFf�f j f 2F gi"X = h

Fh�iF i�X . �

Conjecture 9.15. �dF =

dh�iF for a set F of reloids.

Proposition 9.16. �idRLD(A)= idFCD(P(A�A)).

Proof.�idRLD(A)

�x= idRLD(A) � x=x. �

We can try to develop further theory by applying embedding of reloids into funcoids forresearching of properties of reloids.

Theorem 9.17. Reloid f is monovalued i� funcoid �f is monovalued.

Proof. �f is monovalued,(�f) � (�f)¡1 v 1Dst �f , �(f � f¡1) v 1Dst f , �(f � f¡1) vidFCD(P(Dst f�Dst f)), �(f � f¡1)v �idRLD(Dst f), f � f¡1v idRLD(Dst f), f is monovalued. �

9.5 Embedding reloids into funcoids 141

Chapter 10Continuous morphisms

This chapter uses the apparatus from the section �Partially ordered dagger categories�.

10.1 Traditional de�nitions of continuityIn this section we will show that having a funcoid or reloid "f corresponding to a function f wecan express continuity of it by the formula "f � �v � � "f (or similar formulas) where � and � aresome spaces.

10.1.1 PretopologyLet (A;clA) and (B;clB) be preclosure spaces. Then by de�nition a function f :A!B is continuousi� f clA(X) � clB(fX) for every X 2 PA. Let now � and � be endofuncoids correspondingcorrespondingly to clA and clB. Then the condition for continuity can be rewritten as

"FCD(Ob �;Ob �)f � �v � � "FCD(Ob �;Ob �)f:

10.1.2 Proximity spacesLet � and � be proximity spaces (which I consider a special case of endofuncoids). By de�nitiona function f is a proximity-continuous map (also called equicontinuous) from � to � i�

8X;Y 2P(Ob �): (X [�]�Y )hf iX [�]� hf iY ):

Equivalently transforming this formula we get (writing " instead of "FCD(Ob �;Ob �) for brevity):

8X;Y 2P(Ob �):¡X [�]�Y )hf iY u h� i�hf iX =/ 0F(Dst �)�;

8X;Y 2P(Ob �):¡X [�]�Y )hf iY u h� � "f i�X =/ 0F(Dst �)�;

8X;Y 2P(Ob �): (X [�]�Y )X [� � "f ]� hf iY );8X;Y 2P(Ob �): (X [�]�Y )hf iY [(� � "f)¡1]�X);8X;Y 2P(Ob �): (X [�]�Y )hf iY [("f)¡1 � �¡1]�X);

8X;Y 2P(Ob �):¡X [�]�Y )"Ob �X u h("f)¡1 � �¡1i�hf iY =/ 0F(Ob �)�;

8X;Y 2P(Ob �):¡X [�]�Y )"Ob �X u h("f)¡1 � �¡1 � "f i�Y =/ 0F(Ob �)�;

8X;Y 2P(Ob �): (X [�]�Y )Y [("f)¡1 � �¡1 � "f ]�X);8X;Y 2P(Ob �): (X [�]�Y )X [("f)¡1 � � � "f ]�Y );

�v ("f)¡1 � � � "f:

So a function f is proximity continuous i� �v¡"FCD(Ob �;Ob �)f

�¡1 � � � "FCD(Ob �;Ob �)f .

10.1.3 Uniform spacesUniform spaces are a special case of endoreloids.

Let � and � be uniform spaces. By de�nition a function f is a uniformly continuous map from� to � i�

8"2GR �9� 2GR �8(x; y)2 �: (fx; fy)2 ":

143

Equivalently transforming this formula we get:

8"2GR �9� 2GR �8(x; y)2 �: f(fx; fy)g� ";8"2GR �9� 2GR �8(x; y)2 �: f � f(x; y)g � f¡1� ";

8"2GR �9� 2GR �: f � � � f¡1� ";8"2GR �: "RLD(Ob �;Ob �)f � � �

¡"RLD(Ob �;Ob �)f

�¡1v"RLD(Ob �;Ob �)";

"RLD(Ob �;Ob �)f � � �¡"RLD(Ob �;Ob �)f

�¡1v �:So a function f is uniformly continuous i� "RLD(Ob �;Ob �)f � � �

¡"RLD(Ob �;Ob �)f

�¡1v �.10.2 Our three de�nitions of continuityI have expressed di�erent kinds of continuity with simple algebraic formulas hiding the complexityof traditional epsilon-delta notation behind a smart algebra. Let's summarize these three algebraicformulas:

Let � and � be endomorphisms of some partially ordered precategory. Continuous functionscan be de�ned as these morphisms f of this precategory which conform to the following formula:

f 2C(�; �), f 2Mor(Ob �;Ob �)^ f � �v � � f:

If the precategory is a partially ordered dagger precategory then continuity also can be de�ned intwo other ways:

f 2C0(�; �), f 2Mor(Ob �;Ob �)^ �v f y � � � f ;f 2C00(�; �), f 2Mor(Ob �;Ob �)^ f � � � f yv �:

Remark 10.1. In the examples (above) about funcoids and reloids the �dagger functor� is thereverse of a funcoid or reloid, that is f y= f¡1.

Proposition 10.2. Every of these three de�nitions of continuity forms a wide sub-precategory(wide subcategory if the original precategory is a category).

Proof.

C. Let f 2C(�; �), g 2C(�;�). Then f � �v � � f , g � � v� � g; g � f � �v g � � � f v� � g � f .So g � f 2C(�;�). 1Ob �2C(�; �) is obvious.

C0. Let f 2C0(�; �), g 2C0(�;�). Then �v f y � � � f , � v gy � � � g;

�v f y � gy � � � g � f ; �v (g � f)y � � � (g � f):

So g � f 2C0(�;�). 1Ob �2C0(�; �) is obvious.

C00. Let f 2C00(�; �), g 2C00(�;�). Then f � � � f yv �, g � � � gyv �;

g � f � � � f y � gyv�; (g � f) � � � (g � f)yv�:

So g � f 2C00(�;�). 1Ob �2C00(�; �) is obvious. �

Proposition 10.3. For a monovalued morphism f of a partially ordered dagger category and itsendomorphisms � and �

f 2C0(�; �)) f 2C(�; �)) f 2C00(�; �):

Proof. Let f 2C0(�;�). Then �v f y�� f ; f � �v f � f y�� f v1Dst f �� f =� � f ; f 2C(�;�).Let f 2C(�; �). Then f � �v � � f ; f � � � f yv � � f � f yv � � 1Dst f = �; f 2C00(�; �). �

Proposition 10.4. For an entirely de�ned morphism f of a partially ordered dagger category andits endomorphisms � and �

f 2C00(�; �)) f 2C(�; �)) f 2C0(�; �):

144 Continuous morphisms

Proof. Let f 2C00(�;�). Then f � �� f yv�; f � �� f y� f v� � f ; f � ��1Src f v� � f ; f � �v� � f ;f 2C(�; �).

Let f 2C(�; �). Then f � �v � � f ; f y � f � �v f y � � � f ; 1Src � � �v f y � � � f ; �v f y � � � f ;f 2C0(�; �). �

For entirely de�ned monovalued morphisms our three de�nitions of continuity coincide:

Theorem 10.5. If f is a monovalued and entirely de�ned morphism of a partially ordered daggerprecategory then

f 2C0(�; �), f 2C(�; �), f 2C00(�; �):

Proof. From two previous propositions. �The classical general topology theorem that uniformly continuous function from a uniform

space to an other uniform space is proximity-continuous regarding the proximities generated bythe uniformities, generalized for reloids and funcoids takes the following form:

Theorem 10.6. If an entirely de�ned morphism of the category of reloids f 2C00(�; �) for someendomorphisms � and � of the category of reloids, then (FCD)f 2C0((FCD)�; (FCD)�).

Exercise 10.1. I leave a simple exercise for the reader to prove the last theorem.

10.3 Continuity of a restricted morphismConsider some partially ordered semigroup. (For example it can be the semigroup of funcoids orsemigroup of reloids on some set regarding the composition.) Consider also some lattice (lattice ofobjects). (For example take the lattice of set theoretic �lters.)

We will map every object A to so called restricted identity element IA of the semigroup (forexample restricted identity funcoid or restricted identity reloid). For identity elements we willrequire

1. IA � IB= IAuB;

2. f � IAv f ; IA � f v f .

In the case when our semigroup is �dagger� (that is is a dagger precategory) we will require also(IA)

y= IA.We can de�ne restricting an element f of our semigroup to an object A by the formula f jA=

f � IA.We can de�ne rectangular restricting an element f of our semigroup to objects A and B as

IB � f � IA. Optionally we can de�ne direct product A�B of two objects by the formula (true forfuncoids and for reloids):

f u (A�B)= IB � f � IA:

Square restricting of an element f to an object A is a special case of rectangular restricting andis de�ned by the formula IA � f � IA (or by the formula f u (A�A)).

Theorem 10.7. For every elements f , �, � our semigroup and an object A

1. f 2C(�; �)) f jA2C(IA � � � IA; �);2. f 2C0(�; �)) f jA2C0(IA� � � IA; �);3. f 2C00(�; �)) f jA2C00(IA � � � IA; �).

(Two last items are true for the case when our semigroup is dagger.)

Proof.1. f jA2C(IA � � � IA; �) , f jA�IA � � � IA v � � f jA,f � IA � IA � � � IA v � � f jA,

f � IA � � � IAv � � f � IA( f � IA� �v � � f( f � �v � � f, f 2C(�; �).

2. f jA2C0(IA � � � IA; �), IA � � � IAv (f jA)y � � � f jA(IA � � � IAv (f � IA)y � � � f � IA,IA � � � IAv IA � f y � � � f � IA( �v f y � � � f, f 2C0(�; �).

3. f jA2C00(IA � � � IA; �), f jA�IA � � � IA � (f jA)yv �, f � IA � IA � � � IA � IA � f yv �,f � IA � � � IA � f yv �( f � � � f yv �, f 2C00(�; �). �

10.3 Continuity of a restricted morphism 145

Chapter 11

Connectedness regarding funcoids andreloids

De�nition 11.1. I will call endoreloids and endofuncoids reloids and funcoids with the samesource and destination.[TODO: Move above �continuity� chapter.]

11.1 Some lemmas

Lemma 11.2. If :(A [f ]� B) ^ A [ B 2 dom f t im f then f is closed on "UA for a funcoidf 2FCD(U ;U) for every sets U and A; B 2PU .

Proof. Let A[B 2 dom f t im f . :(A [f ]�B),"UB u hf i"UA=0F(U)) (dom f t im f)u"UB uhf i�A=0F(U)) ((dom f t im f) n "UA)u hf i�A=0F(U),hf i�Av"UA. �

Corollary 11.3. If :(A [f ]�B)Â[B 2dom f t im f then f is closed on "U(A nB) for a funcoidf 2FCD(U ;U) for every sets U and A; B 2PU .

Proof. Let :(A [f ]� B) ^ A [ B 2 dom f t im f . Then :((A n B) [f ]� B) ^ (A n B) [ B 2dom f t im f . �

Lemma 11.4. If :(A [f ]�B)Â[B 2dom f t im f then :(A [fn]�B) for every whole positive n.

Proof. Let :(A [f ]� B) ^ A [ B 2 dom f t im f . From the above lemma hf i�A v "UA."UB u hf i"UA = 0F(U), consequently hf i�A v "U(A n B). Because (by the above corollary)f is closed on "U(A n B), then hf ihf i"UA v "U(A n B), hf ihf ihf i"UA v "U(A n B), etc. Sohfni"UAv"U(A nB), "UB�hfni"UA, :(A [fn]�B). �

11.2 Endomorphism series

De�nition 11.5. S1(�)= �t �2t �3t ::: for an endomorphism � of a precategory with countablejoin of morphisms (that is join de�ned for every countable set of morphisms).

De�nition 11.6. S(�) = �0t S1(�) = �0t �t �2t �3t ::: where �0=1Ob � (identity morphismfor the object Ob �) where Ob � is the object of endomorphism � for an endomorphism � of acategory with countable join of morphisms.

I call S1 and S endomorphism series .We will consider the collection of all binary relations (on a set f), as well as the collection of

all funcoids and the collection of all reloids on a �xed set, as categories with single object f andthe identity morphisms idf, idFCD(f), idRLD(f).

Proposition 11.7. The relation S(�) is transitive for the category of binary relations.

147

Proof.

S(�) �S(�) = �0tS(�)t � �S(�)t �2 �S(�)t :::= (�0t �1t �2t :::)t (�1t �2t �3t :::)t (�2t �3t �4t :::)= �0t �1t �2t :::= S(�):

�

11.3 Connectedness regarding binary relations

Before going to research connectedness for funcoids and reloids we will excurse into the basic specialcase of connectedness regarding binary relations on a set f.

De�nition 11.8. A set A is called (strongly) connected regarding a binary relation � when

8X 2P(dom �) n f;g; Y 2P(im �) n f;g: (X [Y =A)X [�]Y ):

Let f be a set.

De�nition 11.9. Path between two elements a; b2f in a set A�f through binary relation � isthe �nite sequence x0 :::xn where x0=a, xn= b for n2N and xi (�\A�A)xi+1 for every i=0; :::;n¡ 1. n is called path length .

Proposition 11.10. There exists path between every element a2f and that element itself.

Proof. It is the path consisting of one vertex (of length 0). �

Proposition 11.11. There is a path from element a to element b in a set A through a binaryrelation � i� a (S(�\A�A)) b (that is (a; b)2S(�\A�A)).

Proof.

). If a path from a to b exists, then fbg � h(� \ A � A)nifag where n is the path length.Consequently fbg� hS(�\A�A)ifag; a (S(�\A�A)) b.

(. If a (S(�\A�A)) b then there exists n 2N such that a (�\A�A)n b. By de�nition ofcomposition of binary relations this means that there exist �nite sequence x0 ::: xn wherex0= a, xn= b for n2N and xi (�\A�A)xi+1 for every i=0; :::; n¡ 1. That is there is apath from a to b. �

Theorem 11.12. The following statements are equivalent for a binary relation � and a set A:

1. For every a; b2A there is a path between a and b in A through �.

2. S(�\ (A�A))�A�A.

3. S(�\ (A�A))=A�A.4. A is connected regarding �.

Proof.

(1))(2). Let for every a; b 2 A there is a path between a and b in A through �. Thena (S(�\A�A)) b for every a; b2A. It is possible only when S(�\ (A�A))�A�A.

(3))(1). For every two vertices a and b we have a (S(�\A�A)) b. So (by the previoustheorem) for every two vertices a and b there exists a path from a to b.

(3))(4). Suppose :(X [�\ (A�A)]Y ) for some X;Y 2Pfnf;g such that X [Y =A. Thenby a lemma :(X [(�\ (A�A))n]Y ) for every n2N . Consequently :(X [S(�\ (A�A))]Y ).So S(�\ (A�A))=/ A�A.

148 Connectedness regarding funcoids and reloids

(4))(3). If hS(�\ (A�A))ifvg=A for every vertex v then S(�\ (A�A))=A�A. Considerthe remaining case when V =

defhS(�\ (A�A))ifvg�A for some vertex v. Let W =A n V .If cardA= 1 then S(� \ (A�A))� idA=A�A; otherwise W =/ ;. Then V [W =A andso V [�]W what is equivalent to V [�\ (A�A)]W that is h�\ (A�A)iV \W =/ ;. This isimpossible because h�\(A�A)iV = h�\(A�A)ihS(�\(A�A))iV =hS1(�\ (A�A))iV �hS(�\ (A�A))iV =V .

(2))(3). Because S(�\ (A�A))�A�A. �

Corollary 11.13. A set A is connected regarding a binary relation � i� it is connected regarding�\ (A�A).

De�nition 11.14. A connected component of a set A regarding a binary relation F is a maximalconnected subset of A.

Theorem 11.15. The set A is partitioned into connected components (regarding every binaryrelation F ).

Proof. Consider the binary relation a�b,a (S(F ))b^b (S(F ))a. � is a symmetric, re�exive, andtransitive relation. So all points of A are partitioned into a collection of sets Q. Obviously eachcomponent is (strongly) connected. If a setR�A is greater than one of that connected componentsA then it contains a point b2B where B is some other connected component. Consequently R isdisconnected. �

Proposition 11.16. A set is connected (regarding a binary relation) i� it has one connectedcomponent.

Proof. Direct implication is obvious. Reverse is proved by contradiction. �

11.4 Connectedness regarding funcoids and reloids

De�nition 11.17. S1�(�)=df"RLDS1(M) j M 2 xyGR �g for an endoreloid �.

De�nition 11.18. Connectivity reloid S�(�) for an endoreloid � is de�ned as follows:

S�(�)=lf"RLDS(M) j M 2 xyGR �g:

Remark 11.19. Do not mess the word connectivity with the word connectedness which meansbeing connected.11.1

Proposition 11.20. S�(�)= idRLD(Ob �)tS1�(�) for every endoreloid �.

Proof. By the proposition 4.190. �

Proposition 11.21. S�(�)=S(�) if � is a principal reloid.

Proof. S�(�) =dfS(�)g=S(�). �

De�nition 11.22. A �lter A 2 F(Ob �) is called connected regarding an endoreloid � whenS�(�u (A�RLDA))wA�RLDA.

Obvious 11.23. A �lter A 2 F(Ob �) is connected regarding an endoreloid � i� S�(� u(A�RLDA))=A�RLDA.

11.1. In some math literature these two words are used interchangeably.

11.4 Connectedness regarding funcoids and reloids 149

De�nition 11.24. A �lter A2F(Ob �) is called connected regarding an endofuncoid � when

8X ;Y 2F(Ob �) n�0F(Ob �): (X tY =A)X [�]Y):

Proposition 11.25. Let A be a set. The �lter "Ob �A is connected regarding an endofuncoid � i�

8X;Y 2P(Ob �) n f;g: (X [Y =A)X [�]�Y ):

Proof.

). Obvious.

(. It follows from co-separability of �lters. �

Theorem 11.26. The following are equivalent for every set A and binary relation � on a set U :

1. A is connected regarding binary relation �.

2. "UA is connected regarding "RLD(U ;U)�.

3. "UA is connected regarding "FCD(U;U)�.

Proof.

(1),(2). S�¡"RLD(U;U)�u ("UA�RLD "UA)

�=S�

¡"RLD(U ;U)(� \ (A�A))

�= "RLD(U ;U)S(�\

(A � A)). So S�¡"RLD(U;U)� u ("UA �RLD "UA)

�w "UA �RLD "UA , "RLD(U;U)S(� \

(A�A))w"RLD(U;U)(A�A) = "UA�RLD "UA.(1),(3). It follows from the previous proposition. �

Next is conjectured a statement more strong than the above theorem:

Conjecture 11.27. Let A be a �lter on a set U and F is a binary relation on U .A is connected regarding "FCD(U;U)F i� A is connected regarding "RLD(U ;U)F .

Obvious 11.28. A �lter A is connected regarding a reloid � i� it is connected regarding the reloid�u (A�RLDA).

Obvious 11.29. A �lter A is connected regarding a funcoid � i� it is connected regarding thefuncoid �u (A�FCDA).

Theorem 11.30. A �lter A is connected regarding a reloid f i� A is connected regarding everyF 2 h"RLDixyGR f .

Proof.

). Obvious.

(. A is connected regarding "RLDF i� S(F ) =F 0tF 1tF 2t ::: 2A�RLDA.S�(f)=

df"RLDS(F ) j F 2 xyGR f gw

dfA�RLDA j F 2 xyGR f g=A�RLDA. �

Conjecture 11.31. A �lter A is connected regarding a funcoid f i� A is connected regardingevery F 2 h"FCDixyGR f .

The above conjecture is open even for the case when A is a principal �lter.

Conjecture 11.32. A �lter A is connected regarding a reloid f i� it is connected regarding thefuncoid (FCD)f .

The above conjecture is true in the special case of principal �lters:

Proposition 11.33. A �lter "Ob �A (for a set A) is connected regarding an endoreloid f i� it isconnected regarding the endofuncoid (FCD)f .

150 Connectedness regarding funcoids and reloids

Proof. "Ob �A is connected regarding a reloid f i� A is connected regarding every F 2xyGR f thatis when (taken into account that connectedness for "RLDF is the same as connectedness of "FCDF )

8F 2 xyGR f8X ;Y 2F(Ob f) n�0F(Ob f): (X tY = "Ob fA)X ["FCDF ] Y) ,

8X ;Y 2F(Ob f) n�0F(Ob f)8F 2 xyGR f : (X tY = "Ob fA)X ["FCDF ] Y) ,

8X ;Y 2F(Ob f) n�0F(Ob f): (X tY = "Ob fA)8F 2 xyGR f :X ["FCDF ] Y) ,

8X ;Y 2F(Ob f) n�0F(Ob f): (X tY = "Ob fA)X [(FCD)f ] Y)

that is when the set "Ob fA is connected regarding the funcoid (FCD)f . �

Conjecture 11.34. A set A is connected regarding an endofuncoid � i� for every a; b2A thereexists a totally ordered set P �A such that minP = a, maxP = b and

8q 2P n fbg: fx2P j x6 qg [�]� fx2P j x> qg:Weaker condition:

8q2P nfbg:fx2P j x6 qg [�]� fx2P j x> qg_8q2P n fag: fx2P j x< qg [�]� fx2P j x> qg:

11.5 Algebraic properties of S and S�

Theorem 11.35. S�(S�(f))=S�(f) for every endoreloid f .

Proof. S�(S�(f)) =df"RLDS(R) j R 2 xyGR S�(f)g v

df"RLDS(R) j R 2 fS(F ) j F 2

xyGR f gg=df"RLDS(S(R)) j R2 xyGR f g=

df"RLDS(R) j R2 xyGR f g=S�(f).

So S�(S�(f))vS�(f). That S�(S�(f))wS�(f) is obvious. �

Corollary 11.36. S�(S(f)) =S(S�(f))=S�(f) for every endoreloid f .

Proof. Obviously S�(S(f))wS�(f) and S(S�(f))wS�(f).But S�(S(f))vS�(S�(f))=S�(f) and S(S�(f))vS�(S�(f))=S�(f). �

Conjecture 11.37. S(S(f)) =S(f) for

1. every endoreloid f ;

2. every endofuncoid f .

Conjecture 11.38. For every endoreloid f

1. S(f) �S(f) =S(f);

2. S�(f) �S�(f)=S�(f);

3. S(f) �S�(f)=S�(f) �S(f)=S�(f).

Conjecture 11.39. S(f) �S(f)=S(f) for every endofuncoid f .

11.5 Algebraic properties of S and S� 151

Chapter 12Total boundness of reloids

12.1 Thick binary relations

De�nition 12.1. I will call �-thick and denote thick�(E) a Rel-endomorphism E when thereexists a �nite cover S of ObE such that 8A2S:A�A�GRE.

De�nition 12.2. CS(S)=SfA�A j A2Sg for a collection S of sets.

Remark 12.3. CS means �Cartesian squares�.

Obvious 12.4. A Rel-endomorphism is �-thick i� there exists a �nite cover S of ObE such thatCS(S)�GRE.

De�nition 12.5. I will call �-thick and denote thick�(E) a Rel-endomorphism E when thereexists a �nite set B such that hEiB=ObE.

Proposition 12.6. thick�(E)) thick�(E).

Proof. Let thick�(E). Then there exists a �nite cover S of the set Ob E such that 8A 2 S:A�A�GRE. Without loss of generality assume A=/ ; for every A2S. So A�hE ifxAg for somexA for every A2S. So hE ifxA j A2Sg=

SfhEifxAg j A2Sg=ObE and thus E is �-thick. �

Obvious 12.7. Let X be a set, A and B are Rel-endomorphisms on X and B wA. Then:� thick�(A)) thick�(B);

� thick�(A)) thick�(B).

Example 12.8. There is a �-thick Rel-morphism which is not �-thick.

Proof. Consider the Rel-morphism on [0; 1] with the below graph:

¡= f(x;x) j x2 [0; 1]g[ f(x; 0) j x2 [0; 1]g[ f(0;x) j x2 [0; 1]g:

¡ is �-chick because h¡if0g= [0; 1].To prove that ¡ is not �-thick it's enough to prove that every set A such that A�A�¡ is �nite.

153

Suppose for the contrary that A is in�nite. Then A contains more than one non-zero points y,z (y=/ z). Without loss of generality y <z. So we have that (y; z) is not of the form (y; y) nor (0;y) nor (y; 0). Therefore A�A isn't a subset of ¡. �

12.2 Totally bounded endoreloidsThe below is a straightforward generalization of the customary de�nition of totally bounded setson uniform spaces (it's proved below that for uniform spaces the below de�nitions are equivalent).

De�nition 12.9. An endoreloid f is �-totally bounded (totBound�(f)) if every E 2xyGR f is �-thick.

De�nition 12.10. An endoreloid f is �-totally bounded (totBound�(f)) if every E 2 xyGR f is�-thick.

Remark 12.11. We could rewrite the above de�nitions in a more algebraic way like xyGR f �thick� (with thick� would be de�ned as a set rather than as a predicate), but we don't really needthis simpli�cation.

Proposition 12.12. If an endoreloid is �-totally bounded then it is �-totally bounded.

Proof. Because thick�(E)) thick�(E). �

Proposition 12.13. If an endoreloid f is re�exive and Ob f is �nite then f is both �-totallybounded and �-totally bounded.

Proof. It enough to prove that f is �-totally bounded. Really, every E 2 xyGR f is re�exive.Thus fxg�fxg�E for x2Ob f and thus ffxg j x2Ob f g is a sought for �nite cover of Ob f . �

Obvious 12.14.

� A principal endoreloid induced by a Rel-morphism E is �-totally bounded i� E is �-thick.

� A principal endoreloid induced by a Rel-morphism E is �-totally bounded i� E is �-thick.

Example 12.15. There is a �-totally bounded endoreloid which is not �-totally bounded.

Proof. It follows from the example above and properties of principal endoreloids. �

12.3 Special case of uniform spacesDe�nition 12.16. Uniform space is essentially the same as symmetric, re�exive and transitiveendoreloid.

Exercise 12.1. Prove that it is essentially the sameas the standard de�nition of a uniform space (seeWikipediaor PlanetMath).

Theorem 12.17. Let f be such a endoreloid that f � f¡1v f . Then f is �-totally bounded i� itis �-totally bounded.

Proof.

). Proved above.

(. For every "2GR f we have that h"ifc0g; :::; h"ifcng covers the space. h"ifcig� h"ifcig�"� "¡1 because for x2h"ifcig (the same as ci2h"¡1ifxg) we have hh"ifcig� h"ifcigifxg=h"ifcig�h"ih"¡1ifxg=h"�"¡1ifxg. For every "02GR f exists "2GR f such that "�"¡1�"0because f � f¡1v f . Thus for every "0 we have h"ifcig� h"ifcig� "0 and so

h"ifc0g; :::; h"ifcng:

154 Total boundness of reloids

is a sought for �nite cover. �

Corollary 12.18. A uniform space is �-totally bounded i� it is �-totally bounded.

Proof. From the theorem and the de�nition of uniform spaces. �

Thus we can say about just totally bounded uniform spaces (without specifying whether it is �or �).

12.4 Relationships with other properties

Theorem 12.19. Let � and � be endoreloids. Let f be a principal C0(�; �) continuous, mono-valued, surjective reloid. Then if � is �-totally bounded then � is also �-totally bounded.

Proof. Let ' be the monovalued, surjective function, which induces the reloid f .We have �v f¡1 � � � f .Let F 2GR �. Then there exists E 2GR � such that E � '¡1 �F � '.Since � is �-totally bounded, there exists a �nite subset A of Ob � such that hE iA=Ob �.We claim hF ih'iA=Ob �.Indeed let y 2 Ob � be an arbitrary point. Since ' is surjective, there exists x 2 Ob � such

that 'x= y. Since hE iA=Ob � there exists a2A such that aEx and thus a ('¡1 �F � ')x. So('a; y)= ('a; 'x)2F . Therefore y 2 hF ih'iA. �

Theorem 12.20. Let � and � be endoreloids. Let f be a principal C00(�;�) continuous, surjectivereloid. Then if � is �-totally bounded then � is also �-totally bounded.

Proof. Let ' be the surjective binary relation which induces the reloid f .We have f � � � f¡1v �.Let F 2GR �. Then there exists E 2GR � such that '�E � '¡1�F .There exists a �nite cover S of Ob � such that[

fA�A j A2Sg�E:

Thus '� (SfA�A j A2Sg) � '¡1�F that is

Sfh'iA�h'iA j A2Sg�F .

It remains to prove that fh'iA j A2Sg is a cover of Ob �. It is true because ' is a surjectionand S is a cover of Ob �. �

A stronger statement (principality requirement removed):

Conjecture 12.21. The image of a uniformly continuous entirely de�ned monovalued surjectivereloid from a (�-, �-)totally bounded endoreloid is also (�-, �-)totally bounded.

Can we remove the requirement to be entirely de�ned from the above conjecture?

Question 12.22. Under which conditions it's true that join of (�-, �-) totally bounded reloids isalso totally bounded?

12.5 Additional predicates

We may consider also the following predicates expressing di�erent kinds of what is intuitively isunderstood as boundness. Their usefulness is unclear, but I present them for completeness.

� 8E 2GR f 9n2N : thick�(En)

� 8E 2GR f 9n2N : thick�(En)

� 8E 2GR f 9n2N : thick�(E0[ ::: [En)

12.5 Additional predicates 155

� 8E 2GR f 9n2N : thick�(E0[ ::: [En)

� 9n2N : totBound�(fn)

� 9n2N : totBound�(fn)

� 9n2N : totBound�(f0t ::: t fn)

� 9n2N : totBound�(f0t ::: t fn)

� totBound�(S(f))

� totBound�(S(f))

Some of the above de�ned predicates are equivalent:

Proposition 12.23.

� 8E 2GR f 9n2N : thick�(En),9n2N : totBound�(fn).

� 8E 2GR f 9n2N : thick�(En),9n2N : totBound�(fn).

Proof. Because every F 2GR fn is a superset of En for some E 2GR f . �

Proposition 12.24.

� 8E 2GR f 9n2N : thick�(E0[ ::: [En),9n2N : totBound�(f0t ::: t fn).

� 8E 2GR f 9n2N : thick�(E0[ ::: [En),9n2N : totBound�(f0t ::: t fn).

Proof. f0t ::: t fn= f0\ ::: \ fn. Thus every F 2GR(f0\ ::: \ fn) we have F 2 fk, thus F �Ekkfor all k for some Ek2GR f and so F �E0[ ::: [En where E=E0\ ::: \Ek2GR f . �

Proposition 12.25. All predicates in the above list are pairwise equivalent in the case if f is auniform space.

Proof. Because f � f = f . �

156 Total boundness of reloids

Chapter 13

Orderings of �lters in terms of reloids

Whilst the other chapters of this book use �lters to research funcoids and reloids, here the oppositething is discussed, the theory of reloids is used to describe properties of �lters.

In this chapter the word �lter is used to denote a �lter on a set (not on an arbitrary poset) only.

13.1 Equivalent �lters

De�nition 13.1. Two �lters A and B (with possibly di�erent base sets) are equivalent (A� B)i� there exists a set X such that X 2A and X 2B and PX \A=PX \B.

Proposition 13.2. If two �lters with the same base are equivalent they are equal.

Proof. Let A and B be two �lters and PX \ A = PX \ B for some set X such that X 2 Aand X 2 B, and Base(A) = Base(B). Then A = (PX \ A) [ fY 2 PBase(A) j Y � Xg =(PX \B)[fY 2PBase(B) j Y �Xg=B. �

Proposition 13.3. � restricted to small �lters is an equivalence relation.

Proof.


Symmetry. Obvious.

Transitivity. Let A�B and B�C for some small �lters A, B, and C. Then there exist a setX such that X 2A and X 2 B and PX \A=PX \ B and a set Y such that Y 2 B andY 2C and PY \B=PY \C. So X \Y 2A because

PY \PX \A=PY \PX \B=P(X \Y )\B �fX \Y g\B 3X \Y :

Similarly we have X \Y 2 C. Finally P(X \Y )\A=PY \PX \A=PY \PX \B=PX \PY \B=PX \PY \C=P(X \ Y )\C. �

De�nition 13.4. The rebase A�A for a �lter A and a set A (base) such that 9X 2A:X �A isde�ned by the formula

A�A= fX 2PA j 9Y 2A:Y �Xg:

Proposition 13.5. If 9X 2A:X �A then:

1. A�A is a �lter on A;

2. A�A�A.

Proof.

1. We need to prove that fX 2PA j 9Y 2A:Y �Xg is a �lter on A. That it is an upper set isobvious. It is non-empty because 9Y 2A:Y �A and thus A2fX 2PA j 9Y 2A:Y �Xg.Let P ; Q2fX 2PA j 9Y 2A:Y �Xg. Then P ; Q�A and 9P 02A:P 0�P and 9Q02A:Q0�Q. So P \Q�A and P 0\Q0�P \Q. Thus P \Q2fX 2PA j 9Y 2A:Y �Xg.

157

2. (A � A) \ P(A \ Base(A)) = fX 2 PA j 9Y 2 A: Y � Xg \ P(A \ Base(A)) =fX 2 PA j 9Y 2 A: Y � Xg \ PBase(A) = fX 2 P(A \ Base(A)) j X 2 Ag =A\P(A\Base(A)).

Thus A � A � A because A \ Base(A) � X 2 A for some X 2 A and A \ Base(A) �X \Base(A)2fX 2PA j 9Y 2A:Y �Xg=A�A. �

Proposition 13.6. A2A)A�A=PA\A.

Proof. Let A2A. Then A�A= fX 2PA j 9Y 2A:Y �Xg=fX 2PA j X 2Ag=PA\A. �

Lemma 13.7. If A�B then 9Y 2A:Y �X,9Y 2B:Y �X for every �lters A, B, and a set X.

Proof. We will prove 9Y 2A:Y �X)9Y 2B:Y �X (the other direction is similar).We have PK \A=PK \B for some set K such that K 2A, K 2B.9Y 2A:Y �X)9Y 2PK \A:Y �X)9Y 2PK \B: Y �X)9Y 2B:Y �X. �

Proposition 13.8. If A�B then B=A�Base(B) for every �lters A, B.

Proof. PY \A=PY \B for some set Y 2A, Y 2B. There exists a set X 2A such that X 2B.Thus 9X 2A:X �Base(B) and so A�Base(B) is a �lter.

X 2A�Base(B),X 2PBase(B)^9Y 2A:Y �X,X 2PBase(B)^9Y 2B:Y �X,X 2B(the lemma used). �

13.2 Ordering of �lters

Below I will de�ne some categories having �lters (with possibly di�erent bases) as their objectsand some relations having two �lters (with possibly di�erent bases) as arguments induced by thesecategories (de�ned as existence of a morphism between these two �lters).

Theorem 13.9. card a= cardU for every ultra�lter a on U if U is in�nite.

Proof. Let f(X)=X if X 2a and f(X)=U nX if X 2/ a. Obviously f is a surjection from U to a.Every X 2 a appears as a value of f exactly twice, as f(X) and f(U n X). So card a =

(cardU)/2= cardU . �

Corollary 13.10. Cardinality of every two ultra�lters on a set U is the same.

Proof. For in�nite U it follows from the theorem. For �nite case it is obvious. �

De�nition 13.11. f � A= fC 2P(Dst f) j hf¡1iC 2Ag for every �lter A and a Set-morphismf .13.1

Below I'll de�ne some directed multigraphs. By an abuse of notation, I will denote thesemultigraphs the same as (below de�ned) categories based on some of these directed multigraphswith added composition of morphisms (of directed multigraphs edges). As such I will call verticesof these multigraphs objects and edges morphisms.

De�nition 13.12. I will denote GreFunc1 the multigraph whose objects are �lters and whosemorphisms between objects A and B are Set-morphisms from Base(A) to Base(B) such thatB � f �A.

De�nition 13.13. I will denote GreFunc2 the multigraph whose objects are �lters and whosemorphisms between objects A and B are Set-morphisms from Base(A) to Base(B) such thatB= f �A.

13.1. We will assume that f �A is just a set, while it is not yet proved that it is a �lter.

158 Orderings of filters in terms of reloids

De�nition 13.14. LetA be a �lter on a setX and B is a �lter on a set Y . A>1B i� MorGreFunc1(A;B) is not empty.

De�nition 13.15. Let A be a �lter on a set X and B be a �lter on a set Y . A >2 B i�MorGreFunc2(A;B) is not empty.

Proposition 13.16.

1. f 2MorGreFunc1(A;B) i� f is a Set-morphism from Base(A) to Base(B) such that

C 2B(hf¡1iC 2Afor every C 2PBase(B).

2. f 2MorGreFunc2(A;B) i� f is a Set-morphism from Base(A) to Base(B) such that

C 2B,hf¡1iC 2Afor every C 2PBase(B).

Proof.

1. f 2 MorGreFunc1(A; B) , B � f � A , 8C 2 f � A: C 2 B , 8C 2 PBase(B):(hf¡1iC 2A)C 2B).

2. f 2 MorGreFunc2(A; B) , B = f � A , 8C: (C 2 B , C 2 f � A) , 8C 2 PBase(B):(C 2B,C 2 f �A),8C 2PBase(B): (C 2B,hf¡1iC 2A). �

De�nition 13.17. The directed multigraph FuncBij is the directed multigraph got from GreFunc2by restricting to only bijective morphisms.

De�nition 13.18. A �lter A is directly isomorphic to a �lter B i� there is a morphism f 2MorFuncBij(A;B).

Proposition 13.19. f � A= h"FCDf iA for every Set-morphism f :Base(A)!Base(B). [TODO:Make it the primary de�nition instead of the trick with �.]

Proof. For every set C 2PBase(B) we have C 2 f � A, hf¡1iC 2A)9K 2A: hf¡1iC =K)9K 2A: hf ihf¡1iC = hf iK)9K 2A:C �hf iK,9K 2A:C 2 h"FCDf i�K)C 2 h"FCDf iA.

So C 2 f �A)C 2 h"FCDf iA.Let now C 2 h"FCDf iA. Then "Base(A)hf¡1iC w h"FCDf¡1ih"FCDf iA w A and thus hf¡1iC 2

A. �

Corollary 13.20. f 2MorGreFunc1(A;B),Bvh"FCDf iA for every Set-morphism f from Base(A)to Base(B).

Corollary 13.21. f 2MorGreFunc2(A;B),B= h"FCDf iA for every Set-morphism f from Base(A)to Base(B).

Corollary 13.22. A >1 B i� it exists a Set-morphism f : Base(A) ! Base(B) such that B vh"FCDf iA.

Corollary 13.23. A >2 B i� it exists a Set-morphism f : Base(A) ! Base(B) such that B =h"FCDf iA.

Proposition 13.24. For a bijective Set-morphism f :Base(A)!Base(B) the following are equiv-alent:

1. B= f �A.2. 8C 2Base(B): (C 2B,hf¡1iC 2A).3. 8C 2Base(A): (hf iC 2B,C 2A):4. h"FCDf ijA is a bijection from A to B.

13.2 Ordering of filters 159

5. h"FCDf ijA is a function onto B.

6. B= h"FCDf iA.

7. f 2MorGreFunc2(A;B).

8. f 2MorFuncBij(A;B).

Proof.

(1),(2). B = f � A , B = fC 2 PBase(B) j hf¡1iC 2 Ag , 8C 2 Base(B): (C 2 B ,hf¡1iC 2A).

(2),(3). Because f is a bijection.

(2))(5). For every C 2B we have hf¡1iC 2A and thus h"FCDf ijAh"FCDf¡1iC= hf ihf¡1iC=C. Thus h"FCDf ijA is onto B.

(4))(5). Obvious.

(5))(4). We need to prove only that h"FCDf ijA is an injection. But this follows from the factthat f is a bijection.

(4))(3). We have 8C2Base(A): ((h"FCDf ijA)C2B,C2A) and consequently 8C2Base(A):(hf iC 2B,C 2A).

(6),(1). From the last corollary.

(1),(7). Obvious.

(7),(8). Obvious. �

Corollary 13.25. The following are equivalent for every �lters A and B:

1. A is directly isomorphic to B.

2. There is a bijective Set-morphism f :Base(A)!Base(B) such that for every C 2PBase(B)

C 2B,hf¡1iC 2A:

3. There is a bijective Set-morphism f :Base(A)!Base(B) such that for every C 2PBase(B)

hf iC 2B,C 2A:

4. There is a bijective Set-morphism f :Base(A)!Base(B) such that h"FCDf ijA is a bijectionfrom A to B.

5. There is a bijective Set-morphism f :Base(A)!Base(B) such that h"FCDf ijA is a functiononto B.

6. There is a bijective Set-morphism f :Base(A)!Base(B) such that B= h"FCDf iA.

7. There is a bijective morphism f 2MorGreFunc2(A;B).

Proposition 13.26. GreFunc1 and GreFunc2 with function composition are categories.

Proof. Let f : A ! B and g: B ! C be morphisms of GreFunc1. Then B v h"FCDf iA andC vh"FCDgiB. So h"FCD(g � f)iA= h"FCDgih"FCDf iAwh"FCDgiB wC. Thus g � f is a morphism ofGreFunc1. Associativity law is evident. idBase(A) is the identity morphism of GreFunc1 for every�lter A.

Let f :A!B and g:B!C be morphisms of GreFunc2. Then B= h"FCDf iA and C= h"FCDgiB.So h"FCD(g � f)iA = h"FCDgih"FCDf iA = h"FCDgiB = C. Thus g � f is a morphism of GreFunc2.Associativity law is evident. idBase(A) is the identity morphism of GreFunc2 for every �lter A. �

Corollary 13.27. 61 and 62 are preorders.

Theorem 13.28. FuncBij is a groupoid.


Proof. First let's prove it is a category. Let f :A!B and g:B!C be morphisms of FuncBij. Thenf :Base(A)!Base(B) and g:Base(A)!Base(C) are bijections and B=h"FCDf iA and C= h"FCDgiB.Thus g � f :Base(A)!Base(C) is a bijection and C= h"FCD(g � f)iA. Thus g � f is a morphism ofFuncBij. idBase(A) is the identity morphism of FuncBij for every �lter A. Thus it is a category.

It remains to prove only that every morphism f 2 MorFuncBij(A; B) has a reverse (for every�lters A, B). We have f is a bijection Base(A)!Base(B) such that for every C 2PBase(A)

hf iC 2B,C 2A:

Then f¡1:Base(B)!Base(A) is a bijection such that for every C 2PBase(B)

hf¡1iC 2A,C 2B:

Thus f¡12MorFuncBij(B;A). �

Corollary 13.29. Being directly isomorphic is an equivalence relation.

Rudin-Keisler order of ultra�lters is considered in such a book as [37].

Obvious 13.30. For the case of ultra�lters being directly isomorphic is the same as being Rudin-Keisler equivalent.

De�nition 13.31. A �lter A is isomorphic to a �lter B i� there exist sets A2A and B 2B suchthat A�A is directly isomorphic to B�B.

Obvious 13.32. Equivalent �lters are isomorphic.

Theorem 13.33. Being isomorphic (for small �lters) is an equivalence relation.

Proof.

Re�exivity. Because every �lter is directly isomorphic to itself.

Symmetry. If �lter A is isomorphic to B then there exist sets A 2 A and B 2 B such thatA�A is directly isomorphic to B �B and thus B �B is directly isomorphic to A�A. SoB is isomorphic to A.

Transitivity. Let A be isomorphic to B and B be isomorphic to C. Then exist A2A, B12B,B22B, C 2C such that there are bijections f :A!B1 and g:B2!C such that

8X 2PA: (X 2B,hf¡1iX 2A) and 8X 2PB2: (X 2A,hf iX 2B):

Also 8X 2PB2: (X 2B,hgiX 2C).So g � f is a bijection from hf¡1i(B1\B2)2A to hgi(B1\B2)2C such that

X 2A,hf iX 2B,hgihf iX 2C,hg � f iX 2C:

Thus g � f establishes a bijection which proves that A is isomorphic to C. �

Lemma 13.34. Let cardX = card Y , u be an ultra�lter on X and v be an ultra�lter on Y ; letA2u and B 2 v. Let u�A and v�B be directly isomorphic. Then if card(X nA) = card(Y nB)we have u and v directly isomorphic.

Proof. Arbitrary extend the bijection witnessing being directly isomorphic to the sets X nA andY nB. �

Theorem 13.35. If cardX= cardY then being isomorphic and being directly isomorphic are thesame for ultra�lters u on X and v on Y .

Proof. That if two �lters are isomorphic then they are directly isomorphic is obvious.Let ultra�lters u and v be isomorphic that is there is a bijection f :A!B where A2 u, B 2 v

witnessing isomorphism of u and v.If one of the �lters u or v is a trivial ultra�lter then the other is also a trivial ultra�lter and as

it is easy to show they are directly isomorphic. So we can assume u and v are not trivial ultra�lters.


If card(X nA)= card(Y nB) our statement follows from the last lemma.Now assume without loss of generality card(X nA)< card(Y nB).cardB= cardY because card(Y nB)< cardY .It is easy to show that there exists B 0 � B such that card(X n A) = card(Y n B 0) and

cardB 0= cardB.We will �nd a bijection g from B to B 0 which witnesses direct isomorphism of v to v itself.

Then the composition g � f witnesses a direct isomorphism of u�A and v�B 0 and by the lemmau and v are directly isomorphic.

Let D=B 0 nB. We have D2/ v.There exists a set E �B such that cardE > cardD and E 2/ v.We have cardE= card(D[E) and thus there exists a bijection h:E!D[E.Let

g(x)=

�x if x2B nE;h(x) if x2E:

g jBnE and g jE are bijections.im(g jBnE)=B nE; im(g jE) = im h=D [E;

(D[E)\ (B nE)= (D\ (B nE))[ (E \ (B nE)) = ;[;= ;:

Thus g is a bijection from B to (B nE)[ (D[E)=B [D=B 0.To �nish the proof it's enough to show that hgiv= v. Indeed it follows from B nE 2 v. �

Proposition 13.36.

1. For every A2A and B 2B we have A>2B i� A�A>2B�B.

2. For every A2A and B 2B we have A>1B i� A�A>1B�B.

Proof.

1. A>2B i� there exist a bijective Set-morphism f such that B= h"FCDf iA. The equality isobviously preserved replacing A with A�A and B with B �B.

2. A>1B i� there exist a bijective Set-morphism f such that B � h"FCDf iA. The equality isobviously preserved replacing A with A�A and B with B �B. �

Proposition 13.37. For ultra�lters >2 is the same as Rudin-Keisler ordering (as de�ned in [37]).

Proof. x>2 y i� there exist sets A 2 x and B 2 y a bijective Set-morphism f :X! Y such thaty�B= fC 2PY j hf¡1iC 2x�Ag that is when C 2 y�B,hf¡1iC 2x�A what is equivalentto C 2 y,hf¡1iC 2x what is the de�nition of Rudin-Keisler ordering. �

Remark 13.38. The relation of being isomorphic for ultra�lters is traditionally called Rudin-Keisler equivalence.

Obvious 13.39. (>1)� (>2).

De�nition 13.40. Let Q and R be binary relations on the set of �lters. I will denote MonRldQ;Rthe directed multigraph with objects being �lters and morphisms such monovalued reloids f that(dom f)QA and (im f)RB.

I will also denote CoMonRldQ;R the directed multigraph with objects being �lters and mor-phisms such injective reloids f that (im f)QA and (dom f)RB. These are essentially the duals.

Some of these directed multigraphs are categories with reloid composition (see below). By abuseof notation I will denote these categories the same as these directed multigraphs.

Theorem 13.41. For every �lters A and B the following are equivalent:

1. A>1B.2. MorMonRld=;w(A;B) =/ ;.


3. MorMonRldv;w(A;B)=/ ;.

4. MorMonRldv;=(A;B)=/ ;.

5. MorCoMonRld=;w(A;B) =/ ;.

6. MorCoMonRldv;w(A;B)=/ ;.

7. MorCoMonRldv;=(A;B)=/ ;.

Proof.

(1))(2). There exists a Set-morphism f : Base(A)! Base(B) such that B v h"FCDf iA. Wehave

dom ("RLDf)jA=Au 1F(Base(A))=Aand

im ("RLDf)jA=im (FCD)("RLDf)jA=im ("FCDf)jA=h"FCDf iAwB:

Thus ("RLDf)jA is a monovalued reloid such that dom ("RLDf)jA=A and im ("RLDf)jAwB.(2))(3), (4))(3), (5))(6), (7))(6). Obvious.

(3))(1). We have Bvh(FCD)f iA for a monovalued reloid f 2RLD(Base(A);Base(B)). Thenthere exists a Set-morphism F :Base(A)!Base(B) such that B vh"FCDF iA that is A>1B.

(6))(7). dom f jB=B and im f jBvA.(2),(5), (3),(6), (4),(7). By duality. �

Theorem 13.42. For every �lters A and B the following are equivalent:

1. A>2B.2. MorMonRld=;=(A;B)=/ ;.

3. MorCoMonRld=;=(A;B)=/ ;.

Proof.

(1))(2). Let A >2 B that is B = h"FCDf iA for some Set-morphism f : Base(A)! Base(B).Then dom ("RLDf)jA=A and im ("RLDf)jA=im (FCD)("RLDf)jA=im ("FCDf)jA=h"FCDf iA=B. So ("RLDf)jA is a sought for reloid.

(2))(1). By corollary 13.78 below, there exists a Set-morphism F :Base(A)!Base(B) suchthat f = ("RLDF )jA. Thus h"FCDF iA = im("FCDF )jA=im (FCD)("RLDF )jA=im (FCD)f =im f =B. Thus A>2B is testi�ed by the morphism F .

(2),(3). By duality. �

Theorem 13.43. The following are categories (with reloid composition):

1. MonRldv;w;

2. MonRldv;=;

3. MonRld=;=.

4. CoMonRldv;w;

5. CoMonRldv;=;

6. CoMonRld=;=.

Proof. We will prove only the �rst three. The rest follow from duality.[TODO: Check duality.]We need to prove only that composition of morphisms is a morphism, because associativity andexistence of identity morphism are evident. We have:

1. Let f 2MorMonRldv;w(A;B), g2MorMonRldv;w(B;C). Then dom f vA, im f wB, dom gvB,im gwC. So dom(g � f)vA, im(g � f)wC that is g � f 2MorMonRldv;w(A; C).


2. Let f 2MorMonRldv;=(A;B), g2MorMonRldv;=(B;C). Then dom f vA, im f =B, dom gvB,im g= C. So dom(g � f)vA, im(g � f) = C that is g � f 2MorMonRldv;=(A; C).

3. Let f 2MorMonRld=;=(A;B), g2MorMonRld=;=(B;C). Then dom f =A, im f =B, dom g=B,im g= C. So dom(g � f) =A; im(g � f)= C that is g � f 2MorMonRld=;=(A; C). �

De�nition 13.44. Let BijRld be the groupoid of all bijections of the category of reloid triples.Its objects are �lters and its morphisms from a �lter A to �lter B are monovalued injective reloidsf such that dom f =A and im f =B.

Theorem 13.45. Filters A and B are isomorphic i� MorBijRld(A;B) =/ ;.

Proof.

). Let A and B be isomorphic. Then there are sets A2A, B2B and a bijective Set-morphismF :A!B such that hF i:PA\A!PB \B is a bijection.

Obviously f =("RLDF )jA is monovalued and injective.im f =

df"BimG j G 2 ("RLDF )jAg=

df"Bim(H \ F jX) j H 2 ("RLDF )jA; X 2 Ag=d

f"Bim F jP j P 2 Ag =df"BhF iP j P 2 Ag =

df"BhF iP j P 2 PA \ Ag =d

h"Bi(PB \B) =dh"BiB=B.

Thus dom f =A and im f =B.

(. Let f be a monovalued injective reloid such that dom f=A and im f=B. Then there exist afunction F 0 and an injective binary relation F 00 such that F 0;F 002GR f . Thus F =F 0\F 00 isan injection such that F 2GR f . The function F is a bijection from A=domF to B= imF .The function hF i is an injection on PA\A (and moreover on PA). It's simple to show that8X 2PA\A: hF iX 2PB \B and similarly 8Y 2PB \B: hF i¡1Y = hF¡1iY 2PA\A.Thus hF ijPA\A is a bijection PA\A!PB \B. So �lters A and B are isomorphic. �

Proposition 13.46. (>1)= (w) � (>2) (when we limit to small �lters).

Proof. A >1 B i� exists a function f : Base(A) ! Base(B) such that B v h"FCDf iA. But B vh"FCDf iA is equivalent to 9B 02F: (B 0wB^B 0= h"FCDf iA). So A>1B is equivalent to existence ofB 02F such that B 0wB and existence of a function f :Base(A)!Base(B) such that B 0= h"FCDf iA.That is equivalent to A((w) � (>2))B. �

Proposition 13.47. If a and b are ultra�lters then b>1 a, b>2 a.

Proof. We need to prove only b >1 a ) b >2 a. If b >1 a then there exists a monovaluedreloid f : Base(b) ! Base(a) such that dom f = b and im f w a. Then im f = im (FCD)f 2�0F(Base(a))

[ atomsF(Base(a)) because (FCD)f is a monovalued funcoid. So im f = a (taken into

account a=/ 0F(Base(a))) and thus b>2 a. �

Corollary 13.48. For atomic �lters >1 is the same as >2.

Thus I will write simply > for atomic �lters.

13.2.1 Existence of no more than one monovalued injective reloid for agiven pair of ultra�lters

13.2.1.1 The lemmas

The lemmas in this section were provided to me by Robert Martin Solovay in [36]. They are basedon Wistar Comfort's work.

In this section we will assume � is an ultra�lter on a set I and function f :I!I has the propertyX 2 �,hf¡1iX 2 �.

Lemma 13.49. If X 2 � then X \ hf iX 2 �.


Proof. If hf iX2/ � thenX�hf¡1ihf iX2/ � and soX2/ �. ThusX 2�^hf iX2� and consequentlyX \ hf iX 2 �. �

We will say that x is periodic when fn(x) =x for some positive integer x. The least such n iscalled the period of x.

Let's de�ne x� y i� there exist i; j 2N such that f i(x)= f j(y). Trivially it is an equivalencerelation. If x and y are periodic, then x� y i� exists n2N such that fn(y)=x.

Let A= fx2 I j x is periodic with period>1g.We will show that A2/ �. Let's assume A2 �.Let a set D �A contains (by the axiom of choice) exactly one element from each equivalence

class of A de�ned by the relation �.Let � be a function A!N de�ned as follows. Let x 2 A. Let y be the unique element of D

such that x� y. Let �(x) be the least n2N such that fn(y)=x.Let B0= fx2A j �(x) is eveng and B1= fx2A j �(x) is oddg.Let B2= fx2A j �(x)= 0g.

Lemma 13.50. B0\ hf iB0�B2.

Proof. If x 2B0 \ hf iB0 then fn(y) = x for a minimal even n and x= f(x0) where fm(y 0) = x0

for a minimal even m. Thus fn(y)= f(x0) thus y and x0 laying in the same equivalence class andthus y= y 0. So we have fn(y)= fm+1(y). Thus n6m+1 by minimality.

x0 lies on an orbit and thus x0 = f¡1(x) where by f¡1 I mean step backward on our orbit;fm(y) = f¡1(x) and thus x0= fn¡1(y) thus n¡ 1>m by minimality or n=0.

Thus n = m + 1 what is impossible for even n and m. We have a contradiction what provesB0\ hf iB0�;.

Remained the case n=0, then x= f0(y) and thus �(x) = 0. �

Lemma 13.51. B1\ hf iB1= ;.

Proof. Let x2B1\ hf iB1. Then fn(y) =x for an odd n and x= f(x0) where fm(y 0) =x0 for anodd m. Thus fn(y)= f(x0) thus y and x0 laying in the same equivalence class and thus y= y 0. Sowe have fn(y)= fm+1(y). Thus n6m+1 by minimality.

x0 lies on an orbit and thus x0= f¡1(x) where by f¡1 I mean step backward on our orbit;fm(y)= f¡1(x) and thus x0= fn¡1(y) thus n¡1>m by minimality (n=0 is impossible because

n is odd).Thus n = m + 1 what is impossible for odd n and m. We have a contradiction what proves

B1\ hf iB1= ;. �

Lemma 13.52. B2\ hf iB2= ;.

Proof. Let x2B2\ hf iB2. Then x= y and x0= y where x= f(x0). Thus x= f(x) and so x2/ Awhat is impossible. �

Lemma 13.53. A2/ �.

Proof. Suppose A2 �.Since A2 � we have B02 � or B12 �.So either B0 \ hf iB0�B2 or B1 \ hf iB1�B2. As such by the lemma 13.49 we have B2 2 �.

This is incompatible with B2\ hf iB2= ;. So we got a contradiction. �

Let C be the set of points x which are not periodic but fn(x) is periodic for some positive n.

Lemma 13.54. C 2/ �.

Proof. Let � be a function C!N such that �(x) is the least n2N such that fn(x) is periodic.Let C0= fx2C j �(x) is eveng and C1= fx2C j �(x) is oddg.Obviously Cj \ hf iCj = ; for j = 0; 1. Hence by lemma 13.49 we have C0; C1 2/ � and thus

C =C0[C12/ �. �


Let E be the set of x2 I such that for no n2N we have fn(x) periodic.

Lemma 13.55. Let x; y2E be such that f i(x)= f j(y) and f i0(x)= f j

0(y) for some i; j ; i0; j 02N .

Then i¡ j= i0¡ j 0.

Proof. i 7! f i(x) is a bijection.So y= f i¡j(y) and y= f i

0¡j 0(y). Thus f i¡j(y)= f i0¡j 0(y) and so i¡ j= i0¡ j 0. �

Lemma 13.56. E 2/ �.

Proof. Let D 0�E be a subset of E with exactly one element from each equivalence class of therelation � on E.

De�ne the function :E!Z as follows. Let x2E. Let y be the unique element of D 0 such thatx� y. Choose i; j2N such that f i(y)= f j(x). Let (x)= i¡ j. By the last lemma, is well-de�ned.

It is clear that if x2E then f(x)2E and moreover (f(x))= (x) + 1.Let E0= fx2E j (x) is eveng and E1= fx2E j (x) is oddg.We have E0\ hf iE0= ;2/ � and hence E02/ �.Similarly E12/ �.Thus E=E0[E12/ �. �

Lemma 13.57. f is the identity function on a set in �.

Proof. We have shown A; C; E 2/ �. But the points which lie in none of these sets are exactlypoints periodic with period 1 that is �xed points of f . Thus the set of �xed points of f belongsto the �lter �. �

13.2.1.2 The main theorem and its consequences

Theorem 13.58. For every ultra�lter a the morphism (a; a; idaFCD) is the only

1. monovalued morphism of the category of reloid triples from a to a;

2. injective morphism of the category of reloid triples from a to a;

3. bijective morphism of the category of reloid triples from a to a.

Proof. We will prove only (1) because the rest follow from it.Let f be a monovalued morphism from Base(a) to Base(a). Then it exists a Set-morphism F

such that F 2 xyGR f . Trivially h"FCDF ia w a and thus hF iA 2 a for every A 2 a. Thus by thelemma we have that F is the identity function on a set in a and so obviously f is an identity. �

Corollary 13.59. For every two atomic �lters (with possibly di�erent bases) A and B there existsat most one bijective reloid triple from A to B.

Proof. Suppose that f and g are two di�erent bijective reloids from A to B. Then g¡1 � f is notthe identity reloid (otherwise g¡1 � f = iddom f

RLD and so f = g). But g¡1 � f is a bijective reloid (asa composition of bijective reloids) from A to A what is impossible. �

13.3 Rudin-Keisler equivalence and Rudin-Keisler order

Theorem 13.60. Atomic �lters a and b (with possibly di�erent bases) are isomorphic i� a> b^b> a.

Proof. Let a> b^ b> a. Then there are a monovalued reloids f and g such that dom f = a andim f = b and dom g= b and im g= a. Thus g � f and f � g are monovalued morphisms from a to aand from b to b. By the above we have g� f= idaRLD and f � g= idbRLD so g= f¡1 and f¡1� f= idaRLD

and f � f¡1= idbRLD. Thus f is an injective monovalued reloid from a to b and thus a and b areisomorphic. �


The last theorem cannot be generalized from atomic �lters to arbitrary �lters, as it's shown bythe following example:

Example 13.61. A>1B^B>1A but A is not isomorphic to B for some �lters A and B.

Proof. Consider A="R[0; 1] and B=df"R[0; 1+ ") j "> 0g. Then the function f =�x2R:x/2

witnesses both inequalities A>1B and B>1A. But these �lters cannot be isomorphic because onlyone of them is principal. �

Lemma 13.62. Let f0 and f1 be Set-morphisms. Let f(x; y)= (f0x; f1y) for a function f . Then"FCD(Src f0�Src f1;Dst f0�Dst f1)f

�(A�RLDB)= h"FCDf0iA�RLD h"FCDf1iB:

Proof."FCD(Src f0�Src f1;Dst f0�Dst f1)f

�(A �RLD B) =

"FCD(Src f0�Src f1;Dst f0�Dst f1)f

�df"Src f0�Src f1(A � B) j A 2 A; B 2 Bg =

df"Dst f0�Dst f1hf i(A � B) j A 2 A; B 2 Bg =d

f"Dst f0�Dst f1(hf0iA � hf1iB) j A 2 A; B 2 Bg =df"Dst f0hf0iA �RLD "Dst f1hf1iB j A 2

A; B 2 Bg = (theorem 6.79)=df"Dst f0hf0iA j A 2 Ag �RLD d

f"Dst f1hf1iB j A 2 Bg =

h"FCDf0iA�RLD h"FCDf1iB. �

Theorem 13.63. Let f be a monovalued reloid. Then GR f is isomorphic to the �lter dom f .

Proof. Let f be a monovalued reloid. There exists a function F 2GR f . Consider the bijectivefunction p=�x2domF : (x;Fx).hpidom F = F and consequently hpidom f =

d �"RLD(Src f ;Dst f)hpidom K j K 2 GR f

=d �

"RLD(Src f ;Dst f)hpidom(K \ F ) j K 2 GR f=d �

"RLD(Src f ;Dst f)(K \ F ) j K 2 GR f=d �

"RLD(Src f ;Dst f)K j K 2GR f=GR f . Thus p witnesses that GR f is isomorphic to the �lter

dom f . �

Corollary 13.64. The graph of a monovalued reloid with atomic domain is atomic.

Corollary 13.65. GR idARLD is isomorphic to A for every �lter A.

Theorem 13.66. There are atomic �lters incomparable by Rudin-Keisler order.

Proof. See [13]. �

Theorem 13.67. >1 and >2 are di�erent relations.

Proof. Consider a is an arbitrary non-empty �lter. Then a>10F(Base(a)) but not a>20F(Base(a)). �

Proposition 13.68. If a>2 b where a is an ultra�lter then b is also an ultra�lter.

Proof. b= h"FCDf ia for some f :Base(a)!Base(b). So b is an ultra�lter since f is monovalued. �

Corollary 13.69. If a>1 b where a is an ultra�lter then b is also an ultra�lter or 0F(Base(a)).

Proof. bvh"FCDf ia for some f :Base(a)!Base(b). Therefore b0= h"FCDf ia is an ultra�lter. Fromthis our statement follows. �

Proposition 13.70. Principal �lters, generated by sets of the same cardinality, are isomorphic.

Proof. Let A and B be sets of the same cardinality. Then there are a bijection f from A to B.We have hf iA=B and thus A and B are isomorphic. �

Proposition 13.71. If a �lter is isomorphic to a principal �lter, then it is also a principal �lterinduced by a set with the same cardinality.

Proof. Let A be a principal �lter and B is a �lter isomorphic to A. Then there are sets X 2 Aand Y 2B such that there are a bijection f :X!Y such that hf iA=B.

13.3 Rudin-Keisler equivalence and Rudin-Keisler order 167

So minB exists and minB= hf iminA and thus B is a principal �lter (of the same cardinalityas A). �

Proposition 13.72. A �lter isomorphic to a non-trivial ultra�lter is a non-trivial ultra�lter.

Proof. Let a be a non-trivial ultra�lter and a is isomorphic to b. Then a >2 b and thus b is anultra�lter. The �lter b cannot be trivial because otherwise a would be also trivial. �

Theorem 13.73. For an in�nite set U there exist 22cardU

equivalence classes of isomorphic ultra-�lters.

Proof. The number of bijections between any two given subsets of U is no more than(card U)cardU = 2cardU. The number of bijections between all pairs of subsets of U is no morethan 2cardU � 2cardU = 2cardU. Therefore each isomorphism class contains at most 2cardU ultra-�lters. But there are 22

cardUultra�lters. So there are 22

cardUclasses. �

Remark 13.74. One of the above mentioned equivalence classes contains trivial ultra�lters.

Corollary 13.75. There exist non-isomorphic nontrivial ultra�lters on any in�nite set.

13.4 Consequences

Theorem 13.76. The graph of reloid "Afag�RLDF is isomorphic to the �lter F for every set Aand a2A.

Proof. Consider B=fag�Base(F) and f=f(x; (a;x)) j x2Base(F)g. Then f is a bijection fromBase(F) to B.

If X 2F then hf iX �B and hf iX = fag�X 2GR("Afag�RLDF).For every Y 2 GR("Afag �RLD F) \ PB we have Y = fag � X for some X 2 F and thus

Y = hf iX .So hf ijF\PBase(F)=hf ijF is a bijection from PBase(F) to GR("Afag�RLDF)\PB.We have F \ PBase(F) and GR("Afag �RLD F) \ PB directly isomorphic and thus F is

isomorphic to GR("Afag�RLDF). �

Theorem 13.77. If f , g are reloids, f v g and g is monovalued then g jdom f=f .[TODO: A similartheorem for funcoids?]

Proof. It's simple to show that f =F �

f ja j a2atomsF(Src f)(use the fact that kv f ja for some

a2 atomsF(Src f) for every k 2 atoms f and the fact that RLD(Src f ;Dst f) is atomistic).Suppose that g jdom f=/ f . Then there exists a2 atoms dom f such that g ja=/ f ja.Obviously g jawf ja.If g jaAf ja then g ja is not atomic (because f ja=/0RLD(Src f ;Dst f)) what contradicts to a theorem

above. So g ja=f ja what is a contradiction and thus g jdom f=f . �

Corollary 13.78. Every monovalued reloid is a restricted principal monovalued reloid.

Proof. Let f be a monovalued reloid. Then there exists a function F 2GR f . So we have¡"RLD(Src f ;Dst f)F

�jdom f=f: �

Corollary 13.79. Every monovalued injective reloid is a restricted injective monovalued principalreloid.

Proof. Let f be a monovalued injective reloid. There exists a function F such that f =¡"RLD(Src f ;Dst f)F

�jdom f. Also there exists an injection G2GR f .


Thus f = f u¡"RLD(Src f ;Dst f)G

�jdom f=

¡"RLD(Src f ;Dst f)F

�jdom fu

¡"RLD(Src f ;Dst f)G

�jdom f=¡

"RLD(Src f ;Dst f)(F \G)�jdom f. Obviously F \G is an injection. �

Theorem 13.80. If a reloid f is monovalued and dom f is an principal �lter then f is principal.

Proof. f is a restricted principal monovalued reloid. Thus f = F jdom f where F is a principalmonovalued reloid. Thus f is principal. �

Lemma 13.81. If a �lter A is isomorphic to a �lter B then if X is a set then there exists a setY such that "Base(A)X uA is a �lter isomorphic to "Base(B)Y uB.

Proof. Let f be a monovalued injective reloid such that dom f =A, im f =B.By proposition 4.227 we have: "Base(A)X uA=X where X is a �lter complementive to A. Let

Y =A nX .h(FCD)f iX u h(FCD)f iY =0F(Base(B)) by injectivity of f .h(FCD)f iX t h(FCD)f iY= h(FCD)f i(X tY)= h(FCD)f iA=B. So h(FCD)f iX is a �lter com-

plementive to B. So by proposition 4.227 there exists a set Y such that h(FCD)f iX ="Base(B)Y uB.f jX is obviously a monovalued injective reloid with dom(f jX) = "Base(A)X uA and im(f jX) =

"Base(B)Y uB. So "Base(A)X uA is isomorphic to "Base(B)Y uB. �

Example 13.82. A>2B^B>2A but A is not isomorphic to B for some �lters A and B.

Proof. (proof idea by Andreas Blass, rewritten using reloids by me)Let un, hn with n ranging over the set Z be sequences of ultra�lters onN and functions N!N

such that"FCD(N ;N )hn

�un+1 = un and un are pairwise non-isomorphic. (See [6] for a proof that

such ultra�lters and functions exist.)A=

defFf"Zfng�RLD u2n+1 j n2Zg; B=

defFf"Zfng�RLD u2n j n2Zg.

Let the Set-morphisms f ; g:Z �N!Z �N be de�ned by the formulas f(n;x)= (n;h2nx) andg(n;x)= (n¡ 1;h2n¡1x).

Using the fact that every function induces a complete funcoid and a lemma above we get:h"FCDf iA=

Fhh"FCDf iif"Zfng�RLD u2n+1 j n2Zg=

Ff"Zfng�RLDu2n j n2Zg=B.

h"FCDgiB =Fhh"FCDgiif"Zfng �RLD u2n j n 2 Zg =

Ff"Zfn ¡ 1g �RLD u2n¡1 j n 2 Zg =F

f"Zfng�RLD u2n+1 j n2Zg=A.It remains to show that A and B are not isomorphic.Let X 2 "Zfng �RLD u2n+1 for some n 2 Z. Then if "Z�NX u A is an ultra�lter we have

"Z�NX uA= "Zfng�RLDu2n+1 and thus by the theorem 13.76 is isomorphic to u2n+1.If X 2/ "Zfng �RLD u2n+1 for every n 2 Z then (Z � N) n X 2 "Zfng �RLD u2n+1 and thus

(Z �N) nX 2A and thus "Z�NX uA=0Z�N .We have also ("Zf0g �RLD N) u B = ("Zf0g �RLD N ) u

Ff"Zfng �RLD u2n j n 2 Zg =F

f("Zf0g�RLDN)u ("Zfng�RLDu2n) j n2Zg= "Zf0g�RLDu0 (an ultra�lter).Thus every ultra�lter generated as intersecting A with a principal �lter "Z�NX is isomorphic

to some u2n+1 and thus is not isomorphic to u0. By the lemma it follows that A and B are non-isomorphic. �

13.4.1 Metamonovalued reloids

Proposition 13.83. (T

G) � f =Tfg � f j g 2Gg for every function f and a set G of binary

relations.

Proof. (x; z) 2 (T

G) � f , 9y: (fx = y ^ (y; z) 2T

G) , (fx; z) 2T

G , 8g 2 G:(fx; z)2 g,8g 2G9y: (fx= y ^ (y; z)2 g),8g 2G: (x; z)2 g � f, (x; z)2

Tfg � f j g 2Gg. �

Lemma 13.84. (dG) � f =

dfg � f j g2Gg if f is a monovalued principal reloid and G is a set

of reloids (with matching sources and destinations).

Proof. Let f = "RLD' for some monovalued Rel-morphism '.

13.4 Consequences 169

(dG) � f =

df"RLD(g � ') j g 2 xyGR

dGg;

GRdfg � f j g 2Gg=GR

dfdf"RLD(¡ � ') j ¡2 xyGR gg j g 2Gg=GR

d Sff"RLD(¡ �

') j ¡2 xyGR gg j g 2Gg=GRdf"RLD(¡ � ') j ¡2 xyGR

dGg= f(¡0 � ')u ::: u (¡n � ') j ¡i2S

G where i = 0; :::; n for n 2 N g = (proposition above)=f(¡0 u ::: u ¡n) � ' j ¡i 2S

G wherei=0; :::; n for n2N g= f¡ � ' j ¡2 xyGR

dGg.

Thus (dG) � f =

dfg � f j g 2Gg. �

Theorem 13.85.

1. Monovalued reloids are metamonovalued.

2. Injective reloids are metainjective.

Proof. We will prove only the �rst, as the second is dual.Let G be a set of reloids and f be a monovalued reloid.Let f 0 be a principal monovalued continuation of f (so that f = f 0jdom f).By the lemma (

dG) � f 0=

dfg � f 0 j g 2Gg. Restricting this equality to dom f we get:

(dG) � f =

dfg � f j g 2Gg. �

Conjecture 13.86. Every metamonovalued reloid is monovalued.


Chapter 14Counter-examples about funcoids andreloids

For further examples we will use the �lter de�ned by the formula

�=l �

"F(R)(¡"; ") j "2R; " > 0:

I will denote (A) the Fréchet �lter on a set A.

Example 14.1. There exist a funcoid f and a set S of funcoids such that f uFS=/

Fhf u iS.

Proof. Let f=��FCD"F(R)f0g and S=�"FCD(R ;R)((";+1)�f0g) j "2R; ">0

. Then f u

FS=¡

��FCD "F(R)f0g�u "FCD(R ;R)((0; +1)� f0g) =

¡� u "F(R)(0; +1)

��FCD "F(R)f0g=/ 0FCD(R;R)

whileFhf u iS=

F �0FCD(R;R)

=0FCD(R;R). �

Example 14.2. There exist a set R of funcoids and a funcoid f such that f �FR=/

Fhf � iR.

Proof. Let f =��FCD"Rf0g, R= f"Rf0g�FCD"R("; +1) j "2R; " > 0g.We have

FR = "Rf0g �FCD "R(0; +1); f �

FR = "FCD(R;R)(f0g � f0g) =/ 0FCD(R ;R) andF

hf � iR=F �

0FCD(R ;R)=0FCD(R;R). �

Example 14.3. There exist a set R of reloids and a reloid f such that f �FR=/

Fhf � iR.

Proof. Let f =��RLD "Rf0g, R= f"Rf0g�RLD "R("; +1) j "2R; " > 0g.We have

FR = "Rf0g �RLD "R(0; +1); f �

FR = "RLD(R;R)(f0g � f0g) =/ 0RLD(R ;R) andF

hf � iR=F �

0RLD(R;R)=0RLD(R;R). �

Example 14.4. There exist a set R of funcoids and �lters X and Y such that

1. X [FR]Y ^ @f 2R:X [f ]Y;

2. hFRiX AF fhf iX j f 2Rg.

Proof.

1. Take X = � and Y = 1F(R), R =�"FCD(R;R)(("; +1) � R) j " 2 R; " > 0

. ThenF

R= "FCD(R;R)((0;+1)�R). So X [FR]Y and 8f 2R::(X [f ]Y).

2. With the same X and R we have hFRiX =1F(R) and hf iX =0F(R) for every f 2R, thusF

fhf iX j f 2Rg=0F(R). �

Example 14.5.FfA�RLDB j B 2T g=/ A�RLDF T for some �lter A and set of �lters T (with

a common base).

Proof. Take R+= fx2R j x> 0g, A=�, T = f"fxg j x2R+g where "= "R.FT = "R+; A�RLDF T =��RLD "R+.FfA�RLDB j B 2T g=

Ff��RLD "fxg j x2R+g.

We'll prove thatFf��RLD "fxg j x2R+g=/ ��RLD "R+.

Consider K =Sffxg� (¡1/x; 1/x) j x2R+g.

K 2 GR(� �RLD "fxg) and thus K 2 GRFf� �RLD "fxg j x 2 R+g. But K 2/

GR(��RLD "R+). �

171

Theorem 14.6. For a �lter a we have a�RLD av idRLD(Base(a)) only in the case if a=0F(Base(a))

or a is a trivial ultra�lter.

Proof. If a �RLD a v idRLD(Base(a)) then there exists m 2GR(a �RLD a) such that m � idBase(a).Consequently there exist A; B 2GR a such that A�B � idBase(a) what is possible only in the casewhen "Base(a)A= "Base(a)B= a is trivial a ultra�lter or the least �lter. �

Corollary 14.7. Reloidal product of a non-trivial atomic �lter with itself is non-atomic.

Proof. Obviously (a �RLD a) u idRLD(Base(a)) =/ 0F(Base(a)) and (a �RLD a) u idRLD(Base(a)) @a�RLD a. �

Example 14.8. There exist two atomic reloids whose composition is non-atomic and non-empty.

Proof. Let a be a non-trivial ultra�lter on N and x2N . Then

(a �RLD "Nfxg) � ("Nfxg �RLD a) =l �

"RLD(N ;N )((A � fxg) � (fxg � A)) j A 2 a=

l �"RLD(N ;N )(A�A) j A2 a

= a�RLD a

is non-atomic despite of a�RLD "Nfxg and "Nfxg�RLD a are atomic. �

Example 14.9. There exists non-monovalued atomic reloid.

Proof. From the previous example it follows that the atomic reloid "Nfxg �RLD a is not mono-valued. �

Example 14.10. Non-convex reloids exist.

Proof. Let a be a non-trivial ultra�lter. Then idaRLD is non-convex. This follows from the factthat only reloidal products which are below idRLD(Base(a)) are reloidal products of ultra�lters andidaRLD is not their join. �

Example 14.11. (RLD)inf =/ (RLD)outf for a funcoid f .

Proof. Let f = idFCD(N ). Then (RLD)inf =F �

a �RLD a j a 2 atomsF(N )

and (RLD)outf =

idRLD(N ). But as have shown above a �RLD avidRLD(N ) for non-trivial ultra�lter a, and so(RLD)infv(RLD)outf . �

Proposition 14.12. idFCD(U)u"FCD(U;U)((U�U)n idU)= id(U)FCD =/ 0FCD(U;U) for every in�nite set U.

Proof. Note thatid(U)

FCD �X =X u(U) for every �lter X on U.

Let f = idFCD(U), g= "FCD(U;U)((U�U) n idU).Let x be a non-trivial ultra�lter on U. If X 2 x then cardX > 2 (In fact, X is in�nite but we

don't need this.) and consequently hgi�X =1F(U). Thus hgix=1F(U). Consequently

hf u gix= hf ixu hgix=xu 1F(U)=x:

Alsoid(U)

FCD �x=xu(U)=x.Let now x be a trivial ultra�lter. Then hf ix=x and hgix=1F(U) nx. So

hf u gix= hf ixu hgix=xu¡1F(U) nx

�=0F(U):

Alsoid(U)

FCD �x=xu(U)= 0F(U).So hf u gix=

id(U)

FCD �x for every ultra�lter x on U. Thus f u g= id(U)FCD . �

Example 14.13. There exist binary relations f and g such that "FCD(A;B)f u "FCD(A;B)g =/

"FCD(A;B)(f \ g) for some sets A, B such that f ; g �A�B.

172 Counter-examples about funcoids and reloids

Proof. From the proposition above. �

Example 14.14. There exists a principal funcoid which is not a complemented element of thelattice of funcoids.

Proof. I will prove that quasi-complement of the funcoid idFCD(N ) is not its complement. We have:¡idFCD(N )

��=G �

c2FCD(N ;N) j c� idFCD(N )

wG �

"Nf�g�FCD"Nf�g j �; � 2N ; "Nf�g�FCD"Nf�g� idFCD(N )

=Gf"Nf�g�FCD"Nf�g j �; � 2N ; �=/ �g

= "FCD(N ;N )Gff�g�f�g j �; � 2N ; �=/ �g

= "FCD(N ;N )(N �N n idN )

(used corollary 6.107). But by proved above¡idFCD(N )

��u idFCD(N )=/ 0F(N ):

[TODO: Stronger conjecture: ("FCDf)�= f� for every binary relation f .] �

Example 14.15. There exists a funcoid h such that GRh is not a �lter.

Proof. Consider the funcoid h = id(N )FCD . We have (from the proof of proposition 14.12) that

f 2GRh and g 2GRh, but f \ g= ;2/ GRh. �

Example 14.16. There exists a funcoid h=/ 0FCD(A;B) such that (RLD)outh=0RLD(A;B).

Proof. Consider h = id(N )FCD . By proved above h = f u g where f = idFCD(N ) = "FCD(N ;N )idN ,

g= "FCD(N ;N )(N �N n idN ).We have idN ;N �N n idN 2GRh.So (RLD)outh=

d "RLD(N ;N )

�GR hv"RLD(N ;N )(idN \ (N �N n idN )) = 0RLD(N ;N ); and thus

(RLD)outh=0RLD(N ;N ). �

Example 14.17. There exists a funcoid h such that (FCD)(RLD)outh=/ h.

Proof. It follows from the previous example. �

Example 14.18. (RLD)in(FCD)f =/ f for some convex reloid f .

Proof. Let f = idRLD(N ). Then (FCD)f = idFCD(N ). Let a be some non-trivial ultra�lter on N .Then (RLD)in(FCD)f w a�RLD avidRLD(N ) and thus (RLD)in(FCD)fvf . �

Example 14.19. There exist composable funcoids f and g such that

(RLD)out(g � f)=/ (RLD)outg � (RLD)outf:

Proof. Take f = id(N )FCD and g=1F(N )�FCD"Nf�g for some �2N . Then (RLD)outf =0RLD(N ;N )

and thus (RLD)outg � (RLD)outf =0RLD(N ;N ).We have g � f =(N)�FCD"Nf�g.Let's prove (RLD)out((N)�FCD "Nf�g) = (N)�RLD "Nf�g. [TODO: Separate proposition

asserting (RLD)in(X �FCD "Ff�g) = (RLD)out(X �FCD "Ff�g) = X �RLD "Ff�g. Does it hold forevery (co)complete funcoid?]

Really: (RLD)out((N ) �FCD "Nf�g) =d

"RLD(N ;N )�GR((N ) �FCD "Nf�g) =

d�"RLD(N ;N )(K �f�g) j K 2(N)

.

F 2GRd �

"RLD(N ;N )(K �f�g) j K 2(N),F 2GR(

df"NK j K 2(N)g�RLD "Nf�g)

for every F 2 P(N � N ). Thusd �

"RLD(N ;N )(K � f�g) j K 2 (N)=df"NK j K 2

(N)g�RLD "Nf�g=(N)�RLD "Nf�g.

Counter-examples about funcoids and reloids 173

So (RLD)out((N)�FCD"Nf�g)=(N)�RLD "Nf�g.Thus (RLD)out(g � f) =(N)�RLD "Nf�g=/ 0RLD(N ;N ). �

Example 14.20. (FCD) does not preserve �nite meets.

Proof. (FCD)¡idRLD(N )u

¡1RLD(N ;N ) n idRLD(N )

��=(FCD)0RLD(N ;N )=0FCD(N ;N ).

On the other hand, (FCD)idRLD(N )u (FCD)¡1RLD(N ;N ) n idRLD(N )

�= idFCD(N )u"FCD(N ;N )(N �

N n idN )= id(N )FCD =/ 0FCD(N ;N ) (used proposition 8.1). �

Corollary 14.21. (FCD) is not an upper adjoint (in general).

Considering restricting polynomials (considered as reloids) to ultra�lters, it is simple to provethat each that restriction is injective if not restricting a constant polynomial. Does this hold ingeneral? No, see the following example:

Example 14.22. There exists a monovalued reloid with atomic domain which is neither injectivenor constant (that is not a restriction of a constant function).

Proof. (based on [30]) Consider the function F 2NN�N de�ned by the formula (x; y) 7!x.Let !x be a non-trivial ultra�lter on the vertical line fxg�N for every x2N .Let T be the collection of such sets Y that Y \ (fxg�N)2!x for all but �nitely many vertical

lines. Obviously T is a �lter.Let ! 2 atomsT .For every x2N we have some Y 2T for which (fxg�N)\Y =; and thus "N�N (fxg�N)u!=

0F(N�N ).Let g =

¡"RLD(N ;N )F

�j!. If g is constant, then there exist a constant function G 2GR g and

F \G is also constant. Obviously dom "RLD(N�N ;N )(F \G)w !. The function F \G cannot beconstant because otherwise !vdom "RLD(N�N ;N )(F \G)v"N �N (fxg�N ) for some x2N whatis impossible by proved above. So g is not constant.

Suppose that g is injective. Then there exists an injection G2GR g. So domG intersects eachvertical line by atmost one element that is domG intersects every vertical line by the whole line orthe line without one element. Thus domG2T w! and consequently domG2/ ! what is impossible.

Thus g is neither injective nor constant. �

14.1 Second product. Oblique product

De�nition 14.23. A�FRLDB=(RLD)out(A�FCDB) for every �lters A and B. I will call it seconddirect product of �lters A and B.

Remark 14.24. The letter F is the above de�nition is from the word �funcoid�. It signi�es thatit seems to be impossible to de�ne A�FRLDB directly without referring to funcoidal product.

De�nition 14.25. Oblique products of �lters A and B are de�ned asAnB=

d �"RLDf j f 2Rel(Base(A);Base(B)); 8B 2B: "FCDf wA�FCD"Base(B)B

;

AoB=d �

"RLDf j f 2Rel(Base(A);Base(B)); 8A2A: "FCDf w"Base(A)A�FCDB.

Proposition 14.26. A�FRLDB vAnB vA�RLDB for every �lters A, B.

Proof. A n B vd �

"RLDf j f 2 Rel(Base(A); Base(B)); 8A 2 A; B 2 B: "FCDf w"Base(A)A�FCD"Base(B)B

vd �

"Base(A)A�RLD "Base(B)B j A2A; B 2B=A�RLDB.

A n B wdf"RLDf j f 2 Rel(Base(A); Base(B)); "FCDf w A �FCD Bg =

df"RLDf j f 2

xyGR(A�FCDB)g=(RLD)out(A�FCDB)=A�FRLDB. �

Conjecture 14.27. A�FRLDB@AnB for some �lters A, B.

174 Counter-examples about funcoids and reloids

A stronger conjecture:

Conjecture 14.28. A �FRLD B @ A n B @ A �RLD B for some �lters A, B. Particularly, is thisformula true for A=B=�u"R(0;+1)?

The above conjecture is similar to Fermat Last Theorem as having no value by itself but beingsomehow challenging to prove it (not expected to be as hard as FLT however).

Example 14.29. AnB@A�RLDB for some �lters A, B.

Proof. It's enough to prove AnB=/ A�RLDB.Let �+=�u"R(0;+1). Let A=B=�+.Let K =(6)jR�R.Obviously K 2/ GR(A�RLDB).A n B v "RLD(Base(A);Base(B))K and thus K 2 GR(A n B) because "FCD(Base(A);Base(B))K w

�+�FCD"Base(B)B=A�FCD"Base(B)B for B=(0;+1).Thus AnB=/ A�RLDB. �

Example 14.30. A �FRLD B @ A �RLD B for some �lters A, B. [TODO: Does it hold for someprincipal �lters A, B?]

Proof. This follows from the above example. �

Proposition 14.31. (AnB)u (AoB) =A�FRLDB for every �lters A, B.

Proof. (A n B) u (A o B) vdf"RLDf j f 2 Rel(Base(A); Base(B)); "FCDf w A �FCD Bg =d

f"RLDf j f 2 xyGR(A�FCDB)g=(RLD)out(A�FCDB) =A�FRLDB.To �nish the proof we need to show AnB wA�FRLDB and AoB wA�FRLDB. By symmetry

it's enough to show AnB wA�FRLDB what is proved above. �

Example 14.32. (AnB)t (AoB)@A�RLDB for some �lters A, B.

Proof. (based on [8]) Let A=B=(N). It's enough to prove (AnB)t (AoB)=/ A�RLDB.Let X 2A, Y 2B that is X 2(N), Y 2(N).Removing one element x from X produces a set P . Removing one element y from Y produces

a set Q. Obviously P 2(N), Q2(N).Obviously (P �N )[ (N �Q)2GR((AnB)t (AoB)).(P �N)[ (N �Q)+X �Y because (x; y)2X �Y but (x; y)2/ (P �N)[ (N �Q).Thus (P �N)[ (N �Q)2/ GR(A�RLDB) by properties of �lter bases. �

Example 14.33. (RLD)out(FCD)f =/ f for some convex reloid f .

Proof. Let f =A�RLDB where A and B are from example 14.30.(FCD)(A�RLDB)=A�FCDB by proposition 8.9.So (RLD)out(FCD)(A�RLDB) = (RLD)out(A�FCDB)=A�FRLDB=/ A�RLDB. �

14.1 Second product. Oblique product 175

Chapter 15

Pointfree funcoids

This chapter is based on [28].This is a routine chapter. There is almost nothing creative here. I just generalize theorems

about funcoids to the maximum extent for pointfree funcoids (de�ned below) preserving the proofidea. The main idea behind this chapter is to �nd weakest theorem conditions enough for the sametheorem statement as for above theorems for funcoids.

For these who know pointfree topology: Pointfree topology notions of frames and locales is anon-trivial generalization of topological spaces. Pointfree funcoids are di�erent: I just replace theset of �lters on a set with an arbitrary poset, this readily gives the de�nition of pointfree funcoid ,almost no need of creativity here.

Pointfree funcoids are used in the below de�nitions of products of funcoids.

15.1 De�nition

De�nition 15.1. Pointfree funcoid is a quadruple (A;B;�; �) where A and B are posets, �2BA

and � 2AB such that

8x2A; y 2B: (y�/ �x,x�/ �y):

De�nition 15.2. The source Src(A;B; �; �) =A and destination Dst(A;B; �; �) =B for everypointfree funcoid (A;B;�; �).

To every funcoid (A;B; �; �) corresponds pointfree funcoid (PA;PB; �; �). Thus pointfreefuncoids are a generalization of funcoids.

De�nition 15.3. I will denote FCD(A;B) the set of pointfree funcoids from A to B (that is withsource A and destination B), for every posets A and B.

Proposition 15.4. If A and B have least elements, then FCD(A;B) has least element.[TODO:Move below where order of pointfree funcoids is de�ned.]

Proof. It is (A;B;A�f0Bg;B�f0Ag). �

De�nition 15.5. h(A;B;�; �)i=def� for every pointfree funcoid (A;B;�; �).

De�nition 15.6. (A;B;�; �)¡1=(B;A; �;�) for every pointfree funcoid (A;B;�; �).

Proposition 15.7. If f is a pointfree funcoid then f¡1 is also a pointfree funcoid.

Proof. It follows from symmetry in the de�nition of pointfree funcoid. �

Obvious 15.8. (f¡1)¡1= f for every pointfree funcoid f .

De�nition 15.9. The relation [f ]2P(Src f �Dst f) is de�ned by the formula (for every pointfreefuncoid f and x2Src f , y 2Dst f)

x [f ] y=defy�/ hf ix:

177

Obvious 15.10. x [f ] y, y �/ hf ix, x �/ hf¡1iy for every pointfree funcoid f and x 2 Src f ,y 2Dst f .

Obvious 15.11. [f¡1]=[f ]¡1 for every pointfree funcoid f .

Theorem 15.12. Let A and B be posets. Then:

1. If A is separable, for given value of hf i there exists no more than one f 2FCD(A;B).

2. If A and B are separable, for given value of [f ] there exists no more than one f 2FCD(A;B).

Proof. Let f ; g 2FCD(A;B).

1. Let hf i= hgi. Then for every x 2A, y 2B we have x�/ hf¡1iy, y �/ hf ix, y �/ hgix,x�/ hg¡1iy and thus by separability of A we have hf¡1iy= hg¡1iy that is hf¡1i= hg¡1i andso f = g.

2. Let [f ]=[g]. Then for every x 2A, y 2B we have y �/ hf ix, x [f ] y, x [g] y, y �/ hgixand thus by separability of B we have hf ix = hgix that is hf i = hgi. Similarly we havehf¡1i= hg¡1i. Thus f = g. �

Proposition 15.13. If Src f and Dst f have least elements, then hf i0Src f = 0Dst f for everypointfree funcoid f . [TODO: Previously I required separability of Dst f . It turned out to be asuper�uous condition. Remove it also in consequences of this proposition.]

Proof. y �/ hf i0Src f , 0Src f �/ hf¡1iy, 0 for every y 2 Dst f . Thus hf i0Src f � hf i0Src f. Sohf i0Src f =0Dst f. �

Proposition 15.14. If Dst f is a separable meet-semilattice then hf i is a monotone function(for a pointfree funcoid f). [FIXME: Added condition to be a meet-semilattice, add it also in allconsequences.] [TODO: Check whether existence of least element of Dst f is required (in theorem3.14).]

Proof. a v b) 8x 2 Dst f : (a �/ hf¡1ix) b �/ hf¡1ix)) 8x 2 Dst f : (x �/ hf ia) x �/ hf ib),?hf ia� ?hf ib)hf iavhf ib (used theorem 3.14 and that it is a separable meet-semilattice). �

Theorem 15.15. Let f be a pointfree funcoid from a starrish join-semilattice Src f to a separablestarrish join-semilattice Dst f . Then hf i(it j) = hf iit hf ij for every i; j 2 Src f .

Proof.

?hf i(it j) =

fy 2Dst f j y�/ hf i(it j)g =

fy 2Dst f j it j�/ hf¡1iyg =

fy 2Dst f j i�/ hf¡1iy _ j�/ hf¡1iyg =

fy 2Dst f j y�/ hf ii_ y�/ hf ijg =

fy 2Dst f j y�/ hf iit hf ijg =

?(hf iit hf ij):

Thus hf i(it j)= hf iit hf ij by separability. �

Proposition 15.16. Let f be a pointfree funcoid. Then:

1. k [f ] it j,k [f ] i_k [f ] j for every i; j2Dst f , k2Src f if Dst f is a starrish join-semilattice.

2. it j [f ]k, i [f ]k_ j [f ]k for every i; j2Src f , k2Dst f if Src f is a starrish join-semilattice.

Proof. 1. k [f ] it j, it j�/ hf ik, i�/ hf ik _ j�/ hf ik, k [f ] i_ k [f ] j.2. Similar. �

178 Pointfree funcoids

15.2 Composition of pointfree funcoidsDe�nition 15.17. Composition of pointfree funcoids is de�ned by the formula

(B;C;�2; �2) � (A;B;�1; �1) = (A;C;�2 ��1; �1 � �2):

De�nition 15.18. I will call funcoids f and g composable when Dst f = Src g.

Proposition 15.19. If f , g are composable pointfree funcoids then g � f is pointfree funcoid.

Proof. Let f =(A;B;�1; �1), g=(B;C;�2; �2). For every x; y 2A we have

y�/ (�2 ��1)x, y�/ �2�1x,�1x�/ �2y,x�/ �1�2y, x�/ (�1 � �2)y:

So (A;C;�2 ��1; �1 � �2) is a pointfree funcoid. �

Obvious 15.20. hg � f i= hgi � hf i for every composable pointfree funcoids f and g.

Theorem 15.21. (g � f)¡1= f¡1 � g¡1 for every composable pointfree funcoids f and g.

Proof.

h(g � f)¡1i= hf¡1i � hg¡1i= hf¡1 � g¡1i;h((g � f)¡1)¡1i= hg � f i= h(f¡1 � g¡1)¡1i:

�

Proposition 15.22. (h � g) � f =h � (g � f) for every composable pointfree funcoids f , g, h.

Proof.h(h � g) � f i= hh � gi � hf i= hhi � hgi � hf i= hhi � hg � f i= hh � (g � f)i.h((h � g) � f)¡1i= hf¡1 � (h� g)¡1i= hf¡1 � g¡1 �h¡1i= h(g � f)¡1 �h¡1i= h(h � (g � f))¡1i. �

15.3 Pointfree funcoid as continuationProposition 15.23. Let f be a pointfree funcoid. Then for every x2Src f , y 2Dst f we have

1. If (Src f ;Z) is a �ltrator with separable core then x [f ] y,8X 2up(Src f ;Z)x:X [f ] y.

2. If (Dst f ;Z) is a �ltrator with separable core then x [f ] y,8Y 2up(Dst f ;Z) y:x [f ]Y .

Proof. We will prove only the second because the �rst is similar.x [f ] y, y�/Dst f hf ix,8Y 2up(Dst f ;Z) y:Y �/Dst f hf ix,8Y 2 up(Dst f ;Z) y:x [f ] Y . �

Corollary 15.24. Let f be a pointfree funcoid and (Src f ; Z0), (Dst f ; Z1) are �ltrators withseparable core. Then

x [f ] y,8X 2 up(Src f ;Z0)x; Y 2 up(Dst f ;Z1) y:X [f ]Y :

Proof. Apply the proposition twice. �

Theorem 15.25. Let f be a pointfree funcoid. Let (Src f ; Z0) be a �nitely meet-closed �ltratorwith separable core which is a meet-semilattice and 8x2 Src f : up(Src f ;Z0)x=/ ; and (Dst f ;Z1) isa primary �ltrator over a boolean lattice.

hf ix=lDst f

hhf iiup(Src f ;Z0)x:

Proof. By the previous proposition for every y 2Dst f :

y�/Dst f hf ix, x [f ] y,8X 2up(Src f ;Z0)x:X [f ] y,8X 2up(Src f ;Z0)x: y�/Dst f hf iX:

15.3 Pointfree funcoid as continuation 179

Let's denote W =�yuDst f hf iX j X 2up(Src f ;Z0)x

. We will prove that W is a generalized �lter

base over Z1. To prove this enough to show that V =�hf iX j X 2 up(Src f ;Z0) x

is a generalized

�lter base.Let P ;Q 2 V . Then P = hf iA, Q= hf iB where A; B 2 up(Src f ;Z0) x; A uZ0B 2 up(Src f ;Z0) x

(used the fact that it is a �nitely meet-closed and theorem 4.44) and R v P uDst f Q for R =hf i(AuZ0B)2V because Dst f is separable by obvious 4.136. So V is a generalized �lter base andthus W is a generalized �lter base.

0Dst f 2/W,dDst f W 3/ 0Dst f by theorem 4.121. That is

8X 2 up(Src f ;Z0)x: y uDst f hf iX =/ 0Dst f, yuDst flDst f

hhf iiup(Src f ;Z0)x=/0Dst f:

Comparing with the above, y uDst f hf ix =/ 0Dst f, y uDst f dDst f hhf iiup(Src f ;Z0)x=/0Dst f. Sohf ix=

dDst fhhf iiup(Src f ;Z0)x because Dst f is separable (obvious 4.136 and the fact that Z1 is aboolean lattice). �

Theorem 15.26. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices.

1. A function �2BZ0 conforming to the formulas (for every I ; J 2Z0)

�0Z0=0B; �(I t J)=�I t�J


hf iX =lBh�iup(A;Z0)X (15.1)

for every X 2A.

2. A relation � 2P(Z0 � Z1) conforming to the formulas (for every I ; J ; K 2 Z0 and I 0; J 0;K 02Z1)

:(0Z0 � I 0); I tJ �K 0, I �K 0_J �K 0;

:(I � 0Z1); K � I 0tJ 0,K � I 0_K �J 0(15.2)


X [f ]Y,8X 2up(A;Z0)X ; Y 2up(B;Z1)Y:X �Y (15.3)

for every X 2A, Y 2B.

Proof. Existence of no more than one such pointfree funcoids and formulas (15.1) and (15.3) followfrom two previous theorems.

2. fY 2Z1 j X �Y g is obviously a free star for every X 2Z0. By properties of �lters on booleanlattices, there exist a unique �lter �X such that @(�X) = fY 2 Z1 j X � Y g for every X 2 Z0.Thus �2BZ0. Similarly it can be de�ned � 2AZ1 by the formula @(�Y )= fX 2Z0 j X �Y g. Let'scontinue the functions � and � to �02BA and � 02AB by the formulas

�0X =lBh�iup(A;Z0)X and � 0Y =

lAh� iup(B;Z1)Y

and � to � 02P(A�B) by the formula

X � 0Y,8X 2up(A;Z0)X ; Y 2 up(B;Z1)Y :X �Y :

Y u�0X =/ 0B,Y udh�iup(A;Z0)X =/ 0B,

dhY u ih�iup(A;Z0)X =/ 0B. Let's prove that

W = hY u ih�iup(A;Z0)X

is a generalized �lter base: To prove it is enough to show that h�iup(A;Z0)X is a generalized �lterbase.

If A; B 2 h�iup(A;Z0) X then exist X1; X2 2 up(A;Z0) X such that A = �X1 and B = �X2.Then �(X1uZ0X2)2 h�iup(A;Z0)X . So h�iup(A;Z0)X is a generalized �lter base and thus W is ageneralized �lter base.


By properties of generalized �lter bases,dhY u ih�iup(A;Z0)X =/ 0B is equivalent to

8X 2 up(A;Z0)X :Y u�X =/ 0B;

what is equivalent to 8X 2 up(A;Z0) X ; Y 2 up(B;Z1) Y : Y uB �X =/ 0B , 8X 2 up(A;Z0) X ;Y 2up(B;Z1)Y :Y 2 @(�X),8X 2up(A;Z0)X ; Y 2up(B;Z1)Y :X �Y . Combining the equivalencieswe get Yu�0X =/ 0B,X � 0Y . Analogously X u� 0Y=/ 0A,X � 0Y. So Yu�0X =/ 0B,X u� 0Y=/ 0A,that is (A;B;�0; � 0) is a pointfree funcoid. From the formula Y u�0X =/ 0B,X � 0Y it follows that[(A;B;�0; � 0)] is a continuation of �.

1. Let de�ne the relation � 2P(Z0�Z1) by the formula X �Y ,Y uB�X =/ 0B.That :(0Z0 � I 0) and :(I � 0Z1) is obvious. We have K � I 0tZ1 J 0, (I 0tZ1 J 0) uB �K =/ 0B,

(I 0 tB J 0) u �K =/ 0B, (I 0 uB �K) t (J 0 uB �K) =/ 0B, I 0 uB �K =/ 0B _ J 0 uB �K =/ 0B,K � I 0 _ K � J 0 and I tZ0 J � K 0 , K 0 uB �(I tZ0 J) =/ 0B , K 0 uB (�I t �J) =/ 0B ,(K 0uB�I)t (K 0uB�J)=/ 0B,K 0uB�I =/ 0B_K 0uB�J =/ 0B, I �K 0_J �K 0.

That is the formulas (15.2) are true.Accordingly the above � can be continued to the relation [f ] for some f 2FCD(A;B).8X 2 Z0; Y 2 Z1: (Y uB hf iX =/ 0B , X [f ] Y , Y uB �X =/ 0B), consequently 8X 2 Z0:

�X = hf iX because our �ltrator is with separable core. So hf i is a continuation of �. �

Proposition 15.27. Let (Src f ;Z0) be a primary �ltrator over a bounded distributive lattice and(Dst f ; Z1) is a primary �ltrator over a boolean lattice. If S is a generalized �lter base on Src fthen hf i

dSrc f S=dDst f hhf iiS for every pointfree funcoid f .

Proof. First the meetsdSrc f

S anddDst f hhf iiS exist by corollary 4.107.

(Src f ; Z0) is a �nitely meet-closed �ltrator by proposition 4.97 and with separable core bytheorem 4.112; thus we can apply theorem 15.25 (upx=/ ; is obvious).hf idSrc f S v hf iX for every X 2 S because Dst f is separable by obvious 4.136 and thus

hf idSrc f S v

dDst f hhf iiS.Taking into account properties of generalized �lter bases:

hf ilSrc f

S =

lDst f

hhf iiupl

S =

lDst f

hhf iifX j 9P 2S:X 2upPg =

lDst f

fhf iX j 9P 2S:X 2upPg w (because Dst f is a separable poset)lDst f

fhf iP j P 2Sg =

lDst f

hhf iiS:�

15.4 The order of pointfree funcoids

De�nition 15.28. The order of pointfree funcoids FCD(A;B) is de�ned by the formula:

f v g,8x2A: hf ixvhgix^8y 2B: hf¡1iyvhg¡1iy:

Proposition 15.29. It is really a partial order on the set FCD(A;B).

Proof.


15.4 The order of pointfree funcoids 181

Transitivity. It follows from transitivity of the order relations on A and B.

Antisymmetry. It follows from antisymmetry of the order relations on A and B. �

Remark 15.30. It is enough to de�ne order of pointfree funcoids on every set FCD(A;B) whereA and B are posets. We do not need to compare pointfree funcoids with di�erent sources ordestinations.

Obvious 15.31. f v g) [f ]�[g] for every f ; g 2 FCD(A;B) for every posets A and B.

Theorem 15.32. If A and B are separable posets then f v g, [f ]�[g].

Proof. From the theorem 15.12. �

Theorem 15.33. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. Then forR2PFCD(A;B) and X 2Z0, Y 2Z1 we have:

1. X [FR]Y ,9f 2R:X [f ]Y ;

2. hFRiX =

Ffhf iX j f 2Rg.

Proof.2. �X=

defFfhf iX j f 2Rg (by corollary 4.107 all joins on B exist). We have �0A=0B;

�(I t Z0J) =Gfhf i(I tZ0 J) j f 2Rg

=Gfhf i(I tA J) j f 2Rg

=Gfhf iI tB hf iJ j f 2Rg

=Gfhf iI j f 2RgtB

Gfhf iJ j f 2Rg

= �I tB�J

(used theorem 15.15). By theorem 15.26 the function � can be continued to hhi for an h2FCD(A;B). Obviously

8f 2R:hw f: (15.4)

And h is the least element of FCD(A;B) for which the condition (15.4) holds. So h=FR.

1. X [FR] Y , Y uB h

FRiX =/ 0B , Y uB

Ffhf iX j f 2 Rg =/ 0B , 9f 2 R:

Y uB hf iX =/ 0B,9f 2R:X [f ]Y (used theorem 4.118). �

Corollary 15.34. If (A;Z0) and (B;Z1) are primary �ltrators over boolean lattices then FCD(A;B) is a complete lattice.

Proof. Apply [26]. �

Theorem 15.35. Let A and B be starrish join-semilattices. Then for f ; g 2 FCD(A;B):

1. hf t gix= hf ixt hgix for every x2A;

2. [f t g]=[f ][[g].

Proof.1. Let �X =

defhf ixt hgix; �Y=defhf¡1iy t hg¡1iy for every x2A, y 2B. Then

y�/ �x , y�/ hf ix_ y�/ hgix, x�/ hf¡1iy_ x�/ hg¡1iy, x�/ hf¡1iyt hg¡1iy, x�/ �y:

So h=(A;B;�; �) is a pointfree funcoid. Obviously hw f and hw g. If pw f and pw g for somep2FCD(A;B) then hpixwhf ixthgix= hhix and hp¡1iywhf¡1iythg¡1iy= hh¡1iy that is pwh.So f t g=h.


2. x [f t g] y, y�/ hf t gix, y�/ hf ixt hgix, y�/ hf ix_ y�/ hgix,x [f ] y_x [g] y for everyx2A, y 2B. �

Theorem 15.36. Let f be a pointfree funcoid from a separable poset A to a separable poset B.If hf i is an injection, then hf i is an order embedding A!B.

Proof. Suppose xw y but hf ixwhf iy.Then by separability of B there exist z�/ hf iy such that z�hf ix.Thus hf¡1iz�x and hf¡1iz�/ y what is impossible for xw y. �

Corollary 15.37. Let f be a pointfree funcoid from a separable poset A to a separable poset B.If hf i is a bijection A!B, then hf i is an order isomorphism A!B.

15.5 Domain and range of a pointfree funcoid

De�nition 15.38. Let A be a poset. The identity pointfree funcoid idFCD(A)=(A;A; idA; idA).

It is trivial that identity funcoid is really a pointfree funcoid.Let now A be a meet-semilattice.

De�nition 15.39. Let a2A. The restricted identity pointfree funcoid idaFCD(A)

=(A;A;auA;auA ).

Proposition 15.40. The restricted pointfree funcoid is a pointfree funcoid.

Proof. We need to prove that (auAx)�/A y, (auA y)�/A x what is obvious. �

Obvious 15.41.�ida

FCD(A)�¡1

= idaFCD(A).

Obvious 15.42. xhida

FCD(A)iy, a�/A xuA y for every x; y 2A.

De�nition 15.43. I will de�ne restricting of a pointfree funcoid f to an element a2Src f by theformula f ja=

deff � ida

FCD(Src f).

De�nition 15.44. Image of f will be de�ned by the formula im f =Fhhf ii�Src f .

Obvious 15.45. im f w fx for every x2Src f whenever im f is de�ned.

Proposition 15.46. im f = hf i1 if Src f has greatest element 1 and Dst f is a separable poset.

Proof. hf i1 is greater than every hf ix (where x2 Src f) by proposition 15.14 and thus

hf i1=max hhf ii�Src f = im f: �

De�nition 15.47. Domain of a pointfree funcoid f is de�ned by the formula dom f = imf¡1.

Proposition 15.48. hf idom f = im f if f is a pointfree funcoid and Src f has greatest element1 and Dst f is a separable poset.

Proof. y �/ hf idom f , dom f �/ hf¡1iy , hf¡1i1 �/ hf¡1iy , hf¡1iy =/ 0 , 1 �/ hf¡1iy ,y�/ hf i1, y�/ im f for every y 2Dst f .

So hf idom f = im f by separability of Dst f . �

Proposition 15.49. hf ix = hf i(x u dom f) whenever dom f is de�ned, for every x 2 Src f fora pointfree funcoid f whose source is a meet-semilattice with least element and destination is aseparable poset with least element.

15.5 Domain and range of a pointfree funcoid 183

Proof. For every y 2 Dst f we have y �/ hf i(x u dom f) , x u dom f u hf¡1iy =/ 0Src f ,x u imf¡1 u hf¡1iy =/ 0Src f , x u hf¡1iy =/ 0Src f , y �/ hf ix. Thus hf ix = hf i(x u dom f) byseparability of Dst f . �

Proposition 15.50. x�/ dom f, (hf ix is not least) for every pointfree funcoid f and x2Src fif Dst f has greatest element 1 and Src f is a separable poset.

Proof. x�/ dom f, x�/ hf¡1i1Dst f, 1Dst f �/ hf ix, (hf ix is not least). �

Corollary 15.51. dom f =Ffa2 atomsSrc f j hf ia=/ 0Dst fg for every pointfree funcoid f whose

destination is a bounded poset and source is a separable atomistic meet-semilattice.

Proof. For every a2atomsSrc f we have a�/ dom f,a�/ hf¡1i1Dst f,1Dst f�/ hf ia,hf ia=/ 0Dst f.So

dom f =Ffa2 atomsSrc f j a�/ dom f g=

Ffa2 atomsSrc f j hf ia=/ 0Dst fg. �

Proposition 15.52. dom(f ja)=audom f for every pointfree funcoid f and a2Src f where Src fis a separable meet-semilattice and Dst f has greatest element.

Proof. dom(f ja) = im�ida

FCD(Src f) � f¡1�=Dida

FCD(Src f)Ehf¡1i1Dst f = a u hf¡1i1Dst f =

au dom f . �

Proposition 15.53. For every composable pointfree funcoids f and g where the posets Src f andDst f =Src g have greatest elements and Dst f and Dst g are separable:


2. If im f v dom g then dom(g � f) =dom g.

Proof.

1. im(g � f)= hg � f i1Src f = hgihf i1Src f = hgiim f = hgidom g= hgi1Src g= im g.

2. dom(g � f)= im(f¡1 � g¡1) what by the proved is equal to im f¡1 that is dom f . �

15.6 Category of pointfree funcoids

I will de�ne the category pFCD of pointfree funcoids:

� The class of objects are small posets.

� The set of morphisms from A to B is FCD(A;B).

� The composition is the composition of pointfree funcoids.

� Identity morphism for an object A is (A;A; idA; idA).

To prove that it is really a category is trivial.The category of pointfree funcoid triples is de�ned as follows:

� Objects are pairs (A;A) where A is a small poset and A2A.

� The morphisms from an object (A;A) to an object (B;B) are tuples (A;B;A;B; f) wheref 2 FCD(A; B) and dom f v A ^ im f v B. [FIXME: Domain and image are not alwaysde�ned. Even if it's de�ned, the composition law may not hold. We can require insteadf vA�FCDB, but this is de�ned only for posets with least elements.]

� The composition is de�ned by the formula (B; C; g) � (A;B; f)= (A; C; g � f).

� Identity morphism for an object (A;A) is idAFCD(A). [FIXME: De�ned only for meet-semi-

lattices. We can also de�ne a wider precategory without identity.]



15.7 Specifying funcoids by functions or relations on atomic�lters

Theorem 15.54. Let A be an atomic poset and (B; Z1) is a primary �ltrator over a booleanlattice. Then for every f 2FCD(A;B) and X 2A we have

hf iX =GBhhf iiatomsAX :

Proof. For every Y 2Z1 we have

Y �/B hf iX , X �/A hf¡1iY, 9x2 atomsAX :x�/A hf¡1iY, 9x2 atomsAX :Y �/B hf ix:

Thus @ hf iX =Sh@ i hhf iiatomsAX = @

FB hhf iiatomsAX (used theorem 4.132). Consequentlyhf iX =

FB hhf iiatomsAX by the corollary 4.128. �

Proposition 15.55. Let f be a pointfree funcoid. Then for every X 2Src f and Y 2Dst f

1. X [f ]Y,9x2 atomsX :x [f ]Y if Src f is an atomic poset.

2. X [f ]Y,9y 2 atomsY:X [f ] y if Dst f is an atomic poset.

Proof. I will prove only the second as the �rst is similar.If X [f ] Y, then Y �/ hf iX , consequently exists y 2 atoms Y such that y�/ hf iX , X [f ] y. The

reverse is obvious. �

Corollary 15.56. If f is a pointfree funcoid with both source and destination being atomic posets,then for every X 2Src f and Y 2Dst f

X [f ]Y,9x2 atomsX ; y 2 atomsY:x [f ] y:

Proof. Apply the theorem twice. �

Corollary 15.57. If A is a separable atomic poset and B is a separable poset then f 2FCD(A;B)is determined by the values of hf iX for X 2 atomsA.

Proof. y�/ hf ix,x�/ hf¡1iy,9X 2 atomsx:X �/ hf¡1iy,9X 2 atomsx: y�/ hf iX.Thus by separability of B we have hf i is determined by hf iX for X 2 atomsx.By separability of A we infer that f can be restored from hf i (theorem 15.12). �

Theorem 15.58. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices.

1. A function �2BatomsA such that (for every a2 atomsA)

�avl G

�h�i � atomsA�up(A;Z0) a (15.5)


hf iX =Gh�iatomsAX (15.6)

for every X 2A.

2. A relation � 2P(atomsA� atomsB) such that (for every a2 atomsA, b2 atomsB)

8X 2 up(A;Z0) a; Y 2 up(B;Z1) b9x2 atomsAX; y 2 atomsBY :x � y) a � b (15.7)


X [f ]Y,9x2 atomsX ; y 2 atomsY:x � y (15.8)

for every X 2A, Y 2B.

15.7 Specifying funcoids by functions or relations on atomic filters 185

Proof. Existence of no more than one such funcoids and formulas (15.6) and (15.8) follow fromthe theorem 6.65 and corollary 15.24 and the fact that our �ltrators are with separable core.

1. Consider the function �02BZ0 de�ned by the formula (for every X 2Z0)

�0X =Gh�iatomsAX:

Obviously �0 0Z0=0B. For every I ; J 2Z0

�0(I tJ) =Gh�iatomsA(I tJ)

=Gh�i(atomsA I [ atomsA J)

=G

(h�iatomsA I [ h�iatomsAJ)

=Gh�i atomsA I t

Gh�iatomsA J:

= �0 I t�0J:

Let continue �0 till a pointfree funcoid f (by the theorem 15.26): hf iX =dh�0iup(A;Z0)X .

Let's prove the reverse of (15.5):l G

�h�i � atomsA�up(A;Z0) a =

l G�h�i

�hatomsAiup(A;Z0)a

vl G

�h�i�ffagg

=l �¡G

�h�i�fag

=l �G

h�ifag

=l �G

f�ag=lf�ag=�a:

Finally,

�a=l G

�h�i � atomsA�up(A;Z0) a=

lh�0iup(A;Z0) a= hf ia;

so hf i is a continuation of �.2. Consider the relation � 02P(Z0�Z1) de�ned by the formula (for every X 2Z0, Y 2Z1)

X � 0Y ,9x2 atomsAX; y 2 atomsBY :x � y:

Obviously :(X � 0 0Z1) and :(0Z0 � 0Y ).

I tJ � 0 Y , 9x2 atomsA(I t J); y 2 atomsBY :x � y, 9x2 atomsA I [ atomsA J ; y 2 atomsBY :x � y

, 9x2 atomsA I ; y 2 atomsBY :x � y_ 9x2 atomsAJ ; y 2 atomsBY :x � y, I � 0Y _ J � 0 Y ;

similarly X � 0 I t J,X � 0 I _X � 0J . Let's continue � 0 till a funcoid f (by the theorem 15.26):

X [f ]Y,8X 2up(A;Z0)X ; Y 2 up(B;Z1)Y :X � 0Y :

The reverse of (15.7) implication is trivial, so

8X 2 up(A;Z0) a; Y 2up(B;Z1) b9x2 atomsAX; y 2 atomsB Y :x � y, a � b:

8X 2up(A;Z0)a;Y 2up(B;Z1) b9x2atomsAX; y2atomsBY :x� y,8X 2up(A;Z0)a;Y 2up(B;Z1) b:X � 0Y , a [f ] b.

So a � b, a [f ] b, that is [f ] is a continuation of �. �

Theorem 15.59. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. If R 2PFCD(A;B) and x2 atomsA, y 2 atomsB, then

1. hdRix=

dfhf ix j f 2Rg;

2. x [dR] y,8f 2R:x [f ] y.

Proof.


2. Let denote x � y,8f 2R:x [f ] y.

8X 2 up(A;Z0) a; Y 2up(B;Z1) b 9x2 atomsAX; y 2 atomsBY :x � y)8f 2R;X 2up(A;Z0) a; Y 2 up(B;Z1) b 9x2 atomsAX; y 2 atomsBY :x [f ] y)

8f 2R;X 2 up(A;Z0) a; Y 2up(B;Z1) b:X [f ]Y )8f 2R: a [f ] b,

a � b:

So, by the theorem 15.58, � can be continued till [p] for some p2 FCD(A;B).For every q2FCD(A;B) such that 8f 2R: qv f we have x [q] y)8f 2R:x [f ] y,x� y,x [p] y,

so q v p. Consequently p=dR.

From this x [dR] y,8f 2R:x [f ] y.

1. From the former y 2 atomsB hd

Rix, y u hd

Rix =/ 0B , 8f 2 R: y u hf ix =/ 0B ,y 2ThatomsBifhf ix j f 2Rg, y 2 atoms

dfhf ix j f 2Rg for every y 2 atomsB.

B is atomically separable by the corollary 4.138. ThuslR�x=

lfhf ix j f 2Rg: �

15.8 More on composition of pointfree funcoids

Proposition 15.60. [g � f ]=[g]�hf i= hg¡1i¡1� [f ] for every composable pointfree funcoids f andg.

Proof. x [g � f ] y, y�/ hg � f ix, y�/ hgihf ix,hf ix [g] y,x([g]�hf i)y for every x2A, y 2B.Thus [g � f ]=[g]�hf i. [g � f ]=[(f¡1 � g¡1)¡1]=[f¡1 � g¡1]¡1=([f¡1]�hg¡1i)¡1= hg¡1i¡1 � [f ]. �

Theorem 15.61. Let f and g be pointfree funcoids and A = Dst f = Src g is an atomic poset.Then for every X 2Src f and Z 2Dst g

X [g � f ]Z,9y 2 atomsA: (X [f ] y^ y [g]Z):

Proof.

9y 2 atomsA: (X [f ] y ^ y [g]Z) , 9y 2 atomsA: (Z �/ hgiy ^ y�/ hf iX ), 9y 2 atomsA: (y�/ hg¡1iZ ^ y�/ hf iX ), hg¡1iZ �/ hf iX, X [g � f ]Z:

�

Theorem 15.62. Let A, B, C be separable starrish join-semilattices and B is atomic. Then:

1. f � (gt h)= f � g t f �h for g; h2FCD(A;B) and f 2 FCD(B;C).

2. (gt h) � f = g � f th � f for f 2FCD(A;B) and g; h2FCD(B;C).

Proof. I will prove only the �rst equality because the other is analogous.We can apply theorem 15.35.For every X 2A, Y 2C

X [f � (g th)]Z , 9y 2 atomsB: (X [g th] y ^ y [f ]Z), 9y 2 atomsB: ((X [g] y_ X [h] y)^ y [f ]Z), 9y 2 atomsB: ((X [g] y^ y [f ]Z)_ (X [h] y ^ y [f ]Z)), 9y 2 atomsB: (X [g] y ^ y [f ]Z)_ 9y 2 atomsB: (X [h] y ^ y [f ]Z), X [f � g]Z _ X [f �h]Z, X [f � gt f �h]Z:

15.8 More on composition of pointfree funcoids 187

Thus f � (gt h)= f � g t f �h by theorem 15.12. �

Theorem 15.63. Let A, B, C be posets of �lters over some boolean lattices, f 2 FCD(A; B),g 2FCD(B;C), h2FCD(A;C). Then

g � f �/ h, g�/ h � f¡1:

Proof.

g � f �/ h ,9a2 atomsA; c2 atomsC: a [(g � f)uh] c ,

9a2 atomsA; c2 atomsC: (a [g � f ] c^ a [h] c) ,9a2 atomsA; b2 atomsB; c2 atomsC: (a [f ] b^ b [g] c^ a [h] c) ,

9b2 atomsB; c2 atomsC: (b [g] c^ b [h� f¡1] c) ,9b2 atomsB; c2 atomsC: b [g u (h� f¡1)] c ,

g�/ h � f¡1:�

15.9 Direct product of elements

De�nition 15.64. Funcoidal product A�FCDB where A2A, B2B and A and B are posets withleast elements is a pointfree funcoid such that for every X 2A, Y 2B

hA�FCDBiX =

(B if X �/ A;0B if X �A; and h(A�FCDB)¡1iY =

(A if Y �/ B;0A if Y �B:

Proposition 15.65. A�FCDB is really a pointfree funcoid and

X [A�FCDB]Y,X �/ A^Y �/ B:

Proof. Obvious. �

Proposition 15.66. Let A and B be separable bounded posets, f 2 FCD(A;B), A 2 A, B 2B.Then

f vA�FCDB,dom f vA^ im f vB:

Proof. If f v A �FCD B then dom f v dom(A �FCD B) v A, im f v im(A �FCD B) v B. Ifdom f vA^ im f vB then X [f ]Y)Y �/ hf iX )Y �/ B and similarly X [f ]Y)X �/ A.

So [f ]�[A�FCDB] and thus using separability f vA�FCDB. �

Theorem 15.67. Let A, B be sets of �lters over boolean lattices. For every f 2 FCD(A;B) andA2A, B2B

f u (A�FCDB)= idBFCD(B) � f � idA

FCD(A):

Proof. From above FCD(A;B) is a (complete) lattice.

h=def

idBFCD(B) � f � idA

FCD(A). For every X 2A

hhiX =DidB

FCD(B)Ehf iDidA

FCD(A)EX =B u hf i(AuX )

and

hh¡1iX =DidA

FCD(A)Ehf¡1i

DidB

FCD(B)EX =Au hf¡1i(BuX ):


From this, as easy to show, hv f and hvA�FCDB. If gv f ^ gvA�FCDB for a g 2FCD(A;B)then dom g vA, im g vB,

hgiX =Bu hgi(AuX )vBu hf i(AuX )=DidB

FCD(B)Ehf iDidA

FCD(A)EX = hhiX ;

and similarly hg¡1iX v hh¡1iX .g vh. So h= f u (A�FCDB). �

Corollary 15.68. Let A, B be sets of �lters over boolean lattices. For every f 2FCD(A;B) andA2A we have f jA=f u (A�FCD1B).

Proof. f u (A�FCD1B) = id1BFCD(B) � f � idA

FCD(A)= f � idA

FCD(A)= f jA. �

Corollary 15.69. Let A, B be sets of �lters over boolean lattices. For every f 2FCD(A;B) andA2A, B2B we have

f �/ A�FCDB,A [f ]B:

Proof. f �/ A �FCD B , f u (A �FCD B) =/ 0FCD(A;B) ,f u (A �FCD(A;B) B)

�1A =/ 0B ,D

idBFCD(B) � f � idA

FCD(A)E1A=/ 0B,

DidB

FCD(B)Ehf iDidA

FCD(A)E1A=/ 0B,B u hf i(Au 1A) =/ 0B,

Bu hf iA=/ 0B,A [f ]B. �

Theorem 15.70. Let A, B be sets of �lters over boolean lattices. Then the poset FCD(A;B) isseparable.

Proof. Let f ; g2FCD(A;B) and f =/ g. By the theorem 15.12 [f ]=/ [g]. That is there exist x; y2A

such that x [f ] y< x [g] y that is f u (x�FCD y) =/ 0FCD(A;B)< g u (x�FCD y) =/ 0FCD(A;B). ThusFCD(A;B) is separable. �

Theorem 15.71. Let A and B be posets of �lters over boolean lattices. If S 2P(A�B) thenlfA�FCDB j (A;B)2Sg=

ldomS �FCD

limS:

Proof. If x2 atomsA then by the theorem 15.59lfA�FCDB j (A;B)2Sg

�x=

lfhA�FCDBix j (A;B)2Sg:

If xud

domS=/ 0A then

8(A;B)2S: (xuA=/ 0A^ hA�FCDBix=B);fhA�FCDBix j (A;B)2Sg= imS;

if xud

domS=0A then

9(A;B)2S: (xuA=0A^ hA�FCDBix=0B);

fhA�FCDBix j (A;B)2Sg3 0B:

So lfA�FCDB j (A;B)2Sg

�x=

( dimS if xu

ddomS=/ 0A;

0B if xud

domS=0A:

From this by theorem 15.58 the statement of our theorem follows. �

Corollary 15.72. Let A and B be posets of �lters over boolean lattices.For every A0;A12A and B0;B12B

(A0�FCDB0)u (A1�FCDB1)= (A0uA1)�FCD (B0uB1):

Proof. (A0�FCDB0) u (A1�FCDB1) =dfA0�FCDB0;A1�FCDB1g what is by the last theorem

equal to (A0uA1)�FCD (B0uB1). �

15.9 Direct product of elements 189

Theorem 15.73. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices. If A2A thenA�FCD is a complete homomorphism of the lattice A to a the lattice FCD(A;B), if also A=/ 0A

then it is an order embedding.

Proof. Let S 2PA, X 2Z0, x2 atomsA.GhA�FCD iS

�X =

GfhA�FCDBiX j B 2Sg

=

( FS ifX uAA=/ 0A

0B ifX uAA=0A

=A�FCD

GS�X:

ThusFhA�FCD iS=A�FCDF S by theorem 15.25.l

hA�FCD iS�x =

lfhA�FCDBix j B 2Sg

=

( dS if xuAA=/ 0A

0B if xuAA=0A

=A�FCD

lS�x:

ThusdhA�FCD iS=A�FCDd S by theorem 15.54.

If A=/ 0A then obviously the function A�FCD is injective. �

Proposition 15.74. Let A be a meet-semilattice with least element and B be a poset with leastelement. If a is an atom of A, f 2 FCD(A;B) then f ja=a�FCD hf ia.

Proof. Let X 2A.

X u a=/ 0A)hf jaiX = hf ia; X u a=0A)hf jaiX =0B: �

Proposition 15.75. f � (A�FCDB)=A�FCD hf iB for elements A2A and B 2B of some posetsA, B, C with least elements and f 2FCD(B;C).

Proof. Let X 2A, Y 2B.hf � (A�FCDB)iX =

��hf iB if X �/ A0 if X �A

�= hA�FCD hf iBiX .

h(f � (A �FCD B))¡1iY = h(B �FCD A) � f¡1iY =

�(A if hf¡1iY �/ B0 if hf¡1iY �B

�=��

A if Y �/ hf iB0 if Y �hf iB

�=

hhf iB �FCDAiY = h(A�FCD hf iB)¡1iY . �

15.10 Atomic pointfree funcoids

Theorem 15.76. Let A, B be sets of �lters over boolean lattices. A f 2 FCD(A;B) is an atomof the poset FCD(A;B) i� there exist a2 atomsA and b2 atomsB such that f = a�FCD b.

Proof. A and B are atomic by the theorem 4.135.

). Let f be an atom of the poset FCD(A; B). Let's get elements a 2 atoms dom f andb2 atoms hf ia. Then for every X 2A

X �A a)ha�FCD biX =0Bvhf iX ; X �/A a)ha�FCD biX = bvhf iX :

So a�FCD bv f ; because f is atomic we have f = a�FCD b.

(. Let a2 atomsA, b2 atomsB, f 2 FCD(A;B). If b�B hf ia then :(a [f ] b), f u (a�FCD b) =0FCD(A;B) (because A and B are bounded meet-semilattices); if b v hf ia then 8X 2 A:

(X �/ a)hf iX w b), f w a�FCD b. Consequently f u (a�FCD b) = 0FCD(A;B)_ f w a�FCD b;that is a�FCD b is an atomic pointfree funcoid. �


Theorem 15.77. Let A, B be sets of �lters over boolean lattices. Then FCD(A;B) is atomic.

Proof. Let f 2FCD(A;B) and f =/ 0FCD(A;B). Then dom f =/ 0A, thus exists a2 atoms dom f . Sohf ia=/ 0B thus exists b2 atoms hf ia. Finally the atomic pointfree funcoid a�FCD bv f . �

Theorem 15.78. Let A, B be sets of �lters over boolean lattices. Then the poset FCD(A;B) isseparable.

Proof. Let f ; g 2FCD(A;B), f @ g. Then taking into account the theorem 6.65 exists a2atomsA

such that hf ia @ hgia. By corollary 4.138 B is atomically separable. So exists b 2 atomsB suchthat hf iau b=0B and bvhgia. For every x2 atomsA

hf iau ha�FCD bia= hf iau b=0B;x=/ a)hf ixu ha�FCD bix= hf ixu 0B=0B:

Thus hf ixu ha� bix=0B and consequently f u (a�FCD b) = 0FCD(A;B).

ha�FCD bia= bvhgia;x=/ a)ha�FCD bix=0Bvhgix:

Thus ha�FCD bixvhgix and consequently a�FCD bv g.So the lattice of pointfree funcoids is separable by the theorem 3.14. �

Corollary 15.79. Let A, B be sets of �lters over boolean lattices. The poset FCD(A;B) is:

1. separable;

2. atomically separable;

3. conforming to Wallman's disjunction property.

Proof. By the theorem 3.21. �

Remark 15.80. For more ways to characterize (atomic) separability of the lattice of pointfreefuncoids see subsections �Separation subsets and full stars� and �Atomically separable lattices�.

Corollary 15.81. Let (A;Z0) and (B; Z1) be primary �ltrators over boolean lattices. The posetFCD(A;B) is an atomistic lattice.

Proof. By the corollary 15.34 FCD(A; B) is a complete lattice. Let f 2 FCD(A; B). Supposecontrary to the statement to be proved that

Fatoms f @ f . Then there exists a 2 atoms f such

that auF

atoms f =0FCD(A;B) what is impossible. �

Proposition 15.82. Let A, B be sets of �lters over boolean lattices.atoms(f t g) = atoms f [ atoms g for every f ; g 2FCD(A;B).

Proof. (a �FCD b) u (f t g) =/ ; , a [f t g] b, a [f ] b _ a [g] b, (a �FCD b) u f =/ 0FCD(A;B) _(a �FCD b) u g =/ 0FCD(A;B) for every a 2 atomsA and b 2 atomsB (used the corollary 15.69 andtheorem 15.35). �

Theorem 15.83. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. For everyf ; g; h2FCD(A;B), R2PFCD(A;B):

1. f u (gt h)= (f u g)t (f uh);

2. f tdR=

dhf t iR.

Proof. We will take into account that the lattice of funcoids is an atomistic lattice (corollary15.81).

1. atoms(f u (g t h)) = atoms f \ atoms(g t h) = atoms f \ (atoms g [ atoms h) = (atoms f \atoms g)[ (atoms f \ atomsh)= atoms(f u g)[ atoms(f uh) = atoms((f u g)t (f uh)).

15.10 Atomic pointfree funcoids 191

2. atoms(f td

R) = atoms f [ atomsd

R = atoms f [ThatomsiR =

Th(atoms f) [

ihatomsiR=Thatomsihf t iR= atoms

dhf t iR. (Used the following equality.)

h(atoms f)[ ihatomsiR =

f(atoms f)[A j A2 hatomsiRg =

f(atoms f)[A j 9C 2R:A= atomsCg =

f(atoms f)[ (atomsC) j C 2Rg =

fatoms(f tC) j C 2Rg =

fatomsB j 9C 2R:B= f tCg =

fatomsB j B 2 hf t iRg =

hatomsihf t i:�

Corollary 15.84. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices. Then FCD(A;B) is a co-brouwerian lattice.

Proposition 15.85. Let A, B, C be sets of �lters over some boolean lattices and f 2FCD(A;B),g 2FCD(B;C). Let B be an atomic poset. Then

atoms(g � f) = fx �FCD z j x 2 atomsA; z 2 atomsC; 9 y 2 atomsB: (x �FCD y 2 atoms f ^y�FCD z 2 atoms g)g:

Proof. (x�FCDz)u (g� f)=/ 0FCD(A;C),x [g � f ]z,9y2atomsB: (x [f ] y^ y [g]z),9y2atomsB:((x�FCD y) u f =/ 0FCD(A;B) ^ (y �FCD z) u g=/ 0FCD(B;C)) (were used corollary 15.69 and theorem15.61). �

Theorem 15.86. Let f be a pointfree funcoid between sets of �lters on boolean lattices.

1. X [f ]Y,9F 2 atoms f :X [F ]Y for every X 2F(Src f), Y 2F(Dst f);

2. hf iX =FF2atoms f hF iX for every X 2F(Src f).

Proof. 1. 9F 2atoms f :X [F ]Y,9a2atomsF(Src f); b2atomsF(Dst f): (a�FCDb�/ f ^X [a�FCD b]

Y),9a 2 atomsF(Src f); b 2 atomsF(Dst f): (a�FCD b�/ f ^ a �FCD b�/ X �FCD Y),9F 2 atoms f :(F �/ f ^F �/ X �FCDY), f �/ X �FCDY,X [f ]Y .

2. Let Y 2F(Dst f). Suppose Y�/ hf iX . Then X [f ]Y ; 9F 2atoms f :X [F ]Y; 9F 2atoms f :Y�/hF iX ; Y�/

FF 2atoms f hF iX . So hf iX v

FF2atoms f hF iX . The contrary hf iX w

FF2atoms f hF iX

is obvious. �

15.11 Complete pointfree funcoids

De�nition 15.87. Let A and B be posets. A pointfree funcoid f 2FCD(A;B) is complete, whenfor every S 2PA whenever both

FS and

Fhhf iiS are de�ned we have

hf iG

S=Ghhf iiS:

Proposition 15.88. Let A, B be sets of �lters over boolean lattices. A pointfree funcoid f 2FCD(A;B) is complete i� hf ia=

Fhhf iiatoms a for every a2A.

Proof. Direct implication is obvious. The reverse implication:Let S be a set of �lters.hf iF

S =Fhhf ii atoms

FS =

Fhhf ii

ShatomsiS =

F Shhhf iii hatomsi S =

FhFihhhf iii hatomsiS=

FhF�hhf ii �atomsiS=

FfFhhf iiatomsa j a2Sg=

Ffhf ia j a2Sg=F

hhf iiS. �


De�nition 15.89. Let Z0 and Z1 be join-semilattices with least elements. I will call pointfreegeneralized closure such a function �2 (Z1)Z0 that [TODO: It is just a map preserving �nite joins,no need to introduce a new term. It can be generalized for arbitrary posets.]

1. �0Z0=0Z1;

2. 8I ; J 2Z0:�(I tJ)=�I t�J .

De�nition 15.90. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. I will calla co-complete pointfree funcoid a pointfree funcoid f 2 FCD(A;B) such that hf ijZ0 is a pointfreegeneralized closure.

Proposition 15.91. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. Co-complete pointfree funcoids FCD(A;B) bijectively correspond to pointfree generalized closures Z1

Z0,where the bijection is f 7! hf ijZ0.

Proof. It follows from the theorem 15.26. �

Theorem 15.92. Let (A; Z0) be semi�ltered, star-separable, down-aligned �ltrator with �nitelymeet closed, join-closed, and separable core, where Z0 is a complete boolean lattice and both Z0and A are atomistic lattices.

Let (B;Z1) be a star-separable �ltrator.The following conditions are equivalent for every pointfree funcoid f 2FCD(A;B):

1. f¡1 is co-complete;

2. 8S 2PA; J 2Z1: (FA

S [f ] J)9I 2S: I [f ] J);

3. 8S 2PZ0; J 2Z1: (FZ0 S [f ] J)9I 2S: I [f ] J);

4. f is complete;

5. 8S 2PZ0: hf iFZ0 S=

FB hhf iiS.

Proof. First note that the theorem 4.53 applies to the �ltrator (A;Z0).

(3))(1). For every S 2PZ0, J 2Z1GZ0S uA hf¡1iJ =/ 0A)9I 2S: I uA hf¡1iJ =/ 0A; (15.9)

consequently by the theorem 4.53 we have hf¡1iJ 2Z0.

(1))(2). For every S 2PA, J 2Z1 we have hf¡1iJ 2Z0, consequently

8S 2PA; J 2Z1:

GAS�/ hf¡1iJ)9I 2S: I �/ hf¡1iJ

!:

From this follows (2).

(2))(4). Let hf iFZ0 S and

FB hhf iiS be de�ned. We have hf iFA S = hf i

FZ0 S.J uB hf i

FA S =/ 0B ,FA S [f ] J , 9I 2 S: I [f ] J , 9I 2 S: J uB hf iI =/

0B , J uBFB hhf iiS =/ 0B (used theorem 4.53). Thus hf i

FA S =FB hhf iiS by

star-separability of (B;Z1).

(5))(3). Let hf iFZ0 S be de�ned. Then

FB hhf iiS is also de�ned because hf iFZ0 S =FB hhf iiS. Then

FZ0 S [f ] J , J uB hf iFZ0 S =/ 0B, J uB

FB hhf iiS =/ 0B what bytheorem 4.53 is equivalent to 9I 2S: J uB hf iI =/ 0B that is 9I 2S: I [f ] J .

(2))(3), (4))(5). By join-closedness of the core of (A;Z0). �

Theorem 15.93. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices. If R is a setof co-complete pointfree funcoids in FCD(A;B) then

FR is a co-complete pointfree funcoid.

15.11 Complete pointfree funcoids 193

Proof. Let R be a set of co-complete pointfree funcoids. Then for every X 2Z0

GR�X =

GBfhf iX j f 2Rg=

GZ1fhf iX j f 2Rg2Z1

(used the theorem 15.33 and corollary 4.96). �

Let A andB be posets with least elements. I will denote ComplFCD(A;B) and CoComplFCD(A;B) the sets of complete and co-complete funcoids correspondingly from a poset A to a poset B.

Proposition 15.94.

1. Let f 2ComplFCD (A;B) and g 2ComplFCD (B; C) where A and C are posets with leastelements and B is a complete lattice. Then g � f 2ComplFCD (A;C).

2. Let f 2CoComplFCD (A;B) and g2CoComplFCD (B;C) where A, B and C are posets withleast elements and (A;Z0), (B;Z1), (C;Z2) are �ltrators. Then g � f 2CoComplFCD (A;C).

Proof.

1. LetFS and

Fhhg � f iiS be de�ned. Then [FIXME: Is

Fhhf iiS de�ned?]

hg � f iG

S= hgihf iG

S= hgiGhhf iiS=

Ghhgiihhf iiS=

Ghhg � f iiS:

2. hg � f iZ0= hgihf iZ02Z2 because hf iZ02Z1. �

Proposition 15.95. Let (A; Z0) and (B; Z1) be primary �ltrators over boolean lattices. ThenCoComplFCD(A;B) (with induced order) is a complete lattice.

Proof. Follows from the theorem 15.93. �

Theorem 15.96. Let (A;Z0) and (B;Z1) be primary �ltrators where Z0 and Z1 are boolean lattices.Let R be a set of pointfree funcoids from A to B.

g � (FR)=

Ffg � f j g 2Rg=

Fhg � iR if g is a complete pointfree funcoid from B.

Proof. hg � (F

R)iX = hgihF

RiX = hgiFfhf iX j f 2 Rg =

Ffhgihf iX j f 2 Rg =F

fhg � f iX j f 2 Rg= hFfg � f j f 2 RgiX = h

Fhg � iRiX for every X 2 A. So g � (

FR) =F

hg � iR. �

15.12 Completion and co-completionDe�nition 15.97. Let (A;Z0) and (B;Z1) be primary �ltrators over boolean lattices and Z1 is acomplete atomistic lattice.

Co-completion of a pointfree funcoid f 2 FCD (A;B) is pointfree funcoid CoCompl f de�nedby the formula (for every X 2Z0)

hCoCompl f iX =Cor hf iX:

Proposition 15.98. Above de�ned co-completion always exists.

Proof. Existence of Cor hf iX follows from completeness of Z1.We may apply the theorem 15.26 because

Cor hf i(X tZ0 Y ) =Cor(hf iX tB hf iY )=Cor hf iX tZ1Cor hf iY

by proposition 4.156. �

Obvious 15.99. Co-completion is always co-complete.

Proposition 15.100. For above de�ned always CoCompl f v f .



Proposition 15.101. Monovalued pointfree funcoids between sets of �lters on boolean latticesare metamonovalued.[FIXME: Monovalued is de�ned below.]

Proof. h(dG) � f ix= h

dGihf ix=

dg2G hgihf ix=

dg2G hg � f ix=

dg2G (g � f)

�x for every

ultra�lter x2 atomsSrc f. Thus (dG) � f =

dg2G (g � f). �

15.13 Monovalued and injective pointfree funcoids

De�nition 15.102. Let A and B be posets. Let f 2FCD(A;B).The pointfree funcoid f is:

� monovalued when f � f¡1v idFCD(B).

� injective when f¡1 � f v idFCD(A).

Monovaluedness is dual of injectivity.

Proposition 15.103. Let A and B be posets. Let f 2 FCD(A;B).The pointfree funcoid f is:

� monovalued i� f � f¡1v idim fFCD(B), if im f is de�ned and B is a meet-semilattice;

� injective i� f¡1 � f v iddom fFCD(A), if dom f is de�ned and A is a meet-semilattice.

Proof. It's enough to prove f � f¡1v idFCD(B), f � f¡1v idim fFCD(B).

(. Obvious.

). Let f � f¡1v idFCD(B). Then hf � f¡1ix v x; and hf � f¡1ix v im f . Thus hf � f¡1ix vxu im f =

Didim f

FCD(B)Ex.

h(f � f¡1)¡1ix v x and h(f � f¡1)¡1ix = hf � f¡1ix v im f . Thus h(f � f¡1)¡1ix vxu im f =

Didim f

FCD(B)Ex.

Thus f � f¡1v idim fFCD(B). �

Theorem 15.104. Let A be an atomistic meet-semilattice with least element,B be an atomisticbounded meet-semilattice. The following statements are equivalent for every f 2 FCD(A;B):

1. f is monovalued.

2. 8a2 atomsA: hf ia2 atomsB[f0Bg.

3. 8i; j 2A: hf¡1i(iu j)= hf¡1iiu hf¡1ij.

Proof.

(2))(3). Let a2 atomsA, hf ia= b. Then because b2 atomsB[f0Bg

(iu j)u b=/ 0B, iu b=/ 0B^ j u b=/ 0B;a [f ] iu j, a [f ] i^ a [f ] j;

iu j [f¡1] a, i [f¡1] a^ j [f¡1] a;auA hf¡1i(iu j)=/ 0A, au hf¡1ii=/ 0A^ au hf¡1ij=/ 0A;

au hf¡1i(iu j)=/ 0A, au hf¡1iiu hf¡1ij=/ 0A;hf¡1i(iu j)= hf¡1iiu hf¡1ij:

(3))(1). hf¡1ia u hf¡1ib = hf¡1i(a u b) = hf¡1i0B = 0A (by proposition 15.13 because Ais separable by proposition 3.22) for every two distinct a; b 2 atomsB. This is equivalentto :(hf¡1ia [f ] b); b u hf ihf¡1ia = 0B; b u hf � f¡1ia = 0B; :(a [f � f¡1] b). Soa [f � f¡1] b) a= b for every a; b 2 atomsB. This is possible only (corollary 15.56 and thefact that B is atomic) when f � f¡1v idFCD(B).

15.13 Monovalued and injective pointfree funcoids 195

:(2)):(1). Suppose hf ia2/ atomsB[f0Bg for some a2atomsA. Then there exist two atomsp=/ q such that hf iaw p^hf iaw q. Consequently puhf ia=/ 0B; auhf¡1ip=/ 0A; avhf¡1ip;hf � f¡1ip = hf ihf¡1ip w hf ia w q (by proposition 15.14 because B is separable byproposition 3.22); hf � f¡1ipvp and hf � f¡1ip=/ 0B. So it cannot be f � f¡1v idFCD(B). �

Theorem 15.105. Let (A;Z0) and (B;Z1) be primary �ltrators over a boolean lattice. A pointfreefuncoid f 2FCD(A;B) is monovalued i�

8I ; J 2Z1: hf¡1i(I uZ1J)= hf¡1iI u hf¡1iJ:

Proof. A and B are complete lattices (corollary 4.107).(B;Z1) is a �ltrator with separable core by the theorem 4.112.(B;Z1) is �nitely meet-closed by the theorem 4.97.A and B are starrish by corollary 4.114.A is separable by obvious 4.136.We are under conditions of the theorem 15.25.

). Obvious (taking into account that (B;Z1) is �nitely meet-closed).

(. hf¡1i(I u J ) =dhhf¡1iiup(B;Z1)(I u J ) =

dhhf¡1iifI uZ1 J j I 2 up I ; J 2 up J g =d

fhf¡1i(I uZ1 J) j I 2 up I ; J 2 up J g =dfhf¡1iI u hf¡1iJ j I 2 up I ; J 2 up J g =d

fhf¡1iI j I 2 up Ig udfhf¡1iJ j J 2 upJ g= hf¡1iI uA hf¡1iJ (used theorem 15.25,

theorem 4.110, theorem 15.15). �

15.14 Elements closed regarding a pointfree funcoid

Let A be a poset. Let f 2 FCD(A;A).

De�nition 15.106. Let's call closed regarding a pointfree funcoid f such element a 2 A thathf iav a.

Proposition 15.107. If i and j are closed (regarding a pointfree funcoid f 2FCD(A;A)), S is aset of closed elements (regarding f), then

1. it j is a closed element, if A is a separable starrish join-semilattice;

2.dS is a closed element if A is a separable complete lattice.

Proof. hf i(it j)=hf iithf ijv it j (theorem 15.15), hf idSv

dhhf iiSv

dS (used separability

of A twice). Consequently the elements it j anddS are closed. �

Proposition 15.108. If S is a set of elements closed regarding a complete pointfree funcoid fwith separable destination which is a complete lattice, then the element

FS is also closed regarding

our funcoid.

Proof. hf iFS=

Fhhf iiS v

FS. �

15.15 Connectedness regarding a pointfree funcoid

Let A be a poset with least element. Let �2FCD(A;A). [TODO: No necessity for least element.]

De�nition 15.109. An element a2A is called connected regarding a pointfree funcoid � over Awhen

8x; y 2A n f0Ag: (xt y= a) x [�] y):


Proposition 15.110. Let (A; Z) be a co-separable �ltrator with join-closed core. An A 2 Z isconnected regarding a funcoid � i�

8X;Y 2Z n f0Zg: (X tZY =A)X [�] Y ):

Proof.

). Obvious.

(. Follows from co-separability. �

Obvious 15.111. For A being a set of �lters over a boolean lattice, an element a2A is connectedregarding a pointfree funcoid � i� it is connected regarding the funcoid �u (a�FCDa).

15.15 Connectedness regarding a pointfree funcoid 197

Chapter 16

Convergence of funcoids

16.1 Convergence

The following generalizes the well-known notion of a �lter convergent to a point or to a set:

De�nition 16.1. A �lter F 2 F(Dst �) converges to a �lter A 2 F(Src �) regarding a funcoid �(F!

�A) i� F v h�iA.

De�nition 16.2. A funcoid f converges to a �lter A 2 F(Src �) regarding a funcoid � whereDst f =Dst � (denoted f!� A) i� im f vh�iA that is i� im f!� A.

De�nition 16.3. A funcoid f converges to a �lter A2F(Src �) on a �lter B2F(Src f) regardinga funcoid � where Dst f =Dst � i� f jB!

�A.

Remark 16.4. We can de�ne also convergence for a reloid f : f!�A, im f vh�iA or what is the

same f!� A, (FCD)f!� A.

Theorem 16.5. Let f , g be funcoids, �, � be endofuncoids, Dst f = Src g=Ob �, Dst g=Ob �,A2F(Ob �). If f!

�A,

g jh�iA2C(�u (h�iA�FCD h�iA); �);

and h�iA wA, then g � f!� hgiA.

Proof. im f v h�iA; hgiim f v hgih�iA; im(g � f) v hg jh�iAih�iA; im(g � f) v hg jh�iAih� u(h�iA �FCD h�iA)iA; im(g � f)v hg jh�iA�(� u (h�iA �FCD h�iA))iA; im(g � f)v h� � g jh�iAiA;im(g � f)vh� � giA; im(g � f)vh� ihgiA; g � f!

�hgiA. �

Corollary 16.6. Let f , g be funcoids, �, � be endofuncoids, Dst f = Src g=Ob �, Dst g=Ob �,A2F(Ob �). If f!

�A, g 2C(�; �), and h�iA wA then g � f!

�hgiA.

Proof. From the last theorem and theorem 10.7. �

16.2 Relationships between convergence and continuity

Lemma 16.7. Let �, � be endofuncoids, f 2 FCD(Ob �; Ob �), A 2 F(Ob �), Src f = Ob �,Dst f =Ob �. If f 2C(�jA; �) then

f jh�iA!�hf iA,hf � �jAiAv h� � f iA:

Proof. f jh�iA!�hf iA , im f jh�iAvh� ihf iA , hf ih�iA v h� ihf iA , hf � �iA v h� � f iA ,

hf � �jAiA vh� � f iA. �

Theorem 16.8. Let �, � be endofuncoids, f 2 FCD(Ob �; Ob �), A 2 F(Ob �), Src f = Ob �,Dst f =Ob �. If f 2C(�jA; �) then f jh�iA!

� hf iA.

199

Proof. f jh�iA!�hf iA, (by the lemma),hf � �jAiA v h� � f iA( f � �jAv� � f, f 2 C(�jA;

�). �

Corollary 16.9. Let �, � be endofuncoids, f 2 FCD(Ob �; Ob �), A 2 F(Ob �), Src f = Ob �,Dst f =Ob �. If f 2C(�; �) then f jh�iA!

�hf iA.

Theorem 16.10. Let �, � be endofuncoids, f 2 FCD(Ob �;Ob �), A2 F(Ob �) be an ultra�lter,Src f =Ob �, Dst f =Ob �. f 2C(�jA; �) i� f jh�iA!

�hf iA.

Proof. f jh�iA!�hf iA, (by the lemma),hf � �jAiA v h� � f iA, (used the fact that A is an

ultra�lter),f � �jAv� � f jA,f � �jAv� � f, f 2C(�jA; �). �

16.3 Limit

De�nition 16.11. lim�f=a i� f!�"Src �fag for a T2-separable funcoid � and a non-empty funcoid

f such that Dst f =Dst �.

It is de�ned correctly, that is f has no more than one limit.

Proof. Let lim� f = a and lim� f = b. Then im f vh�i�fag and im f vh�i�fbg.Because f =/ 0FCD(Src f ;Dst f) we have im f =/ 0F(Dst f); h�i�fagu h�i�fbg=/ 0F(Dst f); "Src �fbgu

h�¡1ih�i�fag=/ 0F(Src �); "Src �fbgu h�¡1 � �i"Src �fag=/ 0F(Src �); fag [�¡1 � �]� fbg. Because � isT2-separable we have a= b. �

De�nition 16.12. limB�f = lim� (f jB).

Remark 16.13. We can also in an obvious way de�ne limit of a reloid.

16.4 Generalized limit

[TODO: Refer readers to http://portonmath.tiddlyspace.com/]

16.4.1 De�nitionLet � and � be endofuncoids. Let G be a transitive permutation group on Ob �.

For an element r 2G we will denote "r= "FCD(Ob �;Ob �)r.We require that � and every r2G commute, that is

� � "r= "r � �:We require for every y 2Ob �

� wh� i�fyg�FCD h� i�fyg: (16.1)

Proposition 16.14. Formula (16.1) follows from � w � � �¡1.

Proof. Let � w � � �¡1. Then h� i�fyg �FCD h� i�fyg = h� i"Ob �fyg �FCD h� i"Ob �fyg = � �("Ob �fyg �FCD "Ob �fyg) � �¡1 = � � "FCD(Ob �;Ob �)(fyg � fyg) � �¡1 v � � idFCD(Ob �) � �¡1 =� � �¡1v �. �

Remark 16.15. The formula (16.1) usually works if � is a proximity. It does not work if � is apretopology or preclosure.

We are going to consider (generalized) limits of arbitrary functions acting from Ob � to Ob �.(The functions in consideration are not required to be continuous.)

200 Convergence of funcoids

Remark 16.16. Most typically G is the group of translations of some topological vector space.

Generalized limit is de�ned by the following formula:

De�nition 16.17. xlim f=deff� � f � "r j r2Gg for any funcoid f .

Remark 16.18. Generalized limit technically is a set of funcoids.

We will assume that dom f wh�i�fxg.

De�nition 16.19. xlimxf =def

xlim f jh�i�fxg.

Obvious 16.20. xlimxf = f� � f jh�i�fxg�"r j r 2Gg.

Remark 16.21. xlimxf is the same for funcoids � and Compl �.

The function � will de�ne an injection from the set of points of the space � (�numbers�, �points�,or �vectors�) to the set of all (generalized) limits (i.e. values which xlimxf may take).

De�nition 16.22. �(y)=deffh�i�fxg�FCD h� i�fyg j x2Dg.

Proposition 16.23. �(y)= f(h�i�fxg�FCD h� i�fyg) � "r j r 2Gg for every (�xed) x2D.

Proof. (h�i�fxg �FCD h� i�y) � "r = h"r¡1ih�i�fxg �FCD h� i�y = h�ih"r¡1i�fxg �FCD h� i�y =h�i�fr¡1xg�FCD h� i�y 2fh�i�fxg�FCD h� i�fyg j x2Dg.

Reversely h�i�fxg�FCD h� i�fyg=(h�i�fxg�FCD h� i�fyg) � "e where e is the identify elementof G. �

Proposition 16.24. �(y) = xlim¡h�i�fxg �FCD "Base(Ob �)fyg

�(for every x). Informally: Every

�(y) is a generalized limit of a constant funcoid.

Proof. xlim¡h�i�fxg �FCD "Base(Ob �)fyg

�=�� ¡h�i�fxg �FCD "Base(Ob �)fyg

�� "r j r 2 G

=

f(h�i�fxg�FCD h� i�fyg) � "r j r2Gg= �(y). �

Theorem 16.25. If f jh�i�fxg2C(�; �) and h�i�fxgw"Ob �fxg then xlimx f = �(fx).

Proof. f jh�i�fxg��v � � f jh�i�fxgv� � f ; thus hf ih�i�fxgv h� ihf i�fxg; consequently we have

� wh� ihf i�fxg�FCD h� ihf i�fxgw hf ih�i�fxg�FCD h� ihf i�fxg:

� � f jh�i�fxg w(hf ih�i�fxg�FCD h� ihf i�fxg) � f jh�i�fxg =

h(f jh�i�fxg)¡1ihf ih�i�fxg�FCD h� ihf i�fxg widdom f jh�i�fxg

FCD �h�i�fxg�FCD h� ihf i�fxg w

dom f jh�i�fxg�FCDh� ihf i�fxg =

h�i�fxg�FCD h� ihf i�fxg:

im(� � f jh�i�fxg)= h� ihf i�fxg;

� � f jh�i�fxg vh�i�fxg�FCD im(� � f jh�i�fxg) =

h�i�fxg�FCD h� ihf i�fxg:

So � � f jh�i�fxg=h�i�fxg�FCD h� ihf i�fxg.Thus xlimx f = f(h�i�fxg�FCD h� ihf i�fxg) � "r j r 2Gg= �(fx). �

Remark 16.26. Without the requirement of h�i�fxgw"Ob �fxg the last theorem would not workin the case of removable singularity.

16.4 Generalized limit 201

Theorem 16.27. Let � v � � �. If f jh�i�fxg!�"Ob �fyg then xlimx f = �(y).

Proof. im f jh�i�fxgvh� i�fyg; hf ih�i�fxgv h� i�fyg;

� � f jh�i�fxg w(h� i�fyg�FCD h� i�fyg) � f jh�i�fxg =

h(f jh�i�fxg)¡1ih� i�fyg�FCD h� i�fyg =idh�i�fxg

FCD � f¡1�h� i�fyg�FCD h� i�fyg w

idh�i�fxgFCD � f¡1

�hf ih�i�fxg�FCD h� i�fyg =

idh�i�fxgFCD �

hf¡1 � f ih�i�fxg�FCD h� i�fyg widh�i�fxg

FCD �idh�i�fxg

FCD �h�i�fxg�FCD h� i�fyg =

h�i�fxg�FCD h� i�fyg:

On the other hand, f jh�i�fxgvh�i�fxg�FCD h� i�fyg;� � f jh�i�fxgvh�i�fxg�FCD h� ih� i�fygv h�i�fxg�FCD h� i�fyg.So � � f jh�i�fxg=h�i�fxg�FCD h� i�fyg.xlimxf = f� � f jh�i�fxg�"r j r 2Gg= f(h�i�fxg�FCD h� i�fyg) � "r j r2Gg= �(y). �

Corollary 16.28. If limh�i�fxg� f = y then xlimx f = �(y).

We have injective � if h� i�fy1gu h� i�fy2g=0F(Ob �) for every distinct y1; y22Ob � that is if �is T2-separable.

202 Convergence of funcoids

Chapter 17

Multifuncoids and staroids

17.1 Product of two funcoids

17.1.1 Lemmas

Lemma 17.1. Let A, B, C be sets, f 2FCD(A;B), g 2FCD(B;C), h2FCD(A;C). Then

g � f �/ h, g�/ h � f¡1:

Proof. Proposition 6.70. �

Lemma 17.2. Let A, B, C be sets, f 2RLD(A;B), g 2RLD(B;C), h2RLD(A;C). Then

g � f �/ h, g�/ h � f¡1:


17.1.2 De�nition

De�nition 17.3. I will call a quasi-invertible category a partially ordered dagger category suchthat it holds

g � f �/ h, g�/ h � f y (17.1)

for every morphisms f 2Mor(A;B), g 2Mor(B;C), h2Mor(A;C), where A, B, C are objects ofthis category.

Inverting this formula, we get f y � gy�/ hy, gy�/ f � hy. After replacement of variables, thisgives: f y � g�/ h, g�/ f �h.

As it follows from above, the category of funcoids and the category of reloids are quasi-invertible(taking f y= f¡1). Moreover the category of pointfree funcoids between lattices of �lters on booleanlattices is quasi-invertible (theorem 15.63). [TODO: Say that Rel is quasi-invertible.]

Exercise 17.1. Prove that every ordered groupoid is quasi-invertible category if we de�ne the dagger as theinverse morphism.

De�nition 17.4. The cross-composition product of morphisms f and g of a quasi-invertiblecategory is the pointfree funcoid Mor(Src f ;Src g)!Mor(Dst f ;Dst g) de�ned by the formulas (forevery a2Mor(Src f ; Src g) and b2Mor(Dst f ;Dst g)):

f �(C) g�a= g � a � f y and

¡f �(C) g

�¡1�b= gy � b� f:

We need to prove that it is really a pointfree funcoid that is that

b�/f �(C) g

�a, a�/

¡f �(C) g

�¡1�b:This formula means b�/ g � a � f y, a�/ gy � b � f and can be easily proved applying the formula(17.1) two times.

203

Proposition 17.5. a�f �(C) g

�b, a � f y�/ gy � b.

Proof. From the de�nition. �


�b, f

�a�(C) b

�g.

Proof. f�a�(C) b

�g, f � ay�/ by � g, a � f y�/ gy � b, a

�f �(C) g

�b. �

Theorem 17.7.¡f �(C) g

�¡1= f y�(C) gy.

Proof. For every funcoids a2Mor(Src f ; Src g) and b2Mor(Dst f ;Dst g) we have:¡f �(C) g

�¡1�b= gy � b� f =f y�(C) gy

�b.¡¡

f �(C) g�¡1�¡1�a= f �(C) g�a= g � a� f y=

¡f y�(C) gy

�¡1�a. �

Theorem 17.8. Let f , g be pointfree funcoids between �lters on boolean lattices. Then for every�lters A02F(Src f), B02F(Src g)

f �(C) g�(A0�FCDB0)= hf iA0�FCD hgiB0:

Proof. [TODO: No reason to restrict to atomic �lters?] For every atom a1�FCDb1 (a12atomsDst f,b12 atomsDst g) (see theorem 15.76) of the lattice of funcoids we have:

a1�FCDb1�/f �(C) g

�(A0�FCDB0),A0�FCDB0

�f �(C) g

�a1�FCDb1, (A0�FCDB0)� f¡1�/

g¡1 � (a1�FCD b1),hf iA0�FCDB0�/ a1�FCD hg¡1ib1,hf iA0�/ a1^ hg¡1ib1�/ B0,hf iA0�/ a1^hgiB0�/ b1, hf iA0�FCD hgiB0�/ a1�FCD b1. Thus

f �(C) g

�(A0�FCD B0) = hf iA0�FCD hgiB0

because the lattice FCD(F(Dst f);F(Dst g)) is atomically separable (corollary 15.79). �

Corollary 17.9. A0�FCDB0�f �(C) g

�A1�FCDB1,A0 [f ]A1^B0 [g]B1 for every A02F(Src f),

A12F(Dst f), B02F(Src g), B12F(Dst g) where Src f , Dst f , Src g, Dst g are boolean lattices.

Proof. A0 �FCD B0�f �(C) g

�A1 �FCD B1 , A1 �FCD B1 �/

f �(C) g

�(A0 �FCD B0) ,

A1�FCDB1�/ hf iA0�FCD hgiB0,A1�/ hf iA0^B1�/ hgiB0,A0 [f ]A1^B0 [g]B1. �

17.2 Function spaces of posets

De�nition 17.10. Let Ai be a family of posets indexed by some set domA. We will de�ne orderof families of posets by the formula

av b,8i2domA: aiv bi:

I will call this new poset A=Q

A the function space of posets and the above order product order .

Proposition 17.11. The function space for posets is also a poset.

Proof.


Antisymmetry. Obvious.

Transitivity. Obvious. �

Obvious 17.12. A has least element i� each Ai has a least element. In this case

minA=Y

i2domA

minAi:

Proposition 17.13. a�/ b,9i2domA: ai�/ bi for every a; b2Q

A if every Ai has least element.

204 Multifuncoids and staroids

Proof. If domA=;, then a=b=0, a�b and thus the theorem statement holds. Assume domA=/ ;.a�/ b,9c2

QAn�0QA: (cva^ cv b),9c2

QAn�0QA8i2domA: (civai^ civ bi), (for

the reverse implication take cj = 0Aj for i =/ j),9i 2 dom A; c 2 Ai n f0Aig: (c v ai ^ c v bi),9i2domA: ai�/ bi. �

Proposition 17.14.

1. If Ai are join-semilattices then A is a join-semilattice and

AtB=�i2domA:AitBi: (17.2)

2. If Ai are meet-semilattices then A is a meet-semilattice and

AuB=�i2 domA:AiuBi:

Proof. It is enough to prove the formula (17.2).It's obvious that �i2 domA:AitBiwA; B.Let C wA; B. Then (for every i 2 dom A) Ci wAi and Ci wBi. Thus Ci wAi tBi that is

C w�i2 domA:AitBi. �

Corollary 17.15. If Ai are lattices then A is a lattice.

Obvious 17.16. If Ai are distributive lattices then A is a distributive lattice.

Proposition 17.17. If Ai are boolean lattices thenQ

A is a boolean lattice.

Proof. We need to prove only that every element a2Q

A has a complement. But this complementis evidently �i2dom a: ai. �

Proposition 17.18. If every Ai is a poset then for every S 2PQ

A

1.FS=�i2domA:

Ffxi j x2Sg whenever every

Ffxi j x2Sg exists;

2.dS=�i2domA:

dfxi j x2Sg whenever every

dfxi j x2Sg exists.

Proof. It's enough to prove the �rst formula.(�i2domA:

Ffxi j x2Sg)i=

Ffxi j x2Sgwxi for every x2S and i2 domA.

Let y w x for every x 2 S. Then yi w xi for every i 2 dom A and thus yi wFfxi j x 2 Sg =

(�i2domA:Ffxi j x2Sg)i that is y w�i2domA:

Ffxi j x2Sg.

ThusFS=�i2domA:

Ffxi j x2Sg by the de�nition of join. �

Corollary 17.19. If Ai are posets then for every S 2PQ

A

1.FS=�i2domA:

Ffxi j x2Sg whenever

FS exists;

2.dS=�i2domA:

dfxi j x2Sg whenever

dS exists.

Proof. It is enough to prove that (for every i)Ffxi j x2Sg exists whenever

FS exists.

Fix i2domA.Take yi=(

FS)i and let prove that yi is the least upper bound of fxi j x2Sg.

yi is an upper bound of fxi j x2Sg becauseFS wx and thus (

FS)iwxi for every x2S.

Let x2S and for some t2Ai

T (t)=�j 2domA:

�t if i= jxi if i=/ j:

Let twxi. Then T (t)wx for every x2S. So T (t)wFS and consequently t=T (t)iw yi.

So yi is the least upper bound of fxi j x2Sg. �

Corollary 17.20. If Ai are complete lattices then A is a complete lattice.

Obvious 17.21. If Ai are complete (co-)brouwerian lattices then A is a (co-)brouwerian lattice.

17.2 Function spaces of posets 205

Proposition 17.22. If each Ai is a separable poset with least element (for some index set n) thenQA is a separable poset.

Proof. Let a=/ b. Then 9i2 domA: ai=/ bi. So 9x2Ai: (x�/ ai^x� bi) (or vice versa).Take y=(((domA) n fig)�f0g)[f(i;x)g. Then y�/ a and y� b. �

Obvious 17.23. If every Ai is a poset with least element 0i, then the set of atoms ofQ

A is

f(fkg� atomsAk)[ (�i2 (domA) n fkg: 0i) j k 2domAg:

Proposition 17.24. If every Ai is an atomistic poset with least element 0i, thenQ

A is anatomistic poset.

Proof. xi=F

atomsxi for every xi2Ai. Thus

x=�i2 domx:xi=�i2 domx:G

atomsxi=G

i2domx

�j 2domx:

�xi if j= i0i if j=/ i:

Take join two times. �

Corollary 17.25. If Ai are atomistic posets with least elements, thenQ

A is atomically separable.

Proof. Proposition 3.19. �

Proposition 17.26. Let (Ai2n;Zi2n) be a family of �ltrators. Then (Q

A;Q

Z) is a �ltrator.

Proof. We need to prove thatQ

Z is a sub-poset ofQ

A. FirstQ

Z�Q

A because Zi�Ai foreach i2n.

Let A; B 2Q

Z and A vQZB. Then 8i 2 n:Ai vZiBi; consequently 8i 2 n:Ai vAiBi that is

AvQAB. �

Proposition 17.27. Let (Ai2n;Zi2n) be a family of �ltrators.

1. The �ltrator (Q

A;Q

Z) is (�nitely) join-closed if every (Ai;Zi) is (�nitely) join-closed.

2. The �ltrator (Q

A;Q

Z) is (�nitely) meet-closed if every (Ai;Zi) is (�nitely) meet-closed.

Proof. Let every (Ai; Zi) be �nitely join-closed. Let A; B 2Q

Z and A tQZ B exist. Then (by

corollary 17.19) AtQZB=�i2n:AitZiBi=�i2n:AitAiBi=At

QAB.

Let now every (Ai; Zi) be join-closed. Let S 2 PQ

Z andFQ

ZS exist. Then (by corollary

17.19)FQ

Z S=�i2domA:FZi fxi j x2Sg=�i2 domA:

FAi fxi j x2Sg=FQ

A S.The rest follows from symmetry. �

Proposition 17.28. If each (Ai;Zi) where i2n (for some index set n) is a down-aligned �ltratorwith separable core then (

QA;Q

Z) is with separable core.

Proof. Let a=/ b. Then 9i2n: ai=/ bi. So 9x2Zi: (x�/ ai^x� bi) (or vice versa).Take y=((n n fig)�f0g)[f(i;x)g. Then we have y�/ a and y� b and y 2Z. �

Proposition 17.29. Let every Ai be a bounded lattice. Every (Ai; Zi) is a central �ltrator i�(Q

A;Q

Z) is a central �ltrator.

Proof. x 2 Z(Q

A) , 9y 2Q

A:¡x u y = 0

QA ^ x t y = 1

QA�, 9y 2

QA8i 2 dom A:

(xiu yi=0Ai^xit yi=1Ai),8i2domA9y2Ai: (xiu y=0Ai^xit y=1Ai),8i2domA:xi2Z(Ai).[TODO: Finish the proof.] �

Proposition 17.30. For every element a of a product �ltrator (Q

A;Q

Z):

1. up a=Q

i2dom a up ai;

2. downa=Q

i2dom a downai.


Proof. We will prove only the �rst as the second is dual.up a = fc 2

QZ j c w ag = fc 2

QZ j 8i 2 dom a: ci w aig = fc 2

QZ j 8i 2 dom a:

ci2 up aig=Q

i2dom a up ai. �

Proposition 17.31. If every (Ai;Zi) is a �ltered complete lattice �ltrator, then (Q

A;Q

Z) is a�ltered complete lattice �ltrator.

Proof. ThatQ

A is a complete lattice is already proved above. We have for every a2Q

AdQA up a = �i 2 dom A:

dfxi j x 2 up ag = �i 2 dom A:

dfx j x 2 up aig = �i 2 dom A:d

up ai=�i2domA: ai= a. �

Proposition 17.32. If every (Ai2n;Zi2n) is a pre�ltered complete lattice �ltrator with up x=/ ;for every x2Ai (for every i2n), then (

QA;Q

Z) is a pre�ltered complete lattice �ltrator.

Proof. Let a; b 2Q

A and a=/ b. Then there exists i 2 n such that ai=/ bi and so up ai=/ up bi.Consequently

Qi2dom a up ai=/

Qi2dom a up bi (taken into account that up x=/ ; for every x2Ai)

that is up a=/ up b. �

Proposition 17.33. Let every (Ai2n;Zi2n) be a semi�ltered �ltrator with upx=/ ; for every x2Ai(for every i2n). Then (

QA;Q

Z) is a semi�ltered �ltrator. [TODO: Semi�ltered is the same as�ltered, remove one of the two statements (which one? they are not equivalent having di�erenttheorem conditions!)]

Proof. Let every (Ai; Zi) be a semi�ltered �ltrator. Let up a � up b for some a; b 2Q

A. ThenQi2dom a up ai �

Qi2dom a up bi and consequently (taking into account that up x =/ ; for every

x2Ai) up ai� up bi for every i2n. Then 8i2n: aiv bi that is av b. �

Proposition 17.34. Let (Ai;Zi) be �ltrators and each Zi be a complete lattice with upx=/ ; forevery x2Ai (for every i2n). For a2

QA:

1. Cor a=�i2 dom a:Cor ai;

2. Cor0 a=�i2dom a:Cor0 ai.

Proof. We will prove only the �rst, because the second is dual.Cor a=

dQZ upa=�i2doma:

dZi fxi j x2up ag= (upx=/ ; taken into account)=�i2doma:dZi fx j x2 up aig=�i2 dom a:dZi up ai=�i2dom a:Cor ai. �

Proposition 17.35. If each (Ai;Zi) is a �ltrator with (co-)separable core and each Ai has a least(greatest) element, then (

QA;Q

Z) is a �ltrator with (co-)separable core.

Proof. We will prove only for separable core, as co-separable core is dual.x�

QA y, (used the fact that Ai has a least element),8i2domA:xi�Ai yi)8i2domA9X 2

up xi:X �Ai yi,9X 2 upx8i2 domA:Xi�Ai yi,9X 2 upx:X �QA y for every x; y 2

QA. �

Obvious 17.36.

1. If each (Ai;Zi) is a down-aligned �ltrator, then (Q

A;Q

Z) is a down-aligned �ltrator.

2. If each (Ai;Zi) is an up-aligned �ltrator, then (Q

A;Q

Z) is an up-aligned �ltrator.

Proposition 17.37. If every bi is substractive from ai where a and b are n-indexed families ofdistributive lattices with least elements (where n is an index set), then a n b=�i2n: ai n bi.

Proof. We need to prove (�i2n: ai n bi)u b=0 and at b= bt (�i2n: ai n bi).Really, (�i 2 n: ai n bi) u b = �i 2 n: (ai n bi) u bi = 0 and b t (�i 2 n: ai n bi) = �i 2 n:

bit (ai n bi)=�i2n: bit ai= at b. �

Proposition 17.38. If every Ai is a distributive lattice, then a n� b=�i2domA:ai n� bi for everya; b2

QA whenever every ai n� bi is de�ned.

17.2 Function spaces of posets 207

Proof. If some Ai is empty, our statement is obvious. Let's assume Ai=/ ;.We need to prove that �i2domA: ai n� bi=

dfz 2

QA j av bt zg.

To prove it is enough to show ai n� bi =dfzi j z 2

QA; a v b t zg that is ai n� bi =d

fz2Ai j aivbitzg (for the reverse implication take zj=ai for j=/ i) what is true by de�nition. �

Proposition 17.39. If every Ai is a distributive lattice with least element, then a#b=�i2domA:ai#bi for every a; b2

QA whenever every ai#bi is de�ned.

Proof. We need to prove that �i2domA: ai#bi=Ffz 2

QA j z v a^ z� bg.

To prove it is enough to show ai#bi =Ffzi j z 2

QA; z v a ^ z � bg that is ai#bi =F

fz 2Ai j z v ai^8j 2domA: zj� bjg that is ai#bi=Ffz 2Ai j z v ai^ z� big (take zj=0 for

j=/ i) what is true by de�nition. �

Proposition 17.40. Let every Ai be a poset with least element and ai� is de�ned. Then a�=�i2domA: ai

�.

Proof. We need to prove that �i 2 dom A: ai�=

Ffc 2 A j c� ag. To prove this it is enough to

show that ai�=Ffci j c 2

QA; c� ag that is ai�=

Ffci j c 2

QA; 8j 2 dom A: cj � ajg that is

ai�=Ffci j c2

QA; ci� aig (take cj=0 for j=/ i) that is ai�=

Ffc2Ai j c� aig what is true by

de�nition. �

Corollary 17.41. Let every Ai be a poset with greatest element and ai+ is de�ned. Then a+ =

�i2 domA: ai+.

Proof. By duality. �

17.3 De�nition of staroidsLet n be a set. As an example, n may be an ordinal, n may be a natural number, considered as aset by the formula n= f0; :::; n¡ 1g. Let A=Ai2n be a family of posets indexed by the set n.

De�nition 17.42. I will call an anchored relation a pair f =(form f ;GR f) of a family form(f) ofsets indexed by the some index set and a relation GR(f)2P

Qform(f). I call GR(f) the graph

of the anchored relation f . I denote Anch(A) the set of anchored relations of the form A.

De�nition 17.43. An anchored relation on powersets is an anchored relation f such that every(form f)i is a powerset.

I will denote arity f = dom form f .

De�nition 17.44. Every set of anchored relations of the same form constitutes a poset by theformula f v g,GR f �GR g.

Definition 17.45. An anchored relation is an anchored relation between posets when every(form f)i is a poset.

De�nition 17.46. Let f be an anchored relation. For every i 2 arity f and L 2Q

((form f)j(arity f)nfig)(val f)iL= fX 2 (form f)i j L[f(i;X)g2GR f g

(�val� is an abbreviation of the word �value�.)

Obvious 17.47. X 2 (val f)iL,L[f(i;X)g2GR f .

Proposition 17.48. f can be restored knowing form(f) and (val f)i for some i2 arity f .

Proof. GR f = fK 2Q

form f j K 2 GR f g = fL [ f(i; X)g j L 2Q

(form f)j(arity f)nfig;X 2 (form f)i; L[f(i;X)g2GR f g= fL[f(i;X)g j L2

Q(form f)j(arity f)nfig;X 2 (val f)iLg. �


De�nition 17.49. A prestaroid is an anchored relation f between posets such that (val f)iL is afree star for every i2 arity f , L2

Q(form f)j(arity f)nfig.

De�nition 17.50. A staroid is a prestaroid whose graph is an upper set (on the posetQ

form(f)).

Proposition 17.51. If L 2Q

form f and Li= 0(form f)i for some i 2 arity f then L2/ f if f is aprestaroid.

Proof. Let K =Lj(arity f)nfig. We have 02/ (val f)iK; K [f(i; 0)g2/ f ; L2/ f . �

De�nition 17.52. In�nitary anchored relation is such an anchored relation whose arity is in�nite;�nitary anchored relation is such an anchored relation whose arity is �nite.

Next we will de�ne completary staroids . First goes the general case, next simpler case for thespecial case of join-semilattices instead of arbitrary posets.

De�nition 17.53. A completary staroid is an anchored relation between posets conforming to theformulas:

1. 8K 2Q

form f : (K w L0 ^K w L1)K 2GR f),9c 2 f0; 1gn: (�i 2 n:Lc(i)i) 2GR f forevery L0; L12

Qform f .

2. If L2Q

form f and Li=0(form f)i for some i2 arity f then L2/ GR f .

Lemma 17.54. Every graph of completary staroid is an upper set.

Proof. Let f be a completary staroid. Let L0v L1 for some L0; L1 2Q

form f and L0 2GR f .Then taking c=n�f0g we get �i2n:Lc(i)i=�i2n:L0i=L02GR f and thus L12GR f becauseL1wL0^L1wL1. �

Proposition 17.55. A relation between posets whose form is a family of join-semilattices is acompletary staroid i� both:

1. L0tL12GR f,9c2f0; 1gn: (�i2n:Lc(i)i)2GR f for every L0; L12Q

form f .

2. If L2Q

form f and Li=0(form f)i for some i2 arity f then L2/ GR f .

Proof. Let the formulas (1) and (2) hold. Then f is an upper set: Let L0 v L1 for some L0;L12

Qform f and L02 f . Then taking c=n�f0g we get �i2n:Lc(i)i=�i2n:L0i=L02GR f

and thus L1=L0tL12 f .Thus to �nish the proof it is enough to show that

L0tL12GR f,8K 2Y

form f : (K wL0^K wL1)K 2GR f)

under condition that GR f is an upper set. But this is obvious. �

Proposition 17.56. Every completary staroid is a staroid.

Proof. Let f be a completary staroid.Let i 2 arity f , K 2

Qi2(arity f)nfig (form f)i. Let L0=K [ f(i;X0)g, L1 =K [ f(i;X1)g for

some X0; X12Ai.Let

8Z 2Ai: (Z wX0^Z wX1)Z 2 (val f)iK);then

8Z 2Ai: (Z wX0^Z wX1)K [f(i;Z)g2GR f):

If z wL0^ z wL1 then z wK [f(i; zi)g, thus taking into account that GR f is an upper set,

8z 2Y

A: (z wL0^ z wL1)K [f(i; zi)g2GR f):

8z 2Y

A: (z wL0^ z wL1) z 2GR f):

17.3 Definition of staroids 209

Thus, by the de�nition of completary staroid, L02GR f _L12GR f that is

X02 (val f)iK _X12 (val f)iK:

So (val f)iK is a free star (taken into account that zi=0(form f)i) z 2/ GR f and that (val f)iK isan upper set). �

Exercise 17.2. Write a simpli�ed proof for the case if every (formf)i is a join-semilattice.

Lemma 17.57. Every �nitary prestaroid is completary.

Proof. 9c 2 f0; 1gn: (�i 2 n: Lc(i)i) 2GR f,9c 2 f0; 1gn¡1: (f(n ¡ 1; L0(n ¡ 1))g [ (�i 2 n ¡ 1:

Lc(i)i)) 2 GR f _ (f(n ¡ 1; L1(n ¡ 1))g [ (�i 2 n ¡ 1: Lc(i)i)) 2 GR f , 9c 2 f0; 1gn¡1:L0(n ¡ 1) 2 (val f)n¡1(�i 2 n ¡ 1: Lc(i)i) _ L1(n ¡ 1) 2 (val f)n¡1(�i 2 n ¡ 1: Lc(i)i), 9c 2 f0;1gn¡18K 2 (form f)n¡1: (K w L0(n ¡ 1) _ K w L1(n ¡ 1) ) K 2 (val f)n¡1(�i 2 n ¡ 1:Lc(i)i)) , 9c 2 f0; 1gn¡18K 2 (form f)n¡1: (K w L0(n ¡ 1) _ K w L1(n ¡ 1) ) f(n ¡ 1;

K)g[ (�i2n¡ 1:Lc(i)i))2GR f, :::,8K 2Q

form f : (K wL0^K wL1)K 2GR f). �

Exercise 17.3. Prove the simpler special case of the above theorem when the form is a family of join-semilat-tices.

Theorem 17.58. For �nite arity the following are the same:

1. prestaroids;

2. staroids;

3. completary staroids.

Proof. f is a �nitary prestaroid ) f is a �nitary completary staroid.f is a �nitary completary staroid ) f is a �nitary staroid.f is a �nitary staroid ) f is a �nitary prestaroid. �

De�nition 17.59. We will denote the set of staroids, prestaroids, and completary staroids of aform A correspondingly as Strd(A), pStrd(A), and cStrd(A).

17.4 Upgrading and downgrading a set regarding a �ltrator

Let �x a �ltrator (A;Z).

De�nition 17.60. �f = f \Z for every f 2PA (downgrading f).

De�nition 17.61. �f = fL2A j upL� f g for every f 2PZ (upgrading f).

Obvious 17.62. a2�f,up a� f for every f 2PZ and a2A.

Proposition 17.63. ��f = f if f is an upper set for every f 2PZ.

Proof. ��f =�f \Z= fL2Z j upL� f g= fL2Z j L2 f g= f \Z= f . �

17.4.1 Upgrading and downgrading staroidsLet �x a family (A;Z) of �ltrators.

For a graph f of a staroid de�ne �f and �f taking the �ltrator of (Q

A;Q

Z).For a staroid f de�ne: [TODO: De�ne for all anchored relations.]

form�f =Z and GR�f =�GR f ;

form�f =A and GR�f =�GR f:

Proposition 17.64. (val�f)iL=(val f)iL\Zi for every L2Q

Zj(arity f)nfig.


Proof. (val� f)iL= fX 2 Zi j L [ f(i;X)g 2GR f \Q

Zg= fX 2 Zi j L [ f(i;X)g 2GR f g=(val f)iL\Zi. �

Proposition 17.65. Let (Ai;Zi) be �nitely join-closed �ltrators with both the base and the corebeing join-semilattices. If f is a staroid of the form A, then �f is a staroid of the form Z.

Proof. Let f be a staroid.We need to prove that (val �f)iL is a free star. It follows from the last proposition and the

fact that it is �nitely join-closed. �

17.5 Principal staroids

De�nition 17.66. The staroid generated by an anchored relation F is the staroid f = "StrdFon powersets such that " � L 2 GR f ,

QL �/ F and (form f)i = P(form F )i for every

L2Q

i2arity f P(formF )i.

Remark 17.67. Below we will prove that staroid generated by an anchored relation is a staroidand moreover a completary staroid.

De�nition 17.68. A principal staroid is a staroid generated by some anchored relation.

Proposition 17.69. Every principal staroid is a completary staroid.

Proof. That L2/ f if Li=0(form f)i for some i2 arity f is obvious. It remains to proveY(L0tL1)�/ F,9c2f0; 1garity f:

Yi2n

Lc(i)i�/ F:

ReallyQ(L0 t L1) �/ F , 9x 2

Q(L0 t L1): x 2 F , 9x 2

Qi2arity f (form f)i 8i 2 arity f :

(xi 2L0 i[L1 i^ x 2 F ),9x2Q

i2arity f (form f)i 8i 2 arity f : ((xi 2L0 i_ xi 2L1 i) ^ x 2F ),9x 2

Qi2arity f (form f)i

¡9c 2 f0; 1garity f: x 2

Qi2arity fn Lc(i)i ^ x 2 F

�, 9c 2 f0; 1garity f:Q

i2arity f Lc(i)i�/ F . �

De�nition 17.70. The upgraded staroid generated by an anchored relation F is the staroid�"StrdF .

Proposition 17.71. "StrdF =��"StrdF .

Proof. Because GR "StrdF is an upper set. �

Conjecture 17.72. Every upgraded principal staroid is a completary staroid.

Conjecture 17.73. Filtrators of staroids on powersets are join-closed.

17.6 Multifuncoids

De�nition 17.74. Let (Ai;Zi) (where i2n for an index set n) be an indexed family of �ltrators.I call a premultifuncoid sketch f of the form (Ai;Zi) the n-indexed family � of functions where

for every i2n�i:Y

Zj(domA)nfig!Ai:

I denote hf i=�.

De�nition 17.75. A premultifuncoid sketch on powersets is a premultifuncoid sketch such thatevery (Ai;Zi) is the primary �ltrator of �lters on a powerset.

17.6 Multifuncoids 211

De�nition 17.76. I will call a premultifuncoid a premultifuncoid sketch such that for every i;j 2n and L2

QZ

Li�/ �iLj(domL)nfig,Lj�/ �jLj(domL)nfjg: (17.3)

De�nition 17.77. Let A be an indexed family of starrish posets. The prestaroid correspondingto a premultifuncoid f is [f ] de�ned by the formula:

form [f ]=Z and L2GR [f ],Li�/ hf iiLj(domL)nfig:

Proposition 17.78. The prestaroid corresponding to a premultifuncoid is really a prestaroid.

Proof. By the de�nition of starrish posets. �

De�nition 17.79. I will call a multifuncoid a premultifuncoid to which corresponds a staroid.

De�nition 17.80. I will call a completary multifuncoid a premultifuncoid to which correspondsa completary staroid.

Theorem 17.81. Fix some indexed family A of boolean lattices. The the set of premultifuncoidsg for the �ltrator (Fi;Pi) bijectively corresponds to set of prestaroids f of form P= �i 2 domA:Pi by the formulas:

1. f = [g];

2. @ hgiiL=(val f)iL for every i2domA, L2Q

PjdomAnfig.

Proof. Let f be a prestaroid of the form P. If � is de�ned by the formula �i L = hf iiL then@�iL=(val f)iL. Then

Li�/ �iLj(domL)nfig,L2 f,Lj�/ �jLj(domL)nfjg:

For the prestaroid f 0 de�ned by the formula L2 f 0,Li�/ �iLj(domL)nfig we have:

L2 f 0,Li2 @�iLj(domL)nfig,Li2 (val f)iLj(domL)nfig,L2 f ;thus f 0= f .

Let now � be an indexed family of functions �i 2 F(Zi)(domZ)nfig conforming to the formula

(17.3). Let relation f between posets be de�ned by the formula L2 f,Li�/ �iLj(domL)nfig. Then

(val f)iL= fK 2Pi j K�/ �iLj(domL)nfigg= @�iLj(domL)nfig

and thus (val f)iL is a core star that is f is a prestaroid. For the indexed family �0 de�ned by theformula �i0L= hf iiL we have

@�i0L=@ hf iiL= fK 2Pi j K�/ �iLg= @�iL;

thus �0=� (taking into account that Pi is a boolean lattice).We have shown that these are bijections. �

De�nition 17.82. I will denote �f the premultifuncoid corresponding to a prestaroid f (for anindexed family of boolean lattices) by the above theorem.

Theorem 17.83. Fix some indexed family Z of boolean lattices. hf ij(L [ f(i; X t Y )g) =hf ij(L[f(i;X)g)thf ij(L[f(i;Y )g) for every premultifuncoid f for the family (Fi;Pi) of �ltratorsand i; j 2arity f , i=/ j, L2

Qk2Lnfi; jg Zk, X;Y 2Ai. [TODO: It also holds for any �nite number

of arguments.]

Proof. Let i2 arity f and L2Q

k2Lnfi; jg Zk. Let Z 2Zi.

Z �/ hf ij(L [ f(i; X t Y )g) , L [ f(i; X t Y ); (j; Z)g 2 f , X t Y 2 (val f)i(L [ f(j;Z)g),X 2 (val f)i(L[f(j;Z)g_Y 2 (val f)i(L[f(j;Z)g),L[f(i;X); (j;Z)g2 [f ]_L[f(i;Y );(j;Z)g2 [f ],Z �/ hf ij(L[f(i;X)g)_Z �/ hf ij(L[f(i;Y )g).


Thus hf ij(L[f(i;X tY )g)= hf ij(L[f(i;X)g)t hf ij(L[f(i;Y )g). �

Let us consider the �ltrator¡Q

i2arity f F((form f)i);Q

i2arity f (form f)i�.

Theorem 17.84. Let (Ai; Zi) be a family of join-closed down-aligned �ltrators whose both baseand core are join-semilattices. Let f be a staroid of the form Z. Then �f is a staroid of the form A.

Proof. First prove that �f is a prestaroid. We need to prove that 0 2/ (GR �f)i (that isup 0* (GR f)i that is 02/ (GR f)i what is true by the theorem conditions) and that for every X ;Y 2Ai and L2

Qi2(arity f)nfig Ai where i2 arity f

L[f(i;X tY)g2GR�f,L[f(i;X )g2GR�f _L[f(i;Y)g2GR�f:

The reverse implication is obvious. Let L[f(i;X tY)g2GR�f . Then for every L2L and X 2X ,Y 2Y we have and X tZiY wX tAiY thus L[f(i;X tZiY )g2GR f and thus

L[f(i;X)g2GR f _L[f(i;Y )g2GR f

consequently L[f(i;X )g2GR�f _L[f(i;Y)g2GR�f .It is left to prove that �f is an upper set, but this is obvious. �

There is a conjecture similar to the above theorems:

Conjecture 17.85. L 2 � [f ])� [f ]\Q

i2domA atoms Li =/ ; for every multifuncoid f for the�ltrator (Fn;Pn).

Conjecture 17.86. Let f be a set, F be the set of �lters on f, P be the set of principal �lterson f, let n be an index set. Consider the �ltrator (Fn;Pn). Then if f is a completary staroid ofthe form Pn, then �f is a completary staroid of the form Fn.

Obvious 17.87. (F

F )K =Ff2F fK for every set F of premultifuncoid sketches of the same

form A and K 2Q

A whenever everyFf2F fK is de�ned.

17.7 Join of multifuncoids

Premultifuncoid sketches are ordered by the formula f v g,hf i v hgi where v in the right partof this formula is the product order. I will denote u, t,

d,F

(without an index) the order posetoperations on the poset of premultifuncoid sketches.

Remark 17.88. To describe this, the de�nition of product order is used twice. Let f and g bepremultifuncoid sketches of the same form A

hf iv hgi,8i2domA: hf iivhgii and hf iivhgii,8L2Y

Zj(domA)nfig: hf iiLvhgiiL:

Theorem 17.89. f tpFCD(A) g = f t g for every premultifuncoids f and g for the same indexedfamily of starrish join-semilattices �ltrators.

Proof. �i x=deffixt gi x. It is enough to prove that � is a premultifuncoid.

We need to prove:

Li�/ �iLj(domL)nfig,Lj�/ �jLj(domL)nfjg:

Really, Li �/ �i Lj(domL)nfig,Li �/ fi Lj(domL)nfigtgi Lj(domL)nfig,Li �/ fi Lj(domL)nfig_Li �/ gi Lj(domL)nfig,Lj �/ fj Lj(domL)nfjg_Lj �/ gj Lj(domL)nfjg,Lj �/ fj Lj(domL)nfjgtgjLj(domL)nfjg,Lj�/ �jLj(domL)nfjg. �

Theorem 17.90.FpFCD(A) F =

FF for every set F of premultifuncoids for the same indexed

family of join in�nite distributive complete lattices �ltrators.

17.7 Join of multifuncoids 213

Proof. �i x=defF

f2F fix. It is enough to prove that � is a premultifuncoid.We need to prove:

Li�/ �iLj(domL)nfig,Lj�/ �jLj(domL)nfjg:

Really, Li �/ �i Lj(domL)nfig,Li �/Ff2F fi Lj(domL)nfig,9f 2 F : Li �/ fi Lj(domL)nfig,9f 2 F :

Lj�/ fjLj(domL)nfjg,Lj�/Ff2F fjLj(domL)nfjg,Lj�/ �jLj(domL)nfjg. �

Proposition 17.91. The mapping f 7! [f ] is an order embedding, for multifuncoids for indexedfamilies (Ai;Zi) of down-aligned starrish �ltrators with separable �nitely meet-closed core.

Proof. The mapping f 7! [f ] is de�ned because Ai are starrish posets (and (Ai;Zi) is with �nitelymeet-closed core and down-aligned). The mapping is injective because the �ltrators are withseparable cores (fX 2Zi j X �/ hf iAg= fX 2Zi j X �/ hf iBg implies hf iA= hf iB). That f 7! [f ]is a monotone function is obvious. �

Remark 17.92. This order embedding is useful to describe properties of posets of prestaroids.

Theorem 17.93. If f , g are multifuncoids for the �ltrator (Fi;Pi) where Zi are separable starrishposets, then f tpFCD(A) g 2FCD(A).

Proof. Let A2�f tpFCD(A) g

�and B wA. Then for every k 2domA

Ak �/¡f tpFCD(A) g

�Aj(domA)nfkg; Ak �/ (f t g)Aj(domA)nfkg; Ak �/ f(Aj(domA)nfkg) t

g(Aj(domA)nfkg).Thus Ak �/ f(Aj(domA)nfkg) _ Ak �/ g(Aj(domA)nfkg); A 2 [f ]_A 2 [g]; B 2 [f ]_B 2 [g];

Bk�/ f(B j(domA)nfkg)_Bk�/ g(B j(domA)nfkg);Bk �/ f(B j(domA)nfkg) t g(B j(domA)nfkg); Bk �/ (f t g)B j(domA)nfkg=¡

f tpFCD(A) g�B j(domA)nfkg.

Thus B 2�f tpFCD(A) g

�. �

Theorem 17.94. If F is a set of multifuncoids for the same indexed family of join in�nitedistributive complete lattices �ltrators, then

FpFCD(A) F 2FCD(A).

Proof. Let A2hFpFCD(A)

Fiand B wA. Then for every k 2 domA

Ak�/�FpFCD(A) F

�Aj(domA)nfkg=(

FF )Aj(domA)nfkg=

Ff2F f(Aj(domA)nfkg).

Thus 9f 2 F : Ak �/ f(Aj(domA)nfkg); 9f 2 F : A 2 [f ]; B 2 [f ] for some f 2 F ; 9f 2 F :

Bk �/ f(B j(domA)nfkg); Bk �/Ff2F f(B j(domA)nfkg) =

�FpFCD(A) F�B j(domA)nfkg. Thus B 2hFpFCD(A)

Fi. �

Conjecture 17.95. The formula f tFCD(A) g 2 cFCD(A) is not true in general for completarymultifuncoids (even for completary multifuncoids on powersets) f and g of the same form A.

17.8 In�nite product of poset elements

De�nition 17.96. Let Ai be a family of elements of a family Ai of posets. The staroidal productQStrd(A) Ai is de�ned by the formula (for every L2Q

A)

formYStrd(A)

A=A and L2GRYStrd(A)

A,8i2domA:Ai�/ Li:

Proposition 17.97. If Ai are powerset algebras, staroidal product of principal �lters is essentiallyequivalent to Cartesian product. More precisely,

Qi2domAStrd "FAi=�"Strd

QA for an indexed family

A of sets.


Proof. L2GR�"StrdQ A,upL�GR"StrdQ

A,8X 2upL:Q

X�/Q

A,8X 2upL; i2domA:

Xi�/ Ai,8i2 domA:Li�/ "FAi,L2GRQ

i2domAStrd "FAi. �

Corollary 17.98. Staroidal product of principal �lters is an upgraded principal staroid.

Proposition 17.99.QStrd a=��QStrd a if each ai 2 Ai (for i 2 n where n is some index set)

where Ai is a separable poset and (Ai2n;Zi2n) is a down aligned �ltrator.

Proof. GR��QStrd a=�L2

QA j upL�Z\GR

QStrd a=�L2

QA j upL�GR

QStrd a=�

L2Q

A j 8K 2upL:K 2GRQStrd

a=fL2

QA j 8K 2upL; i2n:Ki�/ aig=fL2

QA j 8i2n;

K 2 upL:Ki�/ aig= fL2Q

A j 8i2 n:Li�/ aig=GRQStrd

a (taken into account thatQ

A is aseparable poset). �

Theorem 17.100. Staroidal product is a completary staroid (if our posets are starrish join-semilattices).

Proof. We need to prove

8i2domA:Ai�/ (L0 itL1 i),9c2f0; 1gn8i2 domA:Ai�/ Lc(i) i:

Really, 8i2domA:Ai�/ (L0 itL1 i),8i2domA: (Ai�/ L0 i_Ai�/ L1 i),9c2f0;1gdomA8i2domA:Ai�/ Lc(i) i. �

De�nition 17.101. Let (Ai;Zi) be an indexed family of down-aligned �ltrators.Then for every A2

QA funcoidal product is multifuncoid

QFCD(A)A de�ned by the formula

(for every L2Q

Z): * YFCD(A)

A

+k

L=

�Ak if 8i2 (domA) n fkg:Ai�/ Li0 otherwise:

Proposition 17.102.QStrd(A) A=

hQFCD(A) Ai.

Proof. L 2 GRQStrd(A) A, 8i 2 dom A: Ai �/ Li, 8i 2 (dom A) n fkg: Ai �/ Li ^ Lk �/ Ak,

Lk�/DQFCD(A)

AEkLj(domA)nfkg,L2GR

hQFCD(A)Ai. �

Corollary 17.103. Funcoidal product is a completary multifuncoid.

Proof. It is enough to prove that funcoidal product is a premultifuncoid. Really,

Li�/

* YFCD(A)

A

+i

Lj(domA)nfig,8i2domA:Ai�/ Li,Lj�/

* YFCD(A)

A

+j

Lj(domA)nfjg: �

Theorem 17.104. If our �ltrator (Q

A;Q

Z) is with separable core and A 2Q

Z, then�QStrd(Z) A=

QStrd(A) A.

Proof. GR �QStrd(Z) A =nL 2

QA j up L �

QStrd(Z) Ao= fL 2

QA j 8K 2 up L;

i 2 dom A: Ai �/ Kig = fL 2Q

A j 8i 2 dom A; K 2 up Li: Ai �/ Kg = fL 2Q

A j 8i 2 dom A:

Ai�/ Lig=GRQStrd(A) A. �

Proposition 17.105. Let (Q

A;Q

Z) be a meet-closed �ltrator, A2Q

Z. Then �QStrd(A) A=QStrd(Z)A.

Proof. GR�QStrd(A) A=�GRQStrd(A) A=�fL2Q A j 8i2domA:Ai�/ Lig=fL2

QA j 8i2

domA:Ai�/ Lig\Q

Z= fL2Q

Z j 8i2 domA:Ai�/ Lig=GRQStrd(Z) A. �

17.8 Infinite product of poset elements 215

Corollary 17.106. If each (Fi;Pi) is a powerset �ltrator and A 2Q

P, then �QStrd(F) A is aprincipal staroid.

Proof. Use the �obvious� fact above. �

Theorem 17.107. Let F be a family of sets of �lters on distributive lattices with least elements.Let a2

QF, S 2P

QF, and every PriS be a generalized �lter base,

dS= a. Then

YStrd(F)a=

l( YStrd(F)

A j A2S

):

Proof. ThatQStrd(F) a is a lower bound for

nQStrd(F) A j A2So

is obvious.

Let f be a lower bound fornQStrd(F) A j A2S

o. Thus 8A2S:GR f �GR

QStrd(F) A. Thus

for every A2 S we have L2GR f implies 8i2 domA:Ai�/ Li. Then, by properties of generalized�lter bases, 8i2domA: ai�/ Li that is L2GR

QStrd(F)a.

So f vQStrd(F) a. �

Conjecture 17.108. Let F be a family of sets of �lters on distributive lattices with least elements.Let a2

QF, S 2P

QF be a generalized �lter base,

dS=a, f is a staroid of the form

QF. Then

YStrd(F)a�/ f,8A2S:

YStrd(A)A�/ f:

17.9 On products of staroids

De�nition 17.109.Q(D) F =funcurry z j z 2

QF g (reindexation product ) for every indexed

family F of relations.

De�nition 17.110. Reindexation product of an indexed family F of anchored relations is de�nedby the formulas:

formY(D)

F = uncurry(form �F ) and GRY(D)

F =Y(D)

(GR �F ):

Obvious 17.111.

1. formQ(D) F = f((i; j); (formFi)j) j i2domF ; j 2 arityFig;

2. GRQ(D) F = ff((i; j); (zi)j) j i2domF ; j 2 arityFig j z 2

Q(GR �F )g.

Proposition 17.112.Q(D)

F is an anchored relation if every Fi is an anchored relation.

Proof. We need to prove GRQ (D)F 2P

Qform

¡Q (D)F�that is

GRQ (D)F �

Qform

¡Q (D)F�;

ff((i; j); (zi)j) j i2domF ; j 2arityFig j z2Q

(GR�F )g�Qf((i; j); (formFi)j) j i2domF ;

j 2 arityFig;8z 2

Q(GR �F ); i2domF ; j 2 arityFi: (zi)j 2 (formFi)j.

Really, zi2GRFi�Q

(formFi) and thus (zi)j 2 (formFi)j. �

Obvious 17.113. arityQ(D)

F =`

i2domF arityFi= f(i; j) j i2 domF ; j 2 arityFig.

De�nition 17.114. f �(D) g=Q(D) Jf ; gK.

Lemma 17.115.Q(D) F is an upper set if every Fi is an upper set.


Proof. We need to prove thatQ(D) F is an upper set. Let a2

Q(D) F and an anchored relationbw a of the same form as a. We have a=uncurry z for some z 2

QF that is a(i; j)= (zi)j for all

i2 domF and j 2 domFi where zi 2Fi. Also b(i; j)w a(i; j). Thus (curry b)iw zi; curry b2Q

F

because every Fi is an upper set and so b2Q(D) F . �

Proposition 17.116. Let F be an indexed family of anchored relations and every (formF )i is ajoin-semilattice.

1.Q(D) F is a prestaroid if every Fi is a prestaroid.

2.Q(D) F is a staroid if every Fi is a staroid.

3.Q(D)

F is a completary staroid if every Fi is a completary staroid.

Proof.

1. Let q 2 arityQ(D) F that is q=(i; j) where i2domF , j 2 arityFi; let

L2Y0@ form

Y(D)F

!j¡arityQ(D)F

�nfqg

1Athat is L(i0;j 0) 2

�form

Q(D) F�(i0;j 0)

for every (i0; j 0) 2�arity

Q(D) F�n fqg, that is

L(i0;j 0)2 (formFi)j. We have X 2�form

Q(D)F�(i;j)

,X 2 (formFi)j. So valY(D)

F

!(i;j)

L=

(X 2 (formFi)j j L[f((i; j);X)g2GR

Y(D)F

);

valY(D)

F

!(i;j)

L=�X 2 (formFi)j j 9z 2

Y(GR �F ):L[f((i; j);X)g= uncurry z

;

valY(D)

F

!(i;j)

L=nX 2 (form Fi)j j 9z 2

Y �(GR � F )j¡arityQ(D)F

�nf(i;j)g

�; v 2GR Fi:

(L= uncurry z ^ vj=X)o;�

valQ(D)

F�(i;j)

L =nX 2 (form Fi)j j 9z 2

Q �(GR � F )j¡arityQ(D)F

�nf(i;j)g

�: L =

uncurry z ^9v 2GRFi: vj=Xo.

If 9z 2Q �

(GR � F )j¡arityQ(D)F�nf(i;j)g

�: L = uncurry z is false then�

valQ(D) F

�(i;j)

L= ; is a free star. We can assume it is true. So valY(D)

F

!(i;j)

L= fX 2 (formFi)j j 9v 2GRFi: vj=Xg:

Thus valY(D)

F

!(i;j)

L= fX 2 (formFi)j j 9K 2 (formFi)j(arityFi)nfjg:K [ f(j;X)g 2GR Fig=

fX 2 (formFi)j j 9K 2 (formFi)j(arityFi)nfjg:X 2 (valFi)jKg:

Thus A t B 2�val

Q(D) F�(i;j)

L, 9K 2 (form Fi)j(arityFi)nfjg: A t B 2 (val Fi)jK ,

9K 2 (formFi)j(arityFi)nfjg: (A2 (valFi)jK _B 2 (valFi)jK),9K 2 (formFi)j(arityFi)nfjg:A 2 (val Fi)jK _ 9K 2 (form Fi)j(arityFi)nfjg: B 2 (val Fi)jK , A 2

�val

Q(D) F�(i;j)

L _

B 2�val

Q(D)F�(i;j)

L. Least element 0 is not in�val

Q(D)F�(i;j)

L because K [ f(j;0)g2/ GRFi.

17.9 On products of staroids 217

2. From the lemma.

3. We need to prove

L0tL12GRY(D)

F,9c2f0; 1garityQ(D)F :

�i2 arity

Y(D)F :Lc(i)i

!2GR

Y(D)F

for every L0; L12Q

formQ(D) F that is L0; L12

Quncurry(form �F ).

Really L0tL12GRQ(D)

F,L0tL12funcurry z j z 2Q

(GR �F )g.

9c 2 f0; 1garityQ(D)F :

��i 2 arity

Q(D)F : Lc(i)i: Lc(i)i

�2 GR

Q(D)F , 9c 2 f0;

1garityQ(D)F :

��i 2 arity

Q(D) F : Lc(i)i�2 funcurry z j z 2

Q(GR � F )g , 9c 2 f0;

1garityQ(D)F : curry

��i 2 arity

Q(D) F : Lc(i)i�2Q

(GR � F ) , 9c 2 f0; 1garityQ(D)F :

curry�� (i; j) 2 arity

Q(D) F : Lc(i;j)(i; j)�2Q

(GR � F ) , 9c 2 f0; 1garityQ(D)F :

(�i 2 dom F : (�j 2 dom Fi: Lc(i;j)(i; j))) 2Q

(GR � F ),9c 2 f0; 1garityQ(D)F8i 2 dom F :

(�j 2 dom Fi: Lc(i;j)(i; j)) 2 GR Fi , 8i 2 dom F9c 2 f0; 1gdomFi: (�j 2 dom Fi:Lc(j)(i; j))2GRFi,8i2 domF9c2f0; 1gdomFi: (�j 2domFi: (curry(Lc(j))i)j)2GRFi,8i 2 dom F : curry(L0)i t curry(L1)i 2GR Fi, L0 t L1 2 funcurry z j z 2

Q(GR � F )g,

L0tL12GRQ(D) F . �

For staroids it is de�ned ordinated productQ(ord) as de�ned in the section �Ordinated product�

above.

Obvious 17.117. If f and g are anchored relations and there exists a bijection ' from arity g toarity f such that fF � ' j F 2GR f g=GR g, then:

1. f is a prestaroid i� g is a prestaroid.

2. f is a staroid i� g is a staroid.

3. f is a completary staroid i� g is a completary staroid.

Corollary 17.118. Let F be an indexed family of anchored relations and every (form F )i be ajoin-semilattice.

1.Q(ord) F is a prestaroid if every Fi is a prestaroid.

2.Q(ord)

F is a staroid if every Fi is a staroid.

3.Q(ord) F is a completary staroid if every Fi is a completary staroid.

Proof. Use the fact that GRQ(ord) F =

nF � (

L(dom �F ))¡1 j F 2GR

Q(D) fo. �

De�nition 17.119. f �(ord) g=Q(ord) Jf ; gK.

Remark 17.120. If f and g are binary funcoids, then f �(ord) g is ternary.

Proposition 17.121.QStrd a=��QStrd a if each ai2Ai (for i2 n where n is some index set)

where each (Ai2n;Zi2n) is a down aligned �ltrator with separable core.[TODO: Duplicate with aproposition in �in�nite produict of poset elements� section.]

Proof. GR ��QStrd a =�L 2

QA j up L � Z \

QStrd a=�L 2

QA j up L �

QStrd a=�

L2Q

A j 8K 2upL:K 2QStrd

a= fL2

QA j 8K 2 upL; i2n:Ki�/ aig= fL2

QA j 8i2n;

K 2upL:Ki�/ aig=fL2Q

A j 8i2n:Li�/ aig=GRQStrd

a (taken into account that our �ltratoris with separable core). �


17.10 Star categories

De�nition 17.122. A precategory with star-morphisms consists of

1. a precategory C (the base precategory);

2. a set M (star-morphisms);

3. a function �arity� de�ned on M (how many objects are connected by this star-morphism);

4. a function Objm: aritym!Obj(C) de�ned for every m2M ;

5. a function (star composition) (m; f) 7!StarComp(m; f) de�ned for m2M and f being an(aritym)-indexed family of morphisms of C such that 8i2aritym:Src fi=Objm i (Src fi isthe source object of the morphism fi) such that arityStarComp(m; f)= aritym

such that it holds:

1. StarComp(m; f)2M ;

2. (associativity law )

StarComp(StarComp(m; f); g)= StarComp(m;�i2 aritym: gi� fi):

The meaning of the set M is an extension of C having as morphisms things with arbitrary(possibly in�nite) indexed set Objm of objects, not just two objects as morphisms of C have onlysource and destination.

De�nition 17.123. I will call Objm the form of the star-morphism m.

(Having �xed a precategory with star-morphisms) I will denote StarMor(P ) the set of star-morphisms of the form P .

Proposition 17.124. The sets StarMor(P ) are disjoint (for di�erent P ).

Proof. If two star-morphisms have di�erent forms, they are clearly not equal. �

De�nition 17.125. A category with star-morphisms is a precategory with star-morphisms whosebase is a category and the following equality (the law of composition with identity ) holds for everystar-morphism m:

StarComp(m;�i2 aritym: 1Objm i)=m:

De�nition 17.126. A partially ordered precategory with star-morphisms is a category with star-morphisms, whose base precategory is a partially ordered precategory and every set StarMor(X)is partially ordered for every X, such that:

m0vm1^ f0v f1)StarComp(m0; f0)vStarComp(m1; f1)

for every m0;m12M such that Objm0=Objm1

and indexed families f0 and f1 of morphisms suchthat

8i2 aritym: Src f0 i= Src f1 i=Objm0 i=Objm1 i and 8i2 aritym:Dst f0 i=Dst f1 i:

De�nition 17.127. A partially ordered category with star-morphisms is a category with star-morphisms which is also a partially ordered precategory with star-morphisms.

De�nition 17.128. A quasi-invertible precategory with star-morphisms is a partially orderedprecategory with star-morphisms whose base precategory is a quasi-invertible precategory, suchthat for every index set n, star-morphisms a and b of arity n, and an n-indexed family f ofmorphisms of the base precategory it holds

b�/ StarComp(a; f), a�/ StarComp(b; f y):

(Here f y=�i2 dom f : (fi)y.)

17.10 Star categories 219

De�nition 17.129. A quasi-invertible category with star-morphisms is a quasi-invertible precat-egory with star-morphisms which is a category with star-morphisms.

Each category with star-morphisms gives rise to a category (abrupt category , see a remarkbelow why I call it �abrupt�), as described below. Below for simplicity I assume that the set Mand the set of our indexed families of functions are disjoint. The general case (when they are notnecessarily disjoint) may be easily elaborated by the reader.

� Objects are indexed (by aritym for some m2M) families of objects of the category C andan (arbitrarily chosen) object None not in this set.

� There are the following disjoint sets of morphisms:

1. indexed (by aritym for some m2M) families of morphisms of C;

2. elements of M ;

3. the identity morphism idNone on None.

� Source and destination of morphisms are de�ned by the formulas:

� Src f =�i2 dom f : Src fi;

� Dst f =�i2 dom f :Dst fi;

� Srcm=None;

� Dstm=Objm.

� Compositions of morphisms are de�ned by the formulas:

� g � f =�i2dom f : gi� fi for our indexed families f and g of morphisms;

� f �m= StarComp(m; f) for m2M and a composable indexed family f ;

� m � idNone=m for m2M ;

� idNone � idNone= idNone.

� Identity morphisms for an object X are:

� �i2X: idXi if X =/ None;

� idNone if X =None.

We need to prove it is really a category.

Proof. We need to prove:

1. Composition is associative.

2. Composition with identities complies with the identity law.

Really:

1. (h� g) � f =�i2dom f : (hi� gi) � fi=�i2dom f :hi � (gi � fi)=h� (g � f);g � (f � m) = StarComp(StarComp(m; f); g) = StarComp(m; �i 2 arity m: gi � fi) =

StarComp(m; g � f)= (g � f) �m;f � (m� idNone) = f �m=(f �m) � idNone.

2. m � idNone=m; idDstm �m= StarComp(m;�i2 aritym: idObjm i)=m. �

Remark 17.130. I call the above de�ned category abrupt category because (excluding identitymorphisms) it allows composition with an m 2M only on the left (not on the right) so that themorphism m is �abrupt� on the right.

By Jx0; :::;xn¡1K I denote an n-tuple.

De�nition 17.131. Precategory with star morphisms induced by a dagger precategory C is:

� The base category is C.


� Star-morphisms are morphisms of C.

� arity f = f0; 1g.� Objm= JSrcm;DstmK.� StarComp(m; Jf ; gK)= g �m � f y.

Let prove it is really a precategory with star-morphisms.

Proof. We need to prove the associativity law:

StarComp(StarComp(m; Jf ; gK); Jp; qK)=StarComp(m; Jp � f ; q � gK):Really,

StarComp(StarComp(m; Jf ; gK); Jp; qK) = StarComp(g � m � f y; Jp; qK) = q � g � m � f y � py =q � g �m � (p� f)y= StarComp(m; Jp� f ; q � gK): �

De�nition 17.132. Category with star morphisms induced by a dagger category C is the abovede�ned precategory with star-morphisms.

That it is a category (the law of composition with identity) is trivial.

Remark 17.133. We can carry de�nitions (such as below de�ned cross-composition product) fromcategories with star-morphisms into plain dagger categories. This allows us to research propertiesof cross-composition product of indexed families of morphisms for categories with star-morphismswithout separately considering the special case of dagger categories and just binary star-composi-tion product.

17.10.1 Abrupt of quasi-invertible categories with star-morphisms

De�nition 17.134. The abrupt partially ordered precategory of a partially ordered precategorywith star-morphisms is the abrupt precategory with the following order of morphisms:

� Indexed (by aritym for some m 2M) families of morphisms of C are ordered as functionspaces of posets.

� Star-morphisms (which are morphisms None!Objm for some m 2M) are ordered in thesame order as in the precategory with star-morphisms.

� Morphisms None!None which are only the identity morphism ordered by the unique orderon this one-element set.

We need to prove it is a partially ordered precategory.

Proof. It trivially follows from the de�nition of partially ordered precategory with star-mor-phisms. �

17.11 Product of an arbitrary number of funcoids

In this section it will be de�ned a product of an arbitrary (possibly in�nite) indexed family offuncoids.

17.11.1 Mapping a morphism into a pointfree funcoid

De�nition 17.135. Let's de�ne the pointfree funcoid �f for every morphism f or a quasi-invertible category:

h�f ia= f � a and h(�f)¡1ib= f y � b:

17.11 Product of an arbitrary number of funcoids 221

We need to prove it is really a pointfree funcoid.

Proof. b�/ h�f ia, b�/ f � a, a�/ f y � b, a�/ h(�f)¡1ib. �

Remark 17.136. h�f i=(f �¡) is the Mor-functor17.1 Mor(f ;¡) and we can apply Yoneda lemmato it. (See any category theory book for de�nitions of these terms.)

Obvious 17.137. h�(g � f)ia= g � f � a for composable morphisms f and g or a quasi-invertiblecategory.

17.11.2 General cross-composition product

De�nition 17.138. Let �x a quasi-invertible category with with star-morphisms. If f is anindexed family of morphisms from its base category, then the pointfree funcoid

Q(C) f (cross-composition product of f) from StarMor(�i2dom f :Src fi) to StarMor(�i2dom f :Dst fi) is de�nedby the formulas (for all star-morphisms a and b of these forms):*Y(C)

f

+a= StarComp(a; f) and

* Y(C)f

!¡1+b= StarComp(b; f y):

It is really a pointfree funcoid by the de�nition of quasi-invertible category with star-morphisms.

Theorem 17.139.�Q(C) g

��Q(C) f

�=Q

i2n(C) (gi � fi) for every n-indexed families f and g

of composable morphisms of a quasi-invertible category with star-morphisms.

Proof.DQ

i2n(C) (gi� fi)

Ea= StarComp(a;�i2n: gi� fi) =StarComp(StarComp(a; f); g) andD�Q(C)

g��Q(C)

f�Ea=

DQ(C)gEDQ(C)

fEa=StarComp(StarComp(a; f); g).


Corollary 17.140.�Q(C) fk¡1

�� ::: �

�Q(C) f0�=Q

i2n(C) (fk¡1 � ::: � f0) for every n-indexed

families f0; :::; fn¡1 of composable morphisms of a quasi-invertible category with star-morphisms.

Proof. By math induction. �

17.11.3 Star composition of binary relationsFirst de�ne star composition for an n-ary relation a and an n-indexed family f of binary relationsas an n-ary relation complying with the formulas:

ObjStarComp(a;f)= f�gn;L2 StarComp(a; f),9y 2 a8i2n: yi fiLi

where � is a unique object of the group of small binary relations considered as a category.

Proposition 17.141. b�/ StarComp(a; f),9x2 a; y 2 b8j 2n:xj fj yj.

Proof. b �/ StarComp(a; f) , 9y: (y 2 b ^ y 2 StarComp(a; f)) , 9y: (y 2 b ^ 9x 2 a8j 2 n:xj fj yj),9x2 a; y 2 b8j 2n:xj fj yj. �

Theorem 17.142. The group of small binary relations considered as a category together withthe set of of all n-ary relations (for every small n) and the above de�ned star-composition form aquasi-invertible category with star-morphisms.

17.1. Also called Hom-functor.



1. StarComp(StarComp(m; f); g)= StarComp(m;�i2n: gi� fi);

2. StarComp(m;�i2 aritym: idObjm i)=m;

3. b�/ StarComp(a; f), a�/ StarComp(b; f y)

(the rest is obvious).Really,

1. L2StarComp(a; f),9y 2 a8i2n: yi fiLi.De�ne the relation R(f) by the formula xR(f) y,8i2n:xi fi yi. Obviously

R(�i2n: gi� fi) =R(g) �R(f):

L2StarComp(a; f),9y 2 a: yR(f)L.L 2 StarComp(StarComp(a; f); g) , 9p 2 StarComp(a; f): pR(g) L , 9p; y 2 a:

(yR(f) p ^ pR(g) L) , 9y 2 a: y(R(g) � R(f)) L , 9y 2 a: yR(�i 2 n: gi � fi) L ,L2StarComp(a;�i2n: gi � fi) because p2StarComp(a; f),9y 2 a: yR(f) p.

2. Obvious.

3. It follows from the proposition above. �

Obvious 17.143. StarComp(a[ b; f)=StarComp(a; f)[StarComp(b; f) for n-ary relations a, band an n-indexed family f of binary relations.

Theorem 17.144.DQ(C) f

EQa =

Qi2n hfiiai for every family f = fi2n of binary relations

and a= ai2n where ai is a small set (for each i2n).

Proof. L2DQ(C) f

EQa,L2 StarComp(

Qa; f),9y 2

Qa8i2n: yi fiLi,9y 2

Qa8i2n:

fyig �/ hfi¡1ifLig , 8i 2 n9y 2 ai: fyg �/ hfi¡1ifLig , 8i 2 n: ai �/ hfi¡1ifLig , 8i 2 n:fLig�/ hfiiai,8i2n:Li2 hfiiai,L2

Qi2n hfiiai. �

17.11.4 Star composition of Rel-morphismsDe�ne star composition for an n-ary anchored relation a and an n-indexed family f of Rel-morphisms as an n-ary anchored relation complying with the formulas:

ObjStarComp(a;f)=�i2 aritya:Dst fi;arityStarComp(a; f)= aritya;

L2GRStarComp(a; f),L2 StarComp(GR a;GR � f):

(Here I denote GR(A;B; f)= f for every Rel-morphism f .)

Proposition 17.145. b�/ StarComp(a; f),9x2 a; y 2 b8j 2n:xj fj yj.

Proof. From the previous section. �

Theorem 17.146. Relations with above de�ned compositions form a quasi-invertible categorywith star-morphisms.


1. StarComp(StarComp(m; f); g)= StarComp(m;�i2 aritym: gi� fi);2. StarComp(m;�i2 aritym: idObjm i)=m;


(the rest is obvious).It follows from the previous section. �


Proposition 17.147. StarComp(a t b; f) = StarComp(a; f) t StarComp(b; f) for an n-aryanchored relations a, b and an n-indexed family f of Rel-morphisms.

Proof. It follows from the previous section. �

Theorem 17.148. Cross-composition product of a family ofRel-morphisms is a principal funcoid.

Proof. By the proposition and symmetryQ(C)

f is a pointfree funcoid. Obviously it is a funcoidQi2n Src fi!

Qi2n Dst fi. Its completeness (and dually co-completeness) is obvious. �

17.11.5 Cross-composition product of funcoidsLet a be a an anchored relation of the form A and domA=n.

Let every fi (for all i2n) be a pointfree funcoid with Src fi=Ai.The star-composition of a with f is an anchored relation of the form �i2domA:Dst fi de�ned

by the formula

L2GRStarComp(a; f),9y 2GR a\Yi2n

atomsAi8i2n: yi [fi]Li:

Theorem 17.149. Let Dst fi be a starrish join-semilattice for every i2n.1. If a is a prestaroid then StarComp(a; f) is a staroid.

2. If a is a completary staroid and then StarComp(a; f) is a completary staroid.

Proof.

1. First prove that StarComp(a; f) is a prestaroid. We need to prove that (val StarComp(a;f))jL (for every j 2n) is a free star, that is

fX 2 (form f)j j L[f(j;X)g2GRStarComp(a; f)g

is a free star, that is the following is a free star

fX 2 (form f)j j R(X)g

where R(X),9y 2Q

i2n atomsAi: (8i2n n fjg: yi [fi]Li^ yj [fj]X ^ y 2GR a).R(X),9y 2

Qi2n atomsAi: (8i2n n fjg: yi [fi]Li^ yj [fj]X ^ yj 2 (vala)j(y jnnfjg)),

9y 2Q

i2nnfjg atomsAi; y 0 2 atomsAj: (8i 2 n n fjg: yi [fi] Li ^ y 0 [fj] X ^ y 0 2(val a)j(y jnnfjg)) , 9y 2

Qi2nnfjg atomsAi8i 2 n n fjg: yi [fi] Li ^ 9y 0 2 atomsAj:

(y 0 [fj]X ^ y 02 (val a)j(y jnnfjg)).If 9y2

Qi2nnfjg atomsAi8i2n nfjg: yi [fi]Li is false our statement is obvious. We can

assume it is true.So it is enough to prove that(X 2 (form f)j j 9y 2

Yi2nnfjg

atomsAi; y 02 atomsAj: (y 0 [fj]X ^ y 02 (val a)j(y jnnfjg)))

is a free star. That is

Q=

(X 2 (form f)j j 9y 2

Yi2nnfjg

atomsAi; y 02 (atomsAj)\ (val a)j(y jnnfjg): y 0 [fj]X)

is a free star. 0(form f)j2/ Q is obvious. That Q is an upper set is obvious. It remains to provethat X0tX12Q)X02Q_X12Q for every X0; X12 (form f)j. Let X0tX12Q. Thenthere exist y2

Qi2nnfjg atomsAi, y 02 (atomsAj)\ (vala)j(y jnnfjg) such that y 0 [fj]X0tX1.

Consequently (proposition 15.16) y 0 [fj]X0_ y 0 [fj]X1. But then X02Q_X12Q.To �nish the proof we need to show that GR StarComp(a; f) is an upper set, but this

is obvious.


2. Let a be a completary staroid. Let L0tL12GRStarComp(a; f) that is 9y2Q

i2n atomsAi:(8i 2 n: yi [fi] L0 i t L1 i ^ y 2 a) that is 9c 2 f0; 1gn; y 2

Qi2n atoms Ai: (8i 2 n:

yi [fi] Lc(i) i ^ y 2 a) (taken into account that Dst fi is starrish) that is 9c 2 f0; 1gn:(�i2n:Lc(i)i )2GRStarComp(a; f). So StarComp(a; f) is a completary staroid. �

Lemma 17.150. b�/Anch(A) StarComp(a; f),8A2GR a; B 2GR b; i2 n:Ai [fi]Bi for anchoredrelations a and b, provided that Src fi are atomic posets.

Proof.

b�/Anch(A) StarComp(a; f) ,9x2Anch(A) n f0g: (xv b^xvStarComp(a; f)) ,

9x2Anch(A) n f0g: (xv b^8B 2GRx:B 2GRStarComp(a; f)) ,

9x2Anch(A) n f0g: xv b^8B 2GRx9A2

Yi2domA

atomsAi: (8i2n:Ai [fi]BiÂ2GR a)

!,

9x2Anch(A) n f0g: (xv b^8B 2GRx; A2GR a; i2n:Ai [fi]Bi) ,9x2Anch(A): (xv b^8B 2GRx; A2GR a; i2n:Ai [fi]Bi) ,

8B 2GR b; A2GR a; i2n:Ai [fi]Bi:�

Theorem 17.151. ahQ(C) f

ib,8A 2 a; B 2 b; i 2 n:Ai [fi]Bi for anchored relations a and b,

provided that Src fi and Dst fi are atomic posets.

Proof. From the lemma. �

Conjecture 17.152. b�/ pStrd(A)StarComp(a; f), b�/ pStrd(B)StarComp(a; f) for staroids a and b.

Theorem 17.153. Anchored relations with objects being atomic posets and above de�ned com-positions form a quasi-invertible category with star-morphisms.


1. StarComp(StarComp(m; f); g)= StarComp(m;�i2 aritym: gi� fi);2. StarComp(m;�i2 aritym: idObjm i)=m;



1. L2GRStarComp(a; f),9y 2GR a\Q

i2n atomsAi8i2n: yi [fi]Li.De�ne the relation R(f) by the formula xR(f) y,8i2n:xi [fi] yi. Obviously

R(�i2n: gi� fi) =R(g) �R(f):

L2GRStarComp(a; f),9y 2GR a\Q

i2n atomsAi: yR(f)L.L 2 GR StarComp(StarComp(a; f); g) , 9p 2 GR StarComp(a; f) \

Qi2n atomsAi:

pR(g)L,9p; y2GRa\Q

i2n atomsAi: (yR(f) p^ pR(g)L),9y2GRa\Q

i2n atomsAi:y(R(g) � R(f)) L, 9y 2 GR a \

Qi2n atomsAi: yR(�i 2 n: gi � fi) L , 9y 2 GR a \Q

i2n atomsAi8i 2 n: yi [gi� fi] Li , L 2 GR StarComp(a; �i 2 n: gi � fi) becausep2GRStarComp(a; f),9y 2GR a\

Qi2n atomsAi: yR(f) p.

2. Obvious.

3. It follows from the lemma above. �

Theorem 17.154.DQ(C)

fEQStrd

a =Q

i2nStrd hfiiai for every families f = fi2n of pointfree

funcoids between atomic posets and a= ai2n where ai2Src fi.


Proof. L2GRDQ(C) f

EQStrd a,L2GRStarComp¡QStrd a; f

�,9y2

Qi2domA atomsAi8i2n:

(yi [fi] Li ^ yi �/ ai) , 8i 2 n9y 2 atomsAi: (y [fi] Li ^ y �/ ai) , 8i 2 n: ai [fi] Li , 8i 2 n:

Li�/ hfiiai,L2GRQ

i2nStrd hfiiai. �

Conjecture 17.155. StarComp(a t b; f) = StarComp(a; f) t StarComp(b; f) for anchoredrelations a, b of a form A, where every Ai is a distributive lattice, and an indexed family f ofpointfree funcoids with Src fi=Ai.

17.11.6 Simple product of pointfree funcoids

De�nition 17.156. Let f be an indexed family of pointfree funcoids with every Src fi and Dst fi(for all i2 dom f) being a poset with least element. Simple product of f isY(S)

f =

�x2

Yi2dom f

Src fi:�i2dom f : hfiixi;�y 2Y

i2dom f

Dst fi: �i2dom f : hfi¡1iyi!:

Proposition 17.157. Simple product is a pointfree funcoid

Y(S)f 2FCD

Yi2dom f

Src fi;Y

i2dom f

Dst fi

!:

Proof. Let x 2Q

i2dom f Src fi and y 2Q

i2dom f Dst fi. Then (take into account that Src fiand Dst fi are posets with least elements) y �/

¡�x 2

Qi2dom f Src fi: �i 2 dom f : hfiixi

�x,

y �/ �i 2 dom f : hfiixi, 9i 2 dom f : yi �/ hfiixi, 9i 2 dom f : xi �/ hfi¡1iyi, x �/ �i 2 dom f :

hfi¡1iyi,x�/¡�y 2

Qi2dom f Dst fi:�i2 dom f : hfi¡1iyi

�y. �

Obvious 17.158.DQ(S) f

Ex=�i2dom f : hfiixi for x2

QSrc fi.

Obvious 17.159.�DQ(S) f

Ex�i= hfiixi for x2

QSrc fi.

Proposition 17.160. fi can be restored if we knowQ(S) f if fi is a family of pointfree funcoids

between posets with least elements.

Proof. Let's restore the value of hfiix where i2dom f and x2Src fi.Let xi0=x and xj0 =0 for j=/ i.

Then hfiix= hfiixi0=�DQ(S)

fEx0�i.

We have restored the value of hfii. Restoring the value of hfi¡1i is similar. �

Remark 17.161. In the above proposition it is not required that fi are non-zero.

Proposition 17.162.�Q(S) g

��Q(S) f

�=Q

i2n(S) (gi � fi) for n-indexed families f and g of

composable pointfree funcoids between posets with least elements.

Proof.DQ

i2n(S) (gi� fi)

Ex=�i2dom f : hgi� fiixi=�i2dom f : hgiihfiixi=

DQ(S) gE�i2dom f :

hfiixi=DQ(S) g

EDQ(S) fEx=

D�Q(S) g��Q(S) f

�Ex.

ThusDQ

i2n(S)

(gi� fi)E=D�Q(S)

g��Q(S)

f�E

.D�Q(S) (gi � fi)�¡1E

=D��Q(S) g

��Q(S) f

��¡1Eis similar. �

Corollary 17.163.�Q(S) fk¡1

�� ::: �

�Q(S) f0�=Q

i2n(S) (fk¡1 � ::: � f0) for every n-indexed

families f0; :::; fn¡1 of composable pointfree funcoids between posets with least elements.


17.12 Multireloids

De�nition 17.164. I will call a multireloid of the form A=Ai2n, where every each Ai is a set, apair (f ;A) where f is a �lter on the set

QA.

De�nition 17.165. I will denote Obj(f ;A) =A and GR(f ;A)= f for every multireloid (f ;A).

I will denote RLD(A) the set of multireloids of the form A.The multireloid "RLD(A)F for a relation F is de�ned by the formulas:

Obj "RLD(A)F =A and GR "RLD(A)F = "QAF:

Let a be a multireloid of the form A and domA=n.Let every fi be a reloid with Src fi=Ai.The star-composition of a with f is a multireloid of the form �i2domA:Dst fi de�ned by the

formulas:

arityStarComp(a; f) =n;

GRStarComp(a; f) =l (

"RLD(A)GRStarComp(A;F ) j A2GR a; F 2Yi2n

GR fi

);

Objm StarComp(a; f)=�i2n:Dst fi:

Theorem 17.166. Multireloids with above de�ned compositions form a quasi-invertible categorywith star-morphisms.


1. StarComp(StarComp(m; f); g)= StarComp(m;�i2 aritym: gi� fi);

2. StarComp(m;�i2 aritym: idObjm i)=m;



1. Using properties of generalized �lter bases, StarComp(StarComp(a; f); g) =d �"RLDStarComp(B; G) j B 2 GR StarComp(a; f); G 2

Qi2n GR gi

=d�

"RLDStarComp(StarComp(A; F ); G) j A 2 GR a; F 2Q

i2n fi; G 2Q

i2n gi=d�

"RLDStarComp(A;G �F ) j A2GR a; F 2Q

i2n fi; G2Q

i2n gi=d �

"RLDStarComp(A;H) j A2GR a;H 2

Qi2n �i2n: gi� fi

=StarComp(a;�i2n: gi� fi).

2. StarComp(m; �i 2 arity m: idObjm i) =d �

"RLD(A)StarComp(A; H) j A 2 GR m; H 2Qi2aritym GR idObjm i

=d �

"RLD(A)StarComp(A; �i 2 arity m: Hi) j A 2 GR m; H 2Qi2aritym GR idObjm i

=d �

"RLD(A)StarComp(A; �i 2 arity m: idXi) j A 2 GR m;

X 2Q

i2aritym Objm i=d �

"RLD(A)(A \Q

X) j A 2 GRm; X 2Q

i2aritym Objm i=

d �"RLD(A)A j A2GRm

=m.

3. Using properties of generalized �lter bases,b �/ StarComp(a; f) , 8A 2 GR a; B 2 GR b; F 2

Qi2n GR fi: B �/ StarComp(A;

F ), 8A 2 GR a; B 2 GR b; F 2Q

i2n GR fi: B �/DQ(C)

FEA, 8A 2 GR a; B 2 GR b;

F 2Q

i2n GR fi: A �/D�Q(C)

F�¡1E

B , 8A 2 GR a; B 2 GR b; F 2Q

i2n GR fi:

A�/ StarComp(B;F y), a�/ StarComp(b; f y). �

De�nition 17.167. Let f be a multireloid of the form A. Then for i2domA

PriRLD f =lh"AiihPriiGR f:

17.12 Multireloids 227

Proposition 17.168. PriRLD f = hPriiGR a for every multireloid a and i 2 arity a, given a setA�hPriia.[TODO: Describe it with anchored relations instead.]

Proof. It's enough to show that hPriiGR f is a �lter.That hPriiGR f is an upper set is obvious.Let X; Y 2 hPriiGR f . Then there exist F ; G 2GR f such that X = Pri F , Y = PriG. Then

X \Y �Pri(F \G)2 hPriiGR f . Thus X \ Y 2 hPriiGR f . �

De�nition 17.169.QRLD X =

d �"RLD(�i2domX :Base(X i))

QX j X 2

QXfor every indexed

family X of �lters on powersets.

Proposition 17.170. PrkRLDQRLD x=xk for every indexed family x of proper �lters.

Proof. PrkRLDQRLD

x= hPrkiQRLD

x=xk. �

Conjecture 17.171. GRStarComp(at b; f)=GRStarComp(a; f)tGRStarComp(b; f) if f is areloid and a, b are multireloids of the same form, composable with f .

Theorem 17.172.QRLD A=

F �QRLD a j a2Q

i2domA atomsAifor every indexed family A

of �lters on powersets.

Proof. ObviouslyQRLD Aw

F �QRLD a j a2Q

i2domA atomsAi.

Reversely, let K 2GRF �QRLD a j a2

Qi2domA atomsAi

.

Consequently K 2 GRQRLD

a for every a 2Q

i2domA atoms Ai; K �Q

X and thusK �

SX2

Qa

QX for every X 2

Qa.

ButSX2

Qa

QX=

Qi2domA

ShPriiX �

Qj2domA Zj for some Zj 2Aj because hPriiX 2ai

and our lattice is atomistic. So K 2GRQRLD

A. �

Theorem 17.173. Let a, b be indexed families of �lters on powersets of the same form A. ThenYRLDau

YRLDb=

Yi2domA

RLD

(aiu bi):

Proof. YRLDau

YRLDb =

l("RLD(A)(P \Q) j P 2GR

YRLDa; Q2GR

YRLDb

)=

l �"RLD(A)

¡Yp\Y

q�j p2

Ya; q 2

Yb

=

l8<:"RLD(A)

Yi2domA

(pi\ qi)!j p2

Ya; q 2

Yb

9=; =

l ("RLD(A)

Yr j r 2

Yi2domA

(aiu bi))

=

Yi2domA

RLD

(aiu bi):

�

Theorem 17.174. If S 2PQ

i2domA F(Ai) where A is an indexed family of sets, then

l(YRLD

a j a2S

)=

Yi2domA

RLD l "F(Ai)

�PriS:


Proof. If S = ; thend �QRLD a j a 2 S

=d; = 1RLD(A) and

Qi2domARLD d

"F(Ai)�Pri S =Q

i2domARLD d

"F(Ai)�;=

Qi2domARLD d

;=Q

i2domARLD

1F(Ai)=1RLD(A), thusd �QRLD

a j a2 S=Q

i2domARLD d

"F(Ai)�PriS:

Let S=/ ;.d "F(Ai)

�PriS v

d "F(Ai)

�faig= ai for every a2S because ai2PriS. ThusQ

i2domARLD d

"F(Ai)�PriS v

QRLD a;

l(YRLD

a j a2S

)w

Yi2domA

RLD l "F(Ai)

�PriS:

Now suppose F 2GRQ

i2domARLD d

"F(Ai)�PriS. Then there exist X 2

Qi2domA

d "F(Ai)

�PriS

such that F �Q

X. It is enough to prove that there exist a 2 S such that F 2GRQRLD

a. Forthis it is enough

QX 2GR

QRLD a.Really, Xi2

d "F(Ai)

�PriS thus Xi2 ai for every a2S because PriS �faig.

ThusQ

X 2GRQRLD a. �

De�nition 17.175. I call a multireloid principal i� its graph is a principal �lter. [TODO: Provethat principal multireloids are the same as multireloid corresponding to a relation.]

De�nition 17.176. I call a multireloid convex i� it is a join of reloidal products.

Theorem 17.177. StarComp(a t b; f) = StarComp(a; f) t StarComp(b; f) for multireloids a, band an indexed family f of reloids with Src fi=(form a)i=(form b)i.

Proof. GR(StarComp(a; f) t StarComp(b; f)) =d �

"RLD(form a)StarComp(A; F ) j A 2 GR a;

F 2Q

i2n GR fitd �

"RLD(form b)StarComp(B; F ) j B 2 GR b; F 2Q

i2n GR fi=

d �"RLD(form a)StarComp(A; F ) t "RLD(form b)StarComp(B; F ) j A 2 GR a; B 2 GR b; F 2Q

i2n GR fi=d �

"RLD(form a)(StarComp(A; F ) [ StarComp(B; F )) j A 2 GR a; B 2 GR b;

F 2Q

i2n GR fi=d �

"RLD(form a)(StarComp(A[B;F )) j A2GRa;B2GRb; F 2Q

i2n GR fi=

d �"RLD(form a)StarComp(C;F ) j C 2GR(at b); F 2

Qi2n GR fi

=GRStarComp(at b; f). �

Conjecture 17.178. f vQRLD a , 8i 2 arity f : PriRLD f v ai for every multireloid f and

ai2F((form f)i) for every i2 arity f .

17.12.1 Starred reloidal productTychono� product of topological spaces inspired me the following de�nition, which seems possiblyuseful just like Tychono� product:

De�nition 17.179. Let a be an n-indexed (n is an arbitrary index set) family of �lters on sets.QRLD� a (starred reloidal product) is the reloid of the formQ

i2n Base(ai) induced by the �lter base(Yi2n

��Ai if i2mBase(ai) if i2n nm

�j m is a �nite subset of n;A2

Y(ajm)

):

Obvious 17.180. It is really a �lter base.

Obvious 17.181.QRLD�

awQRLD

a.

Proposition 17.182.QRLD� a=

QRLD a if n is �nite.

Proof. Take m=n to show thatQRLD� av

QRLD a. �

17.12 Multireloids 229

Proposition 17.183.QRLD� a=0RLD(�i2n:Base(ai)) if ai is the non-proper �lter for some i2n.

Proof. Take Ai= ; and m= fig. ThenQ

i2n


�= ;. �

Example 17.184. There exists an indexed family a of principal �lters such thatQRLD�

a is non-principal.

Proof. Let n = N . Let Base(ai) = R and each ai be a principal �lter corresponding to a two-element set.

EveryQ

i2n


�has at least cn> c elements.

There are elementsQRLD

a with cardinality 2n=n. They can't be elements ofQRLD�

a becausen=! < c. �

Corollary 17.185. There exists an indexed family a of principal �lters such thatQRLD� a =/QRLD

a.

Proof. BecauseQRLD a is principal. �

Proposition 17.186. PrkRLDQRLD�

x=xk for every indexed family x of proper �lters.

Proof. PrkRLDQRLD� x= hPrkiGR

QRLD� x=xk. �

17.13 Subatomic product of funcoids

De�nition 17.187. Let f be an indexed family of funcoids. ThenQ(A) f (subatomic product)

is a funcoidQ

i2dom f Src fi!Q

i2dom f Dst fi such that for every a 2 atomsRLD(�i2dom f :Src fi),

b2 atomsRLD(�i2dom f :Dst fi)

a

"Y(A)f

#b,8i2dom f :PriRLD a [fi]PriRLD b:

Proposition 17.188. The funcoidQ(A)

f exists.

Proof. To prove thatQ(A)

f exists we need to prove (for every a 2 atomsRLD(�i2dom f :Src fi),b2 atomsRLD(�i2dom f :Dst fi))

8X 2GR a; Y 2GR b9x 2 atoms "RLD(�i2dom f :Src fi)X; y 2 atoms "RLD(�i2dom f :Dst fi)Y : x

"Y(A)f

#y) a

"Y(A)f

#b:

Let 8X 2 GR a; Y 2 GR b9x 2 atoms "RLD(�i2dom f :Src fi)X; y 2 atoms "RLD(�i2dom f :Dst fi)Y :

xhQ(A) f

iy.

Then

8X 2GR a; Y 2GR b9x2 atoms "RLD(�i2dom f :Src fi)X; y 2 atoms "RLD(�i2dom f :Dst fi)Y 8i2 dom f :

PriRLD x [fi]PriRLD y:

Then because PriRLD x2 atoms "Src fiPriX and likewise for y:8X 2GR a; Y 2GR b8i2dom f9x2 atoms "Src fiPriX; y 2 atoms "Dst fiPriY :x [fi] y.Thus 8X 2GR a; Y 2GR b8i2dom f : "Src fiPriX [fi] "Dst fiPriY ;8X 2GR a; Y 2GR b8i2dom f :PriX [fi]

�PriY .


Then 8X 2 hPriiGR a; Y 2 hPriiGR b:X [fi]�Y .

Thus PriRLD a [fi]PriRLD b. So

8i2dom f :PriRLD a [fi]PriRLD b

and thus a�f �(A) g

�b. �

Remark 17.189. It seems that the proof of the above theorem can be simpli�ed using cross-composition product.

Theorem 17.190.Q

i2n(A) (gi � fi) =

Q(A) g �Q(A) f for indexed (by an index set n) families f

and g of funcoids such that 8i2n:Dst fi= Src gi.

Proof. Let a, b be ultra�lters onQ

i2n Src fi andQ

i2n Dst gi correspondingly,

a

"Yi2n

(A)

(gi� fi)#b,8i2dom f : hPriia [gi � fi] hPriib,8i2dom f9C 2atomsF(Dst fi): (hPriia [fi]C^

C [gi] hPriib),8i 2 dom f9c 2 atomsRLD(�i2n:Dst f): (hPriia [fi] hPriic ^ hPriic [gi] hPriib)(9c 2

atomsRLD(�i2n:Dst f)8i 2 dom f : (hPriia [fi] hPriic ^ hPriic [gi] hPriib),9c 2 atomsRLD(�i2n:Dst f):0@a

"Y(A)f

#c^ c

"Y(A)g

#b

1A, a

"Y(A)g �Y(A)

f

#b:

Let

8i2dom f9c2 atomsRLD(�i2n:Dst f): (hPriia [fi] hPriic^ hPriic [gi] hPriib):

Then there exists c02¡atomsRLD(�i2n:Dst f)�n such that

8i2dom f : (hPriia [fi] hPriici0^ hPriici0 [gi] hPriib):

Then take c00=QRLD c0. Then 8i2 dom f : (hPriia [fi] hPriici00^ hPriici00 [gi] hPriib). Thus

9c2 atomsRLD(�i2n:Dst f)8i2 dom f : (hPriia [fi] hPriic^ hPriic [gi] hPriib):

We have ahQ

i2n(A) (gi � fi)

ib, a

hQ(A) g �Q(A) f

ib. �

Corollary 17.191.�Q(A)

fk¡1

�� ::: �

�Q(A)f0

�=Q

i2n(A)

(fk¡1 � ::: � f0) for every n-indexed

families f0; :::; fn¡1 of composable funcoids.

Proposition 17.192.QRLD a

hQ(A) fiQRLD b,8i2dom f :ai [fi] bi for an indexed family f of

funcoids and indexed families a and b of �lters where ai2F(Src fi), bi2F(Dst fi) for every i2dom f .

Proof. If ai= 0 or bi= 0 for some i our theorem is obvious. We will take ai=/ 0 and bi=/ 0, thusthere exist

x2 atomsYRLD

a; y 2 atomsYRLD

b:QRLD ahQ(A) f

i QRLD b , 9x 2 atomsQRLD a; y 2 atoms

QRLD b: xhQ(A) f

iy , 9x 2

atomsQRLD

a; y 2 atomsQRLD

b8i 2 dom f : hPriix [fi] hPriiy , 8i 2 dom f9x 2 atoms ai;y 2 atoms bi:x [fi] y,8i2 dom f : ai [fi] bi. �

Theorem 17.193.DQ(A)

fEx=

Qi2dom fRLD hfiiPriRLD x for an indexed family f of funcoids and

x2 atomsRLD(�i2dom f :Src fi) for every n2 dom f .

Proof. For every ultra�lter y 2F¡Q

i2dom f Dst fi�we have:

17.13 Subatomic product of funcoids 231

y �/Q

i2dom fRLD hfiiPriRLD x, 8i 2 dom f : PriRLD y �/ hfiiPriRLD x, 8i 2 dom f : PriRLD x [fi]

PriRLD y,xhQ(A) f

iy, y�/

DQ(A) fEx.

ThusDQ(A) f

Ex=

Qi2dom fRLD hfiiPriRLDx. �

Corollary 17.194.f �(A) g

�x= hf i(domx)�RLD hgi(im x).

17.14 On products and projections

Conjecture 17.195. For principal funcoidsQ(C) and

Q(A) coincide with the conventionalproduct of binary relations.

17.14.1 Staroidal productLet f be a staroid, whose form components are boolean lattices.

De�nition 17.196. Staroidal projection of a staroid f is the �lter PrkStrd f corresponding to thefree star

(val f)k¡�i2 (arity f) n fkg: 1(form f)i

�:

Proposition 17.197. Prk GRQStrd x = ?xk if x is an indexed family of proper �lters, and

k 2dom x.

Proof. Prk GRQStrd x=Prk

�L2

ì2domx formxi j 8i2domx:xi�/ Li

= (used the fact that xi

are proper �lters)=fl2 formxk j xk�/ lg= ?xk. �

Proposition 17.198. PrkStrdQStrd x=xk if x is an indexed family of proper �lters, and k2domx.

Proof. @ PrkStrdQStrd x =

¡val

QStrd x�k

¡� i 2 (dom x) n fkg: 1(form x)i

�=�X 2¡

formQStrd x

�kj¡�i2 (dom x) n fkg: 1(formx)i

�[f(k;X)g2GR

QStrd x=�X 2Base xk j

¡8i2

(domx) n fkg: 1(formx)i�/ xi�^X �/ xk

= fX 2Base xk j X �/ xkg= @ xk.

Consequently PrkStrdQStrd x=xk. �

17.14.2 Cross-composition product of pointfree funcoids

De�nition 17.199. Zero pointfree funcoid from a poset A to to a poset B is such a pointfreefuncoid A!B that hf ix is a least element of B for every x2A.

Proposition 17.200. A pointfree funcoid f is zero i� [f ]=;.

Proof. Direct implication is obvious.Let now [f ]=;. Then hf ix� y for every x2Src f , y2Dst f and thus hf ix�hf ix. It is possible

only when hf ix=0Dst f. �

Corollary 17.201. A pointfree funcoid is zero i� its reverse is zero.

Proposition 17.202. Values xi (for every i 2 dom x) can be restored from the value ofQ(C) x

provided that x is an indexed family of non-zero pointfree funcoids if Src fi (for every i2n) is anatomic lattice and every Dst fi is an atomic poset with greatest element.

Proof.DQ(C)

xEQStrd

p=Q

i2nStrd hxiipi by theorem 17.154.


Since xi is non-zero there exist p such that hxiipi is non-zero. Take k 2n, pi0= pi for i=/ k andpk0 = q for an arbitrary value q; then (using the staroidal projections from the previous subsection)

hxkiq=PrkStrdYi2n

Strd

hxiipi0=PrkStrd

*Y(C)x

+YStrdp0:

So the value of x can be restored fromQ(C)

x by this formula. �

17.14.3 Subatomic product

Proposition 17.203. Values xi (for every i 2 dom x) can be restored from the value ofQ(A) x

provided that x is an indexed family of non-zero funcoids.

Proof. Fix k 2dom f . Let for some �lters x and y

a=

(1F(Base(x)) if i=/ k;x if i= k

and b=

(1F(Base(y)) if i=/ k;y if i= k:

Then x [xk] y, ak [xk] bk,8i2 dom f : ai [xi] bi,QRLD a

hQ(A) xiQRLD b. So we have restored

xk fromQ(A) x. �

De�nition 17.204. For every funcoid f :Q

A!Q

B (where A and B are indexed families ofsets) consider the funcoid Prk

(A)f de�ned by the formula

XhPrk

(A)fi�Y ,

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiX if i= k

![f ]

Yi2domB

RLD (

1F(Bi) if i=/ k;"BiY if i= k

!:

Proposition 17.205. Prk(A)

f is really a funcoid.

Proof. :�;hPrk

(A)fi�Y�is obvious.

I [JhPrk

(A)fi�Y ,Y

i2domA

RLD (

1F(Ai) if i=/ k;"Ai(I [J) if i= k

![f ]

Yi2domB

RLD (


!,

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiI t"AiJ if i= k

![f ]

Yi2domB

RLD (


!,

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiI if i= k

!t

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiJ if i= k

![f ]

Yi2domB

RLD (


!,

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiI if i= k

![f ]

Yi2domB

RLD (


!_

Yi2domA

RLD (

1F(Ai) if i=/ k;"AiJ if i= k

![f ]

Yi2domB

RLD (


!,

IhPrk

(A)fi�Y _ J

hPrk

(A)fi�Y :


Proposition 17.206. For every funcoid f :Q

A!Q

B (where A and B are indexed families ofsets) there exists a funcoid Prk

(A)f de�ned by the formula

XhPrk

(A)fiY,

Yi2domA

RLD (

1F(Ai) if i=/ k;X if i= k

![f ]

Yi2domB

RLD (

1F(Bi) if i=/ k;Y if i= k

!:

17.14 On products and projections 233

Proof.

XhPrk

(A)fiY ,

8X 2X ; Y 2Y :XhPrk

(A)fi�Y ,

8X 2X ; Y 2Y :Y

i2domA

RLD (

1F(Ai) if i=/ k;"AiX if i= k

![f ]

Yi2domB

RLD (


!,

8X 2Y

i2domA

RLD (


!; Y 2

Yi2domB

RLD (


!:X [f ]�Y ,

Yi2domA

RLD (


![f ]

Yi2domB

RLD (


!:

�

Remark 17.207. Reloidal product above can be replaced with starred reloidal product, becauseof �nite number of non-maximal multipliers in the products.

Obvious 17.208. Prk(A)Q(A) x=xk provided that x is an indexed family of non-zero funcoids.

17.14.4 Other

Conjecture 17.209. Values xi (for every i 2 dom x) can be restored from the value ofQ(C)

xprovided that x is an indexed family of non-zero reloids.

De�nition 17.210. Displaced productQ(DP) f =�Q(C) f for every indexed family of pointfree

funcoids, where downgrading is de�ned for the �ltrator¡FCD(StarMor(Src � f); StarMor(Dst � f)); h"FCDiP

¡Y(Src � f)�

Y(Dst � f)

��:

Remark 17.211. Displaced product is a funcoid (not just a pointfree funcoid).

Conjecture 17.212. Values xi (for every i 2 dom x) can be restored from the value ofQ(DP) x

provided that x is an indexed family of non-zero funcoids.

De�nition 17.213. Let f 2P¡Z

`Y�where Z is a set and Y is a function.

Prk(D)

f =Prk fcurry z j z 2 f g:

Proposition 17.214. Prk(D)Q(D) F =Fk for every indexed family F of non-empty relations.

Proof. Obvious. �

Corollary 17.215. GRPrk(D)Q(D) F =GRFk and formPrk

(D)Q(D) F = formFk for every indexedfamily F of non-empty anchored relations.

17.15 Relationships between cross-composition and sub-atomic products


�b, dom a [f ] dom b ^ im a [g] im b for funcoids f and g and

atomic funcoids a2FCD(Src f ; Src g) and b2FCD(Dst f ;Dst g).

Proof. a�f �(C) g

�b, a � f¡1�/ g¡1 � b, (dom a�FCD im a) � f¡1�/ g¡1 � (dom b�FCD im b),

hf idom a �FCD im a �/ dom b �FCD hg¡1iim b , hf idom a �/ dom b ^ im a�/ hg¡1iim b ,dom a [f ] dom b^ im a [g] im b. �


Proposition 17.217. XhQ(A) f

iY,8i2dom f :PriRLDX [fi]PriRLDY for every indexed family

f of funcoids and X 2RLD(Src � f), Y 2RLD(Dst � f).

Proof. XhQ(A) f

iY , 9a 2 atoms X ; b 2 atoms Y : a

hQ(A) fib , 9a 2 atoms X ; b 2

atoms Y8i 2 dom f : PriRLD a [fi] PriRLD b, 8i 2 dom f 9x 2 atoms PriRLD X ; y 2 atoms PriRLD Y:xi [fi] yi,8i2 dom f :PriRLDX [fi]PriRLDY. �

Corollary 17.218. X�f �(A) g

�Y,domX [f ]domY ^ imX [g] imY for funcoids f , g and reloids

X 2RLD(Src f ; Src g), and Y 2RLD(Dst f ;Dst g).

Lemma 17.219. For every A2Rel(X ;Y ) (for every sets X , Y ) we have:

f(dom a; im a) j a2 atoms "FCDAg= f(dom a; im a) j a2 atoms "RLDAg:

Proof. Let x 2 f(dom a; im a) j a 2 atoms "RLDAg. Then x0 = dom a and x1 = im a wherea2 atoms "RLDA.

Then x0 = dom (FCD)a and x1 = im (FCD)a and obviously (FCD)a 2 atoms "FCDA. Sox2f(dom a; im a) j a2 atoms "FCDAg.

Let now x 2 f(dom a; im a) j a 2 atoms "FCDAg. Then x0 = dom a and x1 = im a wherea2 atoms "FCDA.

x0 ["FCDA] x1, x0 [(FCD)"RLDA] x1, x0�RLD x1�/ "RLDA. Thus there exists atomic reloid x0

such that x02 atoms "RLDA and dom x0=x0, imx0=x1.So x2f(dom a0; im a0) j a02 atoms "RLDAg. �

Theorem 17.220. "FCDA�f �(C) g

�"FCDB,"RLDA

�f �(A) g

�"RLDB for funcoids f , g, and Rel-

morphisms A: Src f!Src g, and B:Dst f!Dst g.

Proof. "FCDA�f �(C) g

�"FCDB , 9a 2 atoms "FCDA; b 2 atoms "FCDB: a

�f �(C) g

�b ,

9a 2 atoms "FCDA; b 2 atoms "FCDB: (dom a [f ] dom b ^ im a [g] im b)) 9a0 2 atoms dom "FCDA;a12 atoms im "FCDA; b02 atoms dom "FCDB; b12 atoms im "FCDB: (a0 [f ] b0^ a1 [g] b1).

On the other hand:9a0 2 atoms dom "FCDA; a1 2 atoms im "FCDA; b0 2 atoms dom "FCDB; b1 2 atoms im "FCDB:

(a0 [f ] b0 ^ a1 [g] b1) ) 9a0 2 atoms dom "FCDA; a1 2 atoms im "FCDA; b0 2 atoms dom "FCDB;b1 2 atoms im "FCDB: (a0 �FCD b0 �/ f ^ a1 �FCD b1 �/ g)) 9a 2 atoms "FCDA; b 2 atoms "FCDB:(dom a [f ] dom b^ im a [g] im b).

Also using the lemma we have 9a 2 atoms "FCDA; b 2 atoms "FCDB: (dom a [f ] dom b îm a [g] im b),9a2 atoms "RLDA; b2 atoms "RLDB: (dom a [f ] dom b^ im a [g] im b).

So "FCDA�f �(C) g

�"FCDB , 9a 2 atoms "RLDA; b 2 atoms "RLDB: (dom a [f ] dom b ^

im a [g] im b),9a2 atoms "RLDA; b2 atoms "RLDB: a�f �(A) g

�b,"RLDA

�f �(A) g

�"RLDB. �

Corollary 17.221. f �(A) g=��¡f �(C) g

�where downgrading is taken on the �ltrator¡

FCD(FCD(Src � f);FCD(Dst � f));FCD¡PY

(Src � f);PY

(Dst � f)��

and upgrading is taken on the �ltrator¡FCD(RLD(Src � f);RLD(Dst � f));FCD

¡PY

(Src � f);PY

(Dst � f)��:

where we equate n-ary relations with corresponding principal multifuncoids and principal mul-tireloids, when appropriate.

Proof. Leave as an exercise for the reader. �

Conjecture 17.222. "FCDAhQ(C) f

i"FCDB,"RLDA

hQ(A) fi"RLDB for every indexed family

f of funcoids and A2PQ

i2dom f Src fi, B 2PQ

i2dom f Dst fi.

17.15 Relationships between cross-composition and subatomic products 235

Theorem 17.223. For every �lters a0, a1, b0, b1 we have

a0�FCD b0�f �(C) g

�a1�FCD b1, a0�RLD b0

�f �(A) g

�a1�RLD b1:

Proof. a0�RLD b0�f �(A) g

�a1�RLD b1,8A0 2 a0; B0 2 b0; A1 2 a1; B1 2 b1:A0�B0

�f �(A) g

��A1�B1.

A0�B0�f �(A) g

��A1�B1,A0�B0

�f �(C) g

��A1�B1,A0 [f ]

�A1^B0 [g]�B1. (Here by

A0� B0�f �(C) g

��A1� B1 I mean "FCD(Base a;Base b)(A0� B0)

�f �(C) g

�"FCD(Base a;Base b)(A1 �

B1).)Thus it is equivalent to a0 [f ] a1^ b0 [g] b1 that is a0�FCD b0

�f �(C) g

��a1�FCD b1.

(It was used the theorem 17.151.) �

Can the above theorem be generalized for the in�nitary case?

17.16 Coordinate-wise continuity

Theorem 17.224. Let � and � be indexed (by some index set n) families of endomorphisms fora quasi-invertible dagger category with star-morphisms, and fi2Mor(Ob �i;Ob�i) for every i2n.Then:

1. 8i2n: fi2C(�i; �i))Q(C) f 2C

�Q(C) �;Q(C) �

�;

2. 8i2n: fi2C0(�i; �i))Q(C) f 2C0

�Q(C) �;Q(C) �

�;

3. 8i2n: fi2C00(�i; �i))Q(C) f 2C00

�Q(C) �;Q(C) �

�.

Proof. Using the corollary 17.140:

1. 8i 2 n: fi 2 C(�i; �i) , 8i 2 n: fi � �i v �i � fi )Q

i2n(C) (fi � �i) v

Qi2n(C) (�i � fi) ,�Q(C)

f��Q(C)

��v�Q(C)

��Q(C)

f�,Q(C)

f 2C�Q(C)

�;Q(C)

��.

2. 8i 2 n: fi2C0(�i; �i),8i2 n: �iv fiy � �i � fi)

Q(C)�v

Qi2n(C) ¡

fiy � �i � fi

�,Q(C)

�v�Qi2n(C) fi

y��Q

i2n(C) �i

��Q

i2n(C) fi

�,Q(C) �v

�Qi2n(C) fi

�y ��Qi2n(C) �i

��Q

i2n(C) fi

�,Q(C) f 2C0

�Q(C) �;Q(C) �

�.

3. 8i 2 n: fi 2 C00(�i; �i) , 8i 2 n: fi � �i � fiy v �i )

Qi2n(C) ¡

fi � �i � fiy� v Q

i2n(C)

�i ,Qi2n(C) fi �

Qi2n(C) �i �

Qi2n(C) fi

yvQ

i2n(C) �i,

Qi2n(C) fi �

Qi2n(C) �i �

�Qi2n(C) fi

�yv Qi2n(C) �i,Q

i2n(C) fi2C00

�Q(C) �;Q(C) �

�. �

Theorem 17.225. Let � and � be indexed (by some index set n) families of endofuncoids, andfi2 FCD(Ob �i;Ob �i) for every i2n. Then:

1. 8i2n: fi2C(�i; �i))Q(A)

f 2C�Q(A)

�;Q(A)

��;

2. 8i2n: fi2C0(�i; �i))Q(A)

f 2C0�Q(A)

�;Q(A)

��;

3. 8i2n: fi2C00(�i; �i))Q(A) f 2C00

�Q(A) �;Q(A) �

�.

Proof. Similar to the previous theorem. �

Theorem 17.226. Let � and � be indexed (by some index set n) families of pointfree endofuncoidsbetween posets with least elements, and fi2 FCD(Ob �i;Ob �i) for every i2n. Then:

1. 8i2n: fi2C(�i; �i))Q(S)

f 2C�Q(S)

�;Q(S)

��;


2. 8i2n: fi2C0(�i; �i))Q(S) f 2C0

�Q(S) �;Q(S) �

�;

3. 8i2n: fi2C00(�i; �i))Q(S) f 2C00

�Q(S) �;Q(S) �

�.

Proof. Similar to the previous theorem. �

17.17 Counter-examples

Example 17.227. ��f =/ f for some staroid f whose form is an indexed family of �lters on a set.

Proof. Let f = fA2F(f) j "fCorA�/ �g for some in�nite set f where � is some non-principal�lter on f.

A t B 2 f , "fCor(A t B) �/ � , "fCor A t "fCor B �/ � , "fCor A u � =/ 0F(f) _"fCorB u�=/ 0F(f),A2 f _B 2 f .

Obviously 0F(f)2/ f . So f is a free star. But free stars are essentially the same as 1-staroids.�f = @�. ��f = ?�=/ f . �

For the below counter-examples we will de�ne a staroid # with arity#=N and GR#2P(NN )(based on a suggestion by Andreas Blass):

A2GR#, supi2N

card(Ai\ i)=N ^8i2N :Ai=/ ;:

Proposition 17.228. # is a staroid.

Proof. (val#)iL=PN n f;g for every L2 (PN)N nfig if

supi2N nfig

card(Aj \ j)=N ^8j 2N n fig:Lj=/ ;:

Otherwise (val#)iL=;. Thus (val#)iL is a free star. So # is a staroid. [TODO: Show that it's notjust a prestaroid.] �

Proposition 17.229. # is a completary staroid.

Proof. A0 t A1 2 GR # , A0 [ A1 2 GR # , supi2N card((A0i [ A1i) \ i) = N ^ 8i 2 N :A0i[A1i=/ ;, supi2N card((A0i\ i)[ (A1i\ i))=N ^8i2N :A0i[A1i=/ ;.

If A0 i= ; then A0 i\ i= ; and thus A1i\ iwA0i\ i. Thus we can select c(i)2f0; 1g in such away that 8d2f0;1g: card(Ac(i) i\ i)w card(Ad i\ i) and Ac(i)i=/ ;. (Consider the case A0 i; A1 i=/ ;and the similar cases A0 i= ; and A1 i= ;.)

So A0tA12GR#, supi2N card(Ac(i) i\ i)=N ^8i2N :Ac(i)i=/ ;, (�i2n:Ac(i)i)2GR#.Thus # is completary. �

Obvious 17.230. # is non-zero.

Example 17.231. For every family a=ai2N of ultra�ltersQStrd

a is not an atom nor of the posetof staroids neither of the poset of completary staroids of the form �i2N :Base(ai).

Proof. It's enough to prove #wQStrd a.

Let "NRi= ai if ai is principal and Ri=N n i if ai is non-principal.We have 8i2N :Ri2 ai.We have R2/ GR# because supi2N card(Ri\ i)=/ N .R2

QStrd a because 8X 2 ai:X \Ri=/ ;.So #w

QStrda. �

Remark 17.232. At http://mathover�ow.net/questions/60925/special-in�nitary-relations-and-ultra�lters there are a proof for arbitrary in�nite form, not just for N .

17.17 Counter-examples 237

Conjecture 17.233. There exists a non-completary staroid.

Conjecture 17.234. There exists a prestaroid which is not a staroid.

Conjecture 17.235. The set of staroids of the form AB where A and B are sets is atomic.

Conjecture 17.236. The set of staroids of the form AB where A and B are sets is atomistic.

Conjecture 17.237. The set of completary staroids of the form AB where A and B are sets isatomic.

Conjecture 17.238. The set of completary staroids of the form AB where A and B are sets isatomistic.

Example 17.239. StarComp(a; f t g) =/ StarComp(a; f) t StarComp(a; g) in the category ofbinary relations with star-morphisms for some n-ary relation a and an n-indexed families f and gof functions.

Proof. Let n=f0;1g. Let GRa=f(0; 1); (1; 0)g and f = Jf(0; 1)g;f(1; 0)gK, g= Jf(1; 0)g;f(0; 1)gK.For every f0; 1g-indexed family of � of functions:L 2 StarComp(a; �) , 9y 2 a: (y0 �0 L0 ^ y1 �1 L1) , 9y0 2 dom �0; y1 2 dom �1:

(y0 �0L0^ y1 �1L1) for every n-ary relation �.Consequently

L2StarComp(a; f),L0=1^L1=0,L=(0; 1)

that is StarComp(a; f) = f(1; 0)g. SimilarlyStarComp(a; g)= f(0; 1)g.Also

L2StarComp(a; f t g),9y0; y12f0; 1g: ((y0 f0L0_ y0 g0L0)^ (y1 f1L1_ y1 g1L1)):Thus

StarComp(a; f t g)= f(0; 1); (1; 0); (0; 0); (1; 1)g: �

Corollary 17.240. The above inequality is possible also for star-morphisms of funcoids and star-morphisms of reloids.

Proof. Because �nitary funcoids and reloids between �nite sets are essentially the same as �nitaryrelations and our proof above works for binary relations. �

17.18 ConjecturesRemark 17.241. Below I present special cases of possible theorems. The theorems may begeneralized after the below special cases are proved.

Conjecture 17.242. For every two funcoids f and g we have:

1. (RLD)ina�f �(DP) g

�(RLD)inb, a

�f �(C) g

�b for every funcoids a 2 FCD(Src f ; Src g),

b2FCD(Dst f ;Dst g);

2. (RLD)outa�f �(DP) g

�(RLD)outb, a

�f �(C) g



3. (FCD)a�f �(C) g

�(FCD)b , a

�f �(DP) g

�b for every reloids a 2 RLD(Src f ; Src g),

b2RLD(Dst f ;Dst g).

Conjecture 17.243. For every two funcoids f and g we have: [TODO: Haven't yet tried hard tosolve this. Compare theorem 17.220 as a special case.]

1. (RLD)ina�f �(A) g

�(RLD)inb, a

�f �(C) g




2. (RLD)outa�f �(A) g

�(RLD)outb, a

�f �(C) g



3. (FCD)a�f �(C) g

�(FCD)b , a

�f �(A) g

�b for every reloids a 2 RLD(Src f ; Src g),

b2RLD(Dst f ;Dst g).

De�nition 17.244. A staroid on power sets is such a staroid f that every (form f)i is a latticeof all subsets of some set.

Conjecture 17.245.QStrd a�/

QStrd b, b2QStrd a, a 2

QStrd b,8i2 n: ai�/ bi for every n-indexed families a and b of �lters on powersets.

Conjecture 17.246. Let f be a staroid on powersets and a2Q

i2arity f Src fi, b2Q

i2arity f Dst fi.Then YStrd

a

"Y(C)f

#YStrdb,8i2n: ai [fi] bi:

Proposition 17.247. The conjecture 17.246 is a consequence of the conjecture 17.245.

Proof.QStrd a

hQ(C) fiQStrd b,

QStrd b�/DQ(C) f

EQStrd a,QStrd b�/

Qi2nStrd hfiiai,8i2n:

bi�/ hfiiai,8i2n: ai [fi] bi. �

Conjecture 17.248. For every indexed families a and b of �lters and an indexed family f ofpointfree funcoids we have

YStrda

"Y(C)f

#YStrdb,

YRLDa

" Y(DP)

f

#YRLDb:

Conjecture 17.249. For every indexed families a and b of �lters and an indexed family f ofpointfree funcoids we have YStrd

a

"Y(C)f

#YStrdb,

YRLDa

"Y(A)f

#YRLDb:

Strengthening of an above result:

Conjecture 17.250. If a is a completary staroid and Dst fi is a starrish poset for every i2n thenStarComp(a; f) is a completary staroid.

Straightening of above results:

Conjecture 17.251.

1.Q(D) F is a prestaroid if every Fi is a prestaroid.

2.Q(D)

F is a completary staroid if every Fi is a completary staroid.

Conjecture 17.252. If f1 and f2 are funcoids, then there exists a pointfree funcoid f1� f2 suchthat

hf1� f2ix=Gfhf1iX �FCD hf2iX j X 2 atomsxg

for every ultra�lter x.

Conjecture 17.253. Let A=Ai2n be a family of boolean lattices.A relation � 2P

QatomsF(Ai) such that for every a2

QatomsF(Ai)

8A2 a: � \Yi2n

atoms "AiAi=/ ;) a2 � (17.4)

17.18 Conjectures 239

can be continued till the function�f for a unique staroid f of the form �i2n:P(Ai). The funcoidf is completary.

For every X 2Q

i2n F(Ai)

X 2GR�f, � \Yi2n

atomsX i=/ ;: (17.5)

Conjecture 17.254. Let R be a set of staroids of the form �i 2 n: F(Ai) where every Ai is aboolean lattice. If x2

Qi2n atomsF(Ai) then x2GR�d R,8f 2R:x2�f .

17.18.1 Informal questionsDo products of funcoids and reloids coincide with Tychonov topology?

Limit and generalized limit for multiple arguments.Is product of connected spaces connected?Product of T0-separable is T0, of T1 is T1?Relationships between multireloids and staroids.Generalize the section �Specifying funcoids by functions or relations on atomic �lters� from [28].Generalize �Relationships between funcoids and reloids�.Explicitly describe the set of complemented funcoids.


Chapter 18Identity staroids

18.1 Additional propositions

Proposition 18.1.

�hf ikX j X 2 up

�Qi2nnfkgAi;

Qi2nnfkgZi

�X�

is a �lter base on Ak for every

family (Ai; Zi) of �ltrators where i2 n for some index set n (provided that f is a multifuncoid ofthe form Z and k 2n and X 2

Qi2nnfkg Ai).

Proof. Let K; L 2 fhf ik X j X 2 up X g. Then there exist X; Y 2 up X such that K = hf ik X,L= hf ikY . We can take Z 2upX such that Z vX;Y . Then evidently hf ikZ vK and hf ikZ vLand hf ikZ 2fhf ikX j X 2upX g. �

[FIXME: The following proposition seems erroneous and even a nonsense because of di�erencebetween hf i and hf i�. Should remove it after checking that nothing below depends on it. (Checkagain.)]

Proposition 18.2. h�f ikX =dX2upX hf ikX for a �ltrator

�Qi2nnfkg Fi;

Qi2nnfkg Pi

�(i2n

for some index set n) where every Zi is a boolean lattice, k 2n, and X 2Q

i2nnfkg Fi.

Proof. Fk is separable by obvious 4.136. (Fk;Pk) is with separable core by theorem 4.112.Y �/ h�f iiX ,X [f(i;Y)g2GR [�f ],X [f(i;Y)g2�GR [f ],up(X [f(i;Y)g)�GR [f ],

8X 2 up X ; Y 2 up Y : X [ f(i; Y )g 2 GR [f ],8X 2 up X ; Y 2 up Y : Y �/ hf ii X, 8X 2 up X ;Y 2 up Y: Y u hf ii X =/ 0 , 8Y 2 up Y: 0 2/ fY u hf ii X j X 2 up X g , 8Y 2 up Y:0 2/ hY u ifhf ii X j X 2 up X g , (by properties of generalized �lter bases),8Y 2 up Y:dhY u ifhf ii X j X 2 up X g =/ 0, 8Y 2 up Y: Y u

dfhf ii X j X 2 up X g =/ 0, 8Y 2 up Y:

Y �/dX2upX hf iiX,Y �/

dX2upX hf iiX; so h�f iiX =

dX2upX hf iiX. �

18.2 On pseudofuncoidsDe�nition 18.3. Pseudofuncoid from a set A to a set B is a relation f between �lters on A andB such that:

:(I f 0); I tJ fK,I fK_J fK (for every I ;J 2F(A), K2F(B));:(0 f I); K f I tJ ,K f I _K fJ (for every I ;J 2F(B), K2F(A)):

Obvious 18.4. Pseudofuncoid is just a staroid of the form (F(A);F(B)).

Obvious 18.5. [f ] is a pseudofuncoid for every funcoid f .

Example 18.6. If A and B are in�nite sets, then there exist two di�erent pseudofuncoids f andg from A to B such that f \ (P�P) = g \ (P�P) = [c]\(P�P) for some funcoid c.

Remark 18.7. Considering a pseudofuncoid f as a staroid, we get f \ (P�P)=�f .

Proof. Take

f =�(X ;Y) j X 2F(A);Y 2F(B);

\X and

\Y are in�nite

241

and

g= f [f(X ;Y) j X 2F(A);Y 2F(B);X w a;Y w bg

where a and b are nontrivial ultra�lters on A and B correspondingly, c is the funcoid de�ned bythe relation

[c]�= �= f(X;Y ) j X 2PA; Y 2PB;X and Y are in�niteg:

First prove that f is a pseudofuncoid. The formulas :(I f 0) and :(0 f I) are obvious. We haveI tJ fK,

T(I t J ) and

TY are in�nite,

TI [

TJ and

TY are in�nite,(

TI or

TJ is

in�nite)^TY is in�nite,(

TI and

TY are in�nite)_ (

TJ and

TY are in�nite),I fK_J fK.

Similarly K f I t J ,K fI _K fJ . So f is a pseudofuncoid.Let now prove that g is a pseudofuncoid. The formulas :(I g 0) and :(0 g I) are obvious. Let

I t J gK. Then either I t J fK and then I t J gK or I t J w a and then I w a _ J w a thushaving I gK_ J gK. So I t J gK)I gK_J gK. The reverse implication is obvious. We haveI tJ gK,I gK_J gK and similarly K g I tJ ,K g I _K gJ . So g is a pseudofuncoid.

Obviously f =/ g (a g b but not a f b).It remains to prove f \ (P�P)= g\ (P�P)= [c]\(P�P). Really, f \ (P�P)= [c]\(P�P)

is obvious. If ("AX ;"BY )2 g\ (P�P) then either ("AX;"BY )2 f \ (P�P) or X 2upa, Y 2up b,so X and Y are in�nite and thus ("AX ; "BY )2 f \ (P�P). So g\ (P�P)= f \ (P�P). �

Remark 18.8. The above counter-example shows that pseudofuncoids (and more generally, anystaroids on �lters) are �second class� objects, they are not full-�edged because they don't bijectivelycorrespond to funcoids and the elegant funcoids theory does not apply to them.

From the above it follows that staroids on �lters do not correspond (by restriction) to staroidson principal �lters (or staroids on sets).

18.3 Complete staroids and multifuncoids

18.3.1 Complete free stars

De�nition 18.9. Let A be a poset. Complete free stars on A are such S 2 PA that the leastelement (if it exists) is not in S and for every T 2PA

8Z 2A: (8X 2T :Z wX)Z 2S), T \S=/ ;:

Obvious 18.10. Every complete free star is a free star.

Proposition 18.11. S 2PA where A is a poset is a complete free star i� all the following:


2. 8Z 2A: (8X 2T :Z wX)Z 2S))T \S=/ ;.


Proof.

). (1) and (2) are obvious. S is an upper set because S is a free star.

(. We need to prove that

8Z 2A: (8X 2T :Z wX)Z 2S)( T \S=/ ;:

Let X 02T \S. Then 8X 2T :Z wX)Z wX 0)Z 2S because S is an upper set. �

Proposition 18.12. Let S be a complete lattice. S2PA is a complete free star i� all the following:


242 Identity staroids

2.FT 2S)T \S=/ ; for every T 2PS.


Proof.

). We need to prove onlyFT 2S)T \S =/ ;. Let

FT 2S. Because S is an upper set, we

have 8X 2T :Z wX)Z wFT)Z 2S from which we conclude T \S=/ ;.

(. We need to prove only 8Z 2A: (8X 2T :Z wX)Z 2S)) T \S=/ ;.Really, if 8Z 2A: (8X 2T :ZwX)Z 2S) then

FT 2S and thus

FT 2S)T \S=/ ;. �

Proposition 18.13. Let A be a complete lattice. S 2 PA is a complete free star i� the leastelement (if it exists) is not in S and for every T 2PAG

T 2S,T \S=/ ;:

Proof.

). We need to prove onlyFT 2S(T \S=/ ; what follows from that S is an upper set.

(. We need to prove only that S is an upper set. To prove this we can use the fact that S isa free star. �

18.3.1.1 Completely starrish posets

De�nition 18.14. I will call a poset completely starrish when the full star ? a is a complete freestar for every element a of this poset.

Obvious 18.15. Every completely starrish poset is starrish.

Proposition 18.16. Every complete join in�nite distributive lattice is starrish.

Proof. Let A be a join in�nite distributive lattice, a2A. Obviously 02/ ?a (if 0 exists); obviously? a is an upper set. If

FT 2 ?a, then (

FT )u a is non-least that is

Fhau iT is non-least what is

equivalent to aux being non-least for some x2T that is x2 ? a. �

Theorem 18.17. If A is a completely starrish complete lattice lattice then

atomsG

T =[hatomsiT :

for every T 2PA.

Proof. For every atom c we have: c2atomsFT,c�/

FT,

FT 2?c,9X 2T :X 2?c,9X 2T :

X �/ c,9X 2T : c2 atomsX, c2ShatomsiT . �

18.3.2 More on free stars and complete free stars

Obvious 18.18. @F =� ?F for an element F of down-aligned �nitely meet closed �ltrator.

Corollary 18.19. @F =� ?F for every �lter F on a poset.

Proposition 18.20. ?F =� @F for an element F of a �ltrator with separable core.

Proof. X 2� @F,upX �@F,8X 2 upX :X �/ F,X �/ F,X 2 ?F . �

Corollary 18.21. ?F =� @F for every �lter F on a distributive lattice with least element.

Proposition 18.22. For a semi�ltered, star-separable, down-aligned �ltrator (A;Z) with �nitelymeet closed and separable core where Z is a complete boolean lattice and both Z and A are atomisticlattices the following conditions are equivalent for any F 2A:

1. F 2Z.

18.3 Complete staroids and multifuncoids 243

2. @F is a complete free star on Z.

3. ?F is a complete free star on F.

Proof.

(1))(2). That @F does not contain the least element is obvious. That @F is an upper set isobvious. So it remains to apply theorem 4.53.

(2))(3). That ?F does not contain the least element is obvious. That ?F is an upper set isobvious. So it remains to apply theorem 4.53.

(3))(1). Apply theorem 4.53. �

Corollary 18.23. For a �lter F 2F on a complete atomic boolean lattice the following conditionsare equivalent:

1. F 2P.

2. @F is a complete free star on P.

3. ?F is a complete free star on F.

Theorem 18.24. Let Z be a boolean lattice. For any set S 2PP there exists a principal �lterA such that @A=S i� S is a complete free star (on P).

Proof.

). From the previous theorem.

(. 0P 2/ S andFT 2 S, T \ S =/ ;, 9X 2 T :X 2 S. Take A= fX j X 2P n Sg. We will

prove that A is a principal �lter. That A is a �lter follows from properties of free stars. Itremains to show that A is a principal �lter. It follows from the following equivalence:dP A 2 A ,

FP h:iA 2 A ,FP h:iA 2/ S , :9X 2 h:iA: X 2 S , 8X 2 h:iA:

X 2/ S,8X 2A:X 2A, 1. �

Proposition 18.25.

1. If S is a free star on A then �S is a free star on Z, provided that Z is a join-semilattice andthe �ltrator (A;Z) is down-aligned and with �nitely join-closed core.

2. If S is a free star on P then �S is a free star on F, provided that Z is a boolean lattice.

Proof.

1. X tZ Y 2�S,X tZ Y 2 S,X tA Y 2 S,X 2 S _ Y 2 S,X 2�S _ Y 2�S for everyX;Y 2Z; 02/ �S is obvious.

2. There exists a �lter F such that S= @F . For every �lters X ;Y 2F

X tA Y 2 �S, up(X tA Y) � S, 8K 2 up(X tF Y):K 2 @ F , 8K 2 up(X tF Y):K �/ F,X tF Y �/ F,X tF Y 2 ?F,X 2 ?F _ Y 2 ?F,X �/ F _Y �/ F,8X 2 upX :X�/ F _8Y 2upY :Y �/ F,8X 2upX :X 2@F _8Y 2upY:Y 2@F,upX �S_upY �S,X 2�S _Y 2�S;

02�S,up 0�S, 02S what is false. �

Corollary 18.26. If S is a free star on F then �S is a free star on P, provided that Z is a join-semilattice.

Proposition 18.27.

1. If S is a complete free star on A then �S is a complete free star on Z, provided that Z isa complete lattice and the �ltrator (A;Z) is down-aligned and with join-closed core.

2. If S is a complete free star on P then �S is a complete free star on F, provided that Z isa boolean lattice.


Proof.

1.FZ T 2�S,FZ T 2S,

FA T 2S,T \S=/ ;,T \�S=/ ; for every T 2PZ; 02/�S isobvious.

2. There exists a principal �lter F such that S= @F .FA T 2 �S , upFA T 2 S , 8K 2 up

FA T : K 2 @ F , 8K 2 upFA T :

K�/ F,FA T �/ F,

FA T 2 ?F,9K2T :K2 ?F,9K2T :K�/ F,9K2T 8K 2upK:K �/ F,9K2T 8K 2upK:K 2 @F,9K2T : upK�S,9K2T :K2�S,T \�S=/ ;.

02�S,up 0�S, 02S what is false. �

Corollary 18.28. If S is a complete free star on F then �S is a complete free star on P, providedthat Z is a complete lattice.

18.3.3 Complete staroids and multifuncoids

De�nition 18.29. Consider an indexed family Z of posets. A pre-staroid f of the form Z is com-plete in argument k2arity f when (val f)kL is a complete free star for every L2

Qi2(arity f)nfkg Zi.

De�nition 18.30. Consider an indexed family (Ai; Zi) of �ltrators and pre-multifuncoid f isof the form

QZ. Then f is complete in argument k 2 arity f i� hf ikL 2 Zk for every family

L2Q

i2(arity f)nfkg Zi.

Proposition 18.31. Consider an indexed family (Fi;Zi) of primary �ltrators over boolean lattices.Let f be a pre-multifuncoid of the form F and k 2 arity f . The following are equivalent:

1. Pre-multifuncoid f is complete in argument k.

2. Pre-staroid � [f ] is complete in argument k.

Proof. Let L2Q

Z. We have L2GR [f ],Li�/ hf iiLj(domL)nfig;(val [f ])kL= @ hf ikL by the theorem 17.81.So (val [f ])k L is a complete free star i� hf ik L 2 Zk (proposition 18.22) for every L 2Qi2(arity f)nfkg Zi. �

Example 18.32. Consider funcoid f= idFCD(U). It is obviously complete in each its two arguments.Then [f ] is not complete in each of its two arguments because (X ;Y)2 [f ],X �/ Y what does notgenerate a complete free star if one of the arguments (say X ) is a �xed nonprincipal �lter.

Theorem 18.33. Consider a semi�ltered, star-separable, down-aligned �ltrator (A;Z) with �nitelymeet closed and separable core where Z is a complete boolean lattice and both Z and A are atomisticlattices.

Let f be a multifuncoid of the aforementioned form. Let k; l2 arity f and k=/ l. The followingare equivalent:

1. f is complete in the argument k.

2. hf il (L[f(k;FX)g)=

Fx2X hf il (L[f(k;x)g) for everyX 2PZk, L2

Qi2(arity f)nfk;lg Zi.

3. hf il (L[f(k;FX)g)=

Fx2X hf il (L[f(k;x)g) for everyX 2PAk, L2

Qi2(arity f)nfk;lg Zi.

Proof.

(3))(2). Obvious.

(2))(1). Let Y 2Z.FX �/ hf ik(L [ f(l; Y )g), Y �/ hf il (L [ f(k;

FX)g), Y �/

Fx2X hf il (L [ f(k;

x)g), (proposition 4.144),9x2X :Y �/ hf il (L[f(k;x)g),9x2X :x�/ hf ik(L[ (l;Y )).It is equivalent (proposition 18.22 and the fact that [f ] is an upper set) to hf ik(L[f(l;

Y )g) being a principal �lter and thus (val [f ])lL being a complete free star.

18.3 Complete staroids and multifuncoids 245

(1))(3). Y �/ hf il (L[ f(k;FX)g),

FX �/ hf ik(L [ f(l; Y )g),9x 2X : x�/ hf ik(L [ f(l;

Y )g),9x 2X : Y �/ hf il (L[ f(k; x)g), Y �/Fx2X hf il (L[ f(k; x)g) for every principal

Y . �

18.4 Identity staroids and multifuncoids

18.4.1 Identity relationsDenote idA[n]= f(�i2n:x) j x2Ag= fn�fxg j x2Ag the n-ary identity relation on a set A (foreach index set n).

Proposition 18.34.Q

X �/ idA[n],Ti2n Xi\A=/ ;.

Proof.Q

X �/ idA[n],9t2A:n�ftg2Q

X,9t2A8i2n: t2Xi,Ti2n Xi\A=/ ;. �

18.4.2 General de�nitions of identity staroidsConsider a �ltrator (A;Z) and A2A.

I will de�ne below small identity staroids idA[n]Strd and big identity staroids IDA[n]

Strd . That they arereally staroids and even completary staroids (under certain conditions) is proved below.

De�nition 18.35. Consider a �ltrator (A;Z). Let Z be a complete lattice. Let A2A, let n be anindex set.

form idA[n]Strd =Zn; L2GR idA[n]

Strd ,di2nZ Li2 @A.

Obvious 18.36. X 2GRidA[n]Strd ,8A2upA:

di2nZ

XiuA=/ 0 if our �ltrator is with separable core.

De�nition 18.37. The subsetX of a poset A has a nontrivial lower bound (I denote this predicateas MEET(X)) i� there is nonleast a2A such that 8x2X: avx.

De�nition 18.38. Staroid IDA[n]Strd (for any A2A where A is a poset) is de�ned by the formulas:

form IDA[n]Strd =An; L2GR IDA[n]

Strd ,MEET(fLi j i2ng[fAg).

Obvious 18.39. If A is complete lattice, then L2GR IDA[n]Strd ,

dL�/ A.

Obvious 18.40. If A is complete lattice and a is an atom, then L2GR IDa[n]Strd,

dLw a.

Obvious 18.41. If A is a complete lattice then there exists a multifuncoid �IDA[n]Strd such that

�IDA[n]Strd �

kL=

di2n LiuA for every k 2n, L2Annfkg.

Proposition 18.42. If (A;Z) is a meet-closed �ltrator and Z is a complete lattice and A is a meet-semilattice. There exists a multifuncoid � idA[n]

Strd such that� idA[n]

Strd �kL=

di2nZ LiuAA for every

k 2n, L2Znnfkg.

Proof. We need to prove that L[f(k;X)g2 idA[n]Strd ,

di2nZ

LiuAA�/AX. But

l

i2n

Z

LiuAA�/AX,l

i2n

Z

LiuAX �/AA,l

i2n

Z

(L[f(k;X)g)i�/AA,L[f(k;X)g2 idA[n]Strd : �

18.4.3 Identities are staroids

Proposition 18.43. Let A be a complete distributive lattice and A2A. Then IDA[n]Strd is a staroid.


Proof. That L2/ GR IDA[n]Strd if Lk=0 for some k 2n is obvious. It remains to prove

L[f(k;X tY )g2GR IDA[n]Strd ,L[f(k;X)g2GR IDA[n]

Strd _L[f(k;Y )g2GR IDA[n]Strd :

It is equivalent tol

i2nnfkgLiu (X tY )�/ A,

l

i2nnfkgLiuX �/ A_

l

i2nnfkgLiuY �/ A:

Really,di2nnfkg Li u (X t Y ) �/ A ,

�di2nnfkg Li u X

�t�d

i2nnfkg Li u Y��/ A ,

di2nnfkg LiuX �/ A_

di2nnfkg Liu Y �/ A. �

Proposition 18.44. Let (A; Z) be a starrish �ltrator over a complete meet in�nite distributivelattice and A2A. Then idA[n]

Strd is a staroid.

Proof. That L2/ GR idA[n]Strd if Lk=0 for some k 2n is obvious. It remains to prove

L[f(k;X tY )g2GR idA[n]Strd ,L[f(k;X)g2GR idA[n]

Strd _L[f(k;Y )g2GR idA[n]Strd :

It is equivalent to

l

i2nnfkg

Z

Liu (X tY )�/ A,l

i2nnfkg

Z

LiuX �/ A_l

i2nnfkg

Z

LiuY �/ A:

Really,di2nnfkgZ

Li u (X t Y ) �/ A ,�d

i2nnfkgZ

Li u X�t�d

i2nnfkgZ

Li u Y��/ A ,

di2nnfkgZ LiuX �/ A_

di2nnfkgZ Liu Y �/ A. �

Proposition 18.45. Let (A; Z) be a distributive lattice �ltrator with least element and �nitelyjoin-closed core which is a join semilattice. IDA[n]

Strd is a completary staroid for every A2A.

Proof. @A is a free star by theorem 4.47.L0 t Li 2 GR IDA[n]

Strd , 8i 2 n: (L0 t Li) i 2 @ A , 8i 2 n: L0 i t L1 i 2 @ A , 8i 2 n:

(L0 i2@A_L1 i2@A),9c2f0;1gn8i2n:Lc(i) i2@A,9c2f0;1gn: (�i2n:Lc(i) i)2GRIDA[n]Strd . �

Lemma 18.46. X 2GRidA[n]Strd ,Cor0

di2nA Xi�/ A for a join-closed �ltrator (A;Z) such that both

A and Z are complete lattices, provided that A2A.

Proof. X 2GR idA[n]Strd ,

di2nZ Xi�/ A,Cor0

di2nA Xi�/ A. �

Conjecture 18.47. idA[n]Strd is a completary staroid for every set-theoretic �lter A.

Proposition 18.48. Let each (Ai; Zi) for i 2 n (where n is an index set) is a �nitely join-closed�ltrator, such that each Ai and each Zi are join-semilattices. If f is a completary staroid of theform A then �f is a completary staroid of the form Z. [TODO: Move this proposition (and noteits corollary).]

Proof. L0 tZ L1 2 GR � f , L0 tZ L1 2 GR f , L0 tA L1 2 GR f , 9c 2 f0; 1gn:(�i2n:Lc(i) i)2GR f,9c2f0; 1gn: (�i2n:Lc(i) i)2GR�f for every L0; L12

QZ. �

Conjecture 18.49. �idA[n]Strd is a completary staroid if A is a �lter on a set and n is an index set.

18.4.4 Special case of sets and �lters

Proposition 18.50. "ZnX 2GR ida[n]Strd,8A2a:

QX�/ idA[n] for every �lter a on a powerset and

index set n.

18.4 Identity staroids and multifuncoids 247

Proof. 8A 2 a:Q

X �/ idA[n] , 8A 2 a:Ti2n Xi \ A =/ ; , 8A 2 a:

di2nP "Z Xi �/ "A ,d

i2nP ("ZnXi)�/ a,

di2nP ("ZnX)i�/ a,"ZnX 2GR ida[n]. �

Proposition 18.51. Y 2GR idA[n]Strd ,8A2A: Y 2GR "StrdidA[n] for every �lter A on a powerset

and Y 2Pn.

Proof. Take Y = "ZnX .8A 2 A: Y 2 GR "StrdidA[n] , 8A 2 A: "Zn X 2 GR "StrdidA[n] , 8A 2 A:

QX �/ idA[n] ,

"ZnX 2GR idA[n]Strd ,Y 2GR idA[n]

Strd . �

Proposition 18.52. "ZnX 2GR ida[n]Strd,8A2 a9t2A8i2n: t2Xi.

Proof. "ZnX 2GR ida[n]Strd,9A2 a9t2A:n�ftg2

QX,8A2 a9t2A8i2n: t2Xi. �

18.4.5 Relationships between big and small identity staroids

De�nition 18.53. aStrdn =Q

i2nStrd

a for every element a of a poset and an index set n.

Proposition 18.54. �ida[n]Strdv IDa[n]

Strdv aStrdn for every �lter a (on any distributive lattice) and anindex set n.

Proof.

GR� ida[n]Strd �GRIDa[n]

Strd . L 2GR� ida[n]Strd, up L �GR ida[n]

Strd,8L 2 up L: L 2GR ida[n]Strd,

(proposition 4.112),8L 2 up L8A 2 up a:di2nZ

Li �/ A , 8L 2 up L8A 2 up a:di2nZ

LiuA=/ 0)Si2n Li[ a has �nite intersection property,L2GR IDa[n]

Strd.

GRIDa[n]Strd �GR aStrd

n . L 2 GR IDa[n]Strd , MEET(fLi j i 2 ng [ fag) ) 8i 2 a: Li �/ a ,

L2GR aStrda . �

Proposition 18.55. �ida[a]Strd@ IDa[a]

Strd= aStrda for every nontrivial ultra�lter a on a set.

Proof.

GR� ida[a]Strd =/ GRIDa[a]

Strd . Let Li = "Base(a)i. Then trivially L 2 GR IDa[a]Strd. But to disprove

L 2 GR � ida[a]Strd it's enough to show L 2/ GR ida[a]

Strd for some L 2 up L. Really, take

Li=Li= "Base(a)i. Then L2GR ida[a]Strd,8A2 a9t2A8i 2 a: t2 i what is clearly false (we

can always take i2 a such that t2/ i for any point t).

GRIDa[a]Strd =GR aStrd

a . L2GR IDa[a]Strd,8i2n:Liw a,8i2 a:Li�/ a,L2GR aStrd

a . �

Corollary 18.56. aStrda isn't an atom when a is a nontrivial ultra�lter.

Corollary 18.57. Staroidal product of an in�nite indexed family of ultra�lters may be non-atomic.

Proposition 18.58. ida[n]Strd is determined by the value of �ida[n]

Strd. Moreover ida[n]Strd=��ida[n]

Strd.

Proof. Use general properties of upgrading and downgrading (proposition 17.63). �

Lemma 18.59. L 2 GR IDa[n]Strd i�

Si2n Li [ a has �nite intersection property (for primary

�ltrators).

Proof. L2GRIDa[n]Strd,

di2n Lua=/ 0

F,8X 2di2n Lua:X=/ ; what is equivalent of

Si2n Li[a

having �nite intersection property. �

Proposition 18.60. IDa[n]Strd is determined by the value of �IDa[n]

Strd, moreover IDa[n]Strd = ��IDa[n]

Strd

(for primary �ltrators).


Proof. L 2��IDa[n]Strd, up L � �IDa[n]

Strd, up L � IDa[n]Strd, 8L 2 up L: L 2 IDa[n]

Strd, 8L 2 up L:di2n Liu a=/ 0

F,Si2n Li[ a has �nite intersection property,(lemma),L2GR IDa[n]

Strd. �

Proposition 18.61. ida[n]Strdv�IDa[n]

Strd for every �lter a and an index set n.

Proof. ida[n]Strd=��ida[n]

Strdv�IDa[n]Strd. �

Proposition 18.62. ida[a]Strd@�IDa[a]

Strd for every nontrivial ultra�lter a.

Proof. Suppose ida[a]Strd = �IDa[a]

Strd. Then IDa[a]Strd = ��IDa[a]

Strd = � ida[a]Strd what contradicts to the

above. �

Obvious 18.63. L2GR IDa[n]Strd, au

di2n Li=/ 0

F if a is an element of a complete lattice.

Obvious 18.64. L2GR IDa[n]Strd,8i2n:Liw a,8i2n:Li�/ a if a is an ultra�lter on A.

18.4.6 Identity staroids on principal �ltersFor principal �lter "A (where A is a set) the above de�nitions coincide with n-ary identity relation,as formulated in the following propositions:

Proposition 18.65. "Strd idA[n]= id"A[n]Strd .

Proof. L2GR"Strd idA[n],Q

L�/ idA[n],9t2A8i2n: t2Li,Ti2n Li\A=/ ;,L2GRid"A[n]

Strd .Thus "Strd idA[n]= id"A[n]

Strd . �

Corollary 18.66. id"A[n]Strd is a principal staroid.

Question 18.67. Is IDA[n]Strd principal for every principal �lter A on a set and index set n?

Proposition 18.68. "Strd idA[n]v�ID"A[n]Strd for every set A.

Proof. L2GR"Strd idA[n],L2GRid"A[n]Strd ,"A�/

di2nA Li("A�/

di2nZ Li,L2�GRID"A[n]

Strd . �

Proposition 18.69. "Strd idA[n]@�ID"A[n]Strd for some set A and index set n.

Proof. L 2 GR "Strd idA[n] ,di2nZ

Li �/ "A what is not implied bydi2nA

Li �/ "A that isL2�GR ID"A[n]

Strd . (For a counter example take n=N , Li=(0; 1/i), A=R.) �

Proposition 18.70. �"Strd idA[n]=� id"A[n]Strd .

Proof. �"Strd idA[n]=�id"A[n]Strd is obvious from the above. �

Proposition 18.71. �"Strd idA[n]v ID"A[n]Strd .

Proof. X 2GR�"Strd idA[n],upX �GR"Strd idA[n],8Y 2upX :Y 2GR"Strd idA[n],8Y 2upX :Y 2 id"A[n]

Strd ,8Y 2upX :di2nZ Yiu"A=/ 0)

di2nA X iu"A=/ 0,X 2GR ID"A[n]

Strd . �

Proposition 18.72. �"Strd idA[n]@ ID"A[n]Strd for some set A.

Proof. We need to prove �"Strd idA[n]=/ ID"A[n]Strd that is it's enough to prove (see the above proof)

that 8Y 2upX :di2nZ Yiu"A=/ 0:

di2nA X iu"A=/ 0. A counter-example follows:

18.4 Identity staroids and multifuncoids 249

8Y 2 upX :di2nZ Yiu "A=/ 0 does not hold for n=N , X i= "(¡1/i; 0) for i 2 n, A= (¡1; 0).

To show this, it's enough to provedi2nZ

Yi u "A=/ 0 for Yi= "(¡1/ i; 0) but this is obvious sincedi2nZ Yi=0.On the other hand,

di2nA X iu"A=/ 0 for the same X and A. �

The above theorems are summarized in the following diagram:

"Strd idA[n]= id"A[n]Strd

�"Strd idA[n]=� id"A[n]Strd

��

w�ID"A[n]Strd

w

��

ID"A[n]Strd

Remark 18.73. v on the diagram means inequality which can become strict for some A and n.

18.4.7 Identity staroids represented as meets and joins

Proposition 18.74. ida[n]Strd =

df"Strd idA[n] j A 2 ag for every �lter a on a powerset where the

meet may be taken on every of the following posets: anchored relations, staroids.

Proof. That ida[n]Strdv"Strd idA[n] for every A2 a is obvious.

Let f v "Strd idA[n] for every A 2 a. L 2GR f) L 2GR "Strd idA[n])8A 2 a:di2nA Li�/ A)d

i2nA Li�/ a)L2GR ida[n]

Strd. Thus f v ida[n]Strd. �

Proposition 18.75. IDA[n]Strd =

F �IDa[n]

Strd j a2 atomsA=FfaStrdn j a2 atomsAg where the join

may be taken on every of the following posets: anchored relations, staroids, completary staroids,provided that A is a �lter on a set.

Proof. IDA[n]Strd w IDa[n]

Strd for every a2 atomsA is obvious.

Let f w IDa[n]Strd for every a2 atomsA. Then 8L2GR IDa[n]

Strd:L2GR f that is

8L2 form f : (MEET(fLi j i2ng[fag))L2GR f):

But 9a 2 atoms A: MEET(fLi j i 2 ng [ fag), 9a 2 atoms A:di2nA

Li �/ a(di2nA

Li �/ A,L2 IDA[n]

Strd .

So L2 IDA[n]Strd )L2GR f . Thus f w IDA[n]

Strd .Then use the fact that IDa[n]

Strd= aStrdn . �

Proposition 18.76. idA[n]Strd =

F �ida[n]

Strd j a2 atomsAwhere the meet may be taken on every of

the following posets: anchored relations, staroids, provided that A is a �lter on a set.

Proof. idA[n]Strd w ida[n]

Strd for every a2 atomsA is obvious.

Let f w ida[n]Strd for every a2 atomsA. Then 8L2GR ida[n]

Strd:L2GR f that is

8L2 form f :

l

i2n

Z

Li�/ a)L2GR f

!:

But 9a2 atomsA:di2nZ Li�/ a(

di2nZ Li�/ A,L2 idA[n]

Strd .


So L2 idA[n]Strd )L2GR f . Thus f w idA[n]

Strd . �

18.5 Finite case

Theorem 18.77. Let n be a �nite set.

1. idA[n]Strd =�IDA[n]

Strd if A and Z are meet-semilattices and (A;Z) is a �nitely meet-closed �ltrator.

2. IDA[n]Strd =�idA[n]

Strd if (A;Z) is a primary �ltrator over a distributive lattice.

Proof.

1. L 2GR�IDA[n]Strd , L 2GR IDA[n]

Strd ,MEET(fLi j i 2 ng [ fAg),di2nA Li uA=/ 0, (by

�niteness),di2nZ LiuA=/ 0,L2 idA[n]

Strd for every L2Q

Z.

2. L 2 GR � idA[n]Strd , up L � GR idA[n]

Strd , 8K 2 up L: K 2 GR idA[n]Strd , 8K 2 up L:

di2nZ Ki2 @A,8K 2 upL:

di2nZ Ki�/ A, (by �niteness and theorem 4.44),8K 2 upL:

di2nA

Ki�/ A,A2Th?i�d

i2nA

Ki j K 2upL, (by the formula for �nite meet of �lters,

theorem 4.111),A2Th?idi2nA Li,8K 2

di2nA Li:A2?K,8K 2

di2nA Li:A�/ K, (by

separability of core, theorem 4.112),di2nA Li�/ A,L2 IDA[n]

Strd . �

Proposition 18.78. Let (A;Z) be a �nitely meet closed �ltrator. �IDA[n]Strd and idA[n]

Strd are the samefor �nite n.

Proof. Becausedi2domLZ

Li=di2domLA

Li for �nitary L. [FIXME: Are meets de�ned?] �

18.6 Counter-examples and conjectures

The following example shows that the theorem 18.33 can't be strenghtened:

Example 18.79. For some multifuncoid f on powersets complete in argument k the followingformula is false:hf il (L[f(k;

FX)g)=

Fx2X hf il (L[f(k;x)g) for every X 2PPk, L2

Qi2(arity f)nfk;lg Fi.

Proof. Consider multifuncoid f =�id"U [3]Strd where U is an in�nite set (of the formP3) and L=(Y )

where Y is a nonprincipal �lter on U .hf i0(L[f(k;

FX)g)=Y u

FX ;F

x2X hf i0 (L[f(k;x)g) =Fx2X (Y u x).

It can be Y uFX =

Fx2X (Y u x) only if Y is principal: Really: Y u

FX =

Fx2X (Y u x)

implies Y �/FX)

Fx2X (Y u x) =/ 0)9x 2X : Y �/ x and thus Y is principal. But we claimed

above that it is nonprincipal. �

Example 18.80. There exists a staroid f and an indexed family X of principal �lters (witharity f = dom X and (form f)i = Base(Xi) for every i 2 arity f), such that f v

QStrd X andY uX 2/ GR f for some Y 2GR f .

Remark 18.81. Such examples obviously do not exist if both f is a principal staroid and Xand Y are indexed families of principal �lters (because for powerset algebras staroidal product isequivalent to Cartesian product). This makes the above example inspired.

Proof. (Monroe Eskew) Let a be any (trivial or nontrivial) ultra�lter on an in�nite set U . LetA; B 2 a be such that A\B �A; B. In other words, A, B are arbitrary nonempty sets such that;=/ A\B�A; B and a be an ultra�lter on A\B.

18.6 Counter-examples and conjectures 251

Let f be the staroid whose graph consists of functions p:U! a such that either p(n)�A forall but �nitely many n or p(n)�B for all but �nitely many n. Let's prove f is really a staroid.

It's obvious px=/ ; for every x2U . Let k 2U , L2 aU nfkg. It is enough (taking symmetry intoaccount) to prove that

L[f(k;xt y)g2GR f,L[f(k;x)g2GR f _L[f(k; y)g2GR f: (18.1)

Really, L[ f(k; xt y)g 2GR f i� xt y 2 a and L(n)�A for all but �nitely many n or L(n)�Bfor all but �nitely many n; L[ f(k; x)g 2GR f i� x 2 a and L(n)�A for all but �nitely many nor L(n)�B; and similarly for y.

But x t y 2 a, x 2 a _ y 2 a because a is an ultra�lter. So, the formula (18.1) holds, and wehave proved that f is really a staroid.

Take X be the constant function with value A and Y be the constant function with value B.8p2GR f : p�/ X because pi\Xi2 a; so GR f �GR

QStrdX that is f v

QStrdX .

Finally, Y uX 2/ GR f because X uY =�i2U :A\B. �

Some conjectures similar to the above example:

Conjecture 18.82. There exists a completary staroid f and an indexed family X of principal�lters (with arity f =domX and (form f)i=Base(Xi) for every i2arity f), such that f v

QStrd Xand Y uX 2/ GR f for some Y 2GR f .

Conjecture 18.83. There exists a staroid f and an indexed family x of ultra�lters (with arity f =dom x and (form f)i=Base(xi) for every i2 arity f), such that f v

QStrdx and Y ux2/ GR f for

some Y 2GR f .

Other conjectures:

Conjecture 18.84. If staroid 0=/ f v aStrdn for an ultra�lter a and an index set n, then n�fag2GR f . (Can it be generalized for arbitrary staroidal products?)

Conjecture 18.85. The following posets are atomic:

1. anchored relations on powersets;

2. staroids on powersets;

3. completary staroids on powersets.

Conjecture 18.86. The following posets are atomistic:

1. anchored relations on powersets;

2. staroids on powersets;

3. completary staroids on powersets.

The above conjectures seem di�cult, because we know almost nothing about structure of atomicstaroids.

Conjecture 18.87. A staroid on powersets is principal i� it is complete in every argument.

Conjecture 18.88. If a is an ultra�lter, then ida[n]Strd is an atom of the lattice of:

1. anchored relations of the form (PBase(a))n;

2. staroids of the form (PBase(a))n;

3. completary staroids of the form (PBase(a))n.

Conjecture 18.89. If a is an ultra�lter, then �ida[n]Strd is an atom of the lattice of:

1. anchored relations of the form F(Base(a))n;

2. staroids of the form F(Base(a))n;

3. completary staroids of the form F(Base(a))n.

Informal problem: Formulate and prove associativity of staroidal product.


Chapter 19

Postface

See this Web page for my research plans: http://www.mathematics21.org/agt-plans.htmlI deem that now the most important research topics in Algebraic General Topology are:

� to solve the open problems mentioned in this work;

� de�ne and research compactness of funcoids;

� research categories related with funcoids and reloids;

� research multifuncoids and staroids in more details;

� research generalized limit of compositions of functions.

All my research of funcoids and reloids is presented athttp://www.mathematics21.org/algebraic-general-topology.html

19.1 Formalizing this theory

Despite of all measures taken, it is possible that there are errors in this book. While special cases,such as �lters of powersets or funcoids, are most likely correct, general cases (such as �lters onposets or pointfree funcoids) may possibly contain wrong theorem conditions.

Thus it would be good to formalize the theory presented in this book in a proof assistant19.1

such as Coq.If you want to work on formalizing this theory, please let me know.See also https://coq.inria.fr/bugs/show_bug.cgi?id=2957

19.1. A proof assistant is a computer program which checks mathematical proofs written in a formal languageunderstandable by computer.

253

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mathoverflow.net/questions/44055 (version: 2010-10-29).[31] Yu. M. Smirnov. On proximity spaces. Mat. Sb. (N.S.), 31(73):543�574, 1952.[32] Yu. M. Smirnov. On completeness of proximity spaces. Doklady Acad. Nauk SSSR, 88:761�764, 1953.[33] Yu. M. Smirnov. On completeness of uniform spaces and proximity spaces. Doklady Acad. Nauk SSSR,

91:1281�1284, 1953.[34] Yu. M. Smirnov. On completeness of proximity spaces i. Trudy Moscov. Mat. Obsc., 3:271�306, 1954.[35] Yu. M. Smirnov. On completeness of proximity spaces ii. Trudy Moscov. Mat. Obsc., 4:421�438, 1955.[36] R. M. Solovay. Maps preserving measures. At http://math.berkeley.edu/~solovay/Preprints/

Rudin_Keisler.pdf, 2011.[37] Comfort W. W. and Negrepontis S. The theory of ultra�lters. Springer-Verlag, 1974.[38] Eric W. Weisstein. Point-set topology. At http://mathworld.wolfram.com/Point-SetTopology.html from

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[39] Wikipedia. Boolean algebras canonically de�ned � wikipedia, the free encyclopedia, 2009. [Online; accessed3-January-2010].

[40] Wikipedia. Distributive lattice � wikipedia, the free encyclopedia, 2009. [Online; accessed 13-May-2009].[41] Wikipedia. Limit ordinal � wikipedia, the free encyclopedia, 2009. [Online; accessed 26-February-2010].

256 Bibliography

Index

adjointlower . . . . . . . . . . . . . . . . . . . . . . . ?upper . . . . . . . . . . . . . . . . . . . . . . . ?

associativein�nite . . . . . . . . . . . . . . . . . . . . . . ?

atom . . . . . . . . . . . . . . . . . . . . . . . . . ?atomic . . . . . . . . . . . . . . . . . . . . . . . . ?atomistic . . . . . . . . . . . . . . . . . . . . . . . ?ball

closed . . . . . . . . . . . . . . . . . . . . . . . ?open . . . . . . . . . . . . . . . . . . . . . . . . ?

big staroidgenerated . . . . . . . . . . . . . . . . . . . . . ?

boundslower . . . . . . . . . . . . . . . . . . . . . . . ?upper . . . . . . . . . . . . . . . . . . . . . . . ?

category . . . . . . . . . . . . . . . . . . . . . . . ?abrupt . . . . . . . . . . . . . . . . . . . . . . . ?dagger . . . . . . . . . . . . . . . . . . . . . . . ?of funcoid triples . . . . . . . . . . . . . . . . . ?of funcoids . . . . . . . . . . . . . . . . . . . . ?of pointfree funcoid triples . . . . . . . . . . . . ?of pointfree funcoids . . . . . . . . . . . . . . . ?of reloid triples . . . . . . . . . . . . . . . . . . ?of reloids . . . . . . . . . . . . . . . . . . . . . ?partially ordered . . . . . . . . . . . . . . . . . ?quasi-invertible . . . . . . . . . . . . . . . . . . ?with star morphisms

induced by dagger category . . . . . . . . . ?with star-morphisms . . . . . . . . . . . . . . . ?

quasi-invertible . . . . . . . . . . . . . . . . ?category theory . . . . . . . . . . . . . . . . . . ?, ?chain . . . . . . . . . . . . . . . . . . . . . . . . . ?closed

regarding pointfree funcoid . . . . . . . . . . . ?closure

in metric space . . . . . . . . . . . . . . . . . . ?Kuratowski . . . . . . . . . . . . . . . . . . . . ?

co-completionfuncoid

pointfree . . . . . . . . . . . . . . . . . . . . ?of funcoid . . . . . . . . . . . . . . . . . . . . . ?of reloid . . . . . . . . . . . . . . . . . . . . . . ?

co-metacomplete . . . . . . . . . . . . . . . . . . . ?complement . . . . . . . . . . . . . . . . . . . . . ?complemented

element . . . . . . . . . . . . . . . . . . . . . . ?lattice . . . . . . . . . . . . . . . . . . . . . . . ?

complementive . . . . . . . . . . . . . . . . . . . . ?complete

multifuncoid . . . . . . . . . . . . . . . . . . 17staroid . . . . . . . . . . . . . . . . . . . . . 17

complete latticehomomorphism . . . . . . . . . . . . . . . . . . ?

completely starrish . . . . . . . . . . . . . . . . . 14

completionof funcoid . . . . . . . . . . . . . . . . . . . . . ?of reloid . . . . . . . . . . . . . . . . . . . . . . ?

composablefuncoids . . . . . . . . . . . . . . . . . . . . . . ?reloids . . . . . . . . . . . . . . . . . . . . . . . ?

compositionfuncoids . . . . . . . . . . . . . . . . . . . . . . ?of reloids . . . . . . . . . . . . . . . . . . . . . ?

concatenation . . . . . . . . . . . . . . . . . . . . ?connected

regarding endofuncoid . . . . . . . . . . . . . . ?regarding endoreloid . . . . . . . . . . . . . . . ?regarding pointfree funcoid . . . . . . . . . . . ?

connected component . . . . . . . . . . . . . . . . ?connectedness

regarding binary relation . . . . . . . . . . . . . ?connectivity reloid . . . . . . . . . . . . . . . . . . ?contained . . . . . . . . . . . . . . . . . . . . . . . ?contains . . . . . . . . . . . . . . . . . . . . . . . . ?continuity

coordinate-wise . . . . . . . . . . . . . . . . ?, 11generalized . . . . . . . . . . . . . . . . . . ?, ?in metric space . . . . . . . . . . . . . . . . . . ?of restricted morphism . . . . . . . . . . . . ?, ?pre-topology . . . . . . . . . . . . . . . . . ?, ?proximity . . . . . . . . . . . . . . . . . . . ?, ?uniformity . . . . . . . . . . . . . . . . . . ?, ?

convergesregarding funcoid . . . . . . . . . . . . . . ?, ?, ?

coreof �ltrator . . . . . . . . . . . . . . . . . . . . . ?

core part . . . . . . . . . . . . . . . . . . . . . . . ?dual . . . . . . . . . . . . . . . . . . . . . . . . ?

core star . . . . . . . . . . . . . . . . . . . . . . . ?currying . . . . . . . . . . . . . . . . . . . . . . ?, ?De Morgan's laws . . . . . . . . . . . . . . . . . . ?

in�nite . . . . . . . . . . . . . . . . . . . . . . ?decomposition of composition . . . . . . . . . . ?, ?

of reloids . . . . . . . . . . . . . . . . . . . ?, ?destination . . . . . . . . . . . . . . . . . . . . . . ?di�erence . . . . . . . . . . . . . . . . . . . . . . . ?directly isomorphic . . . . . . . . . . . . . . . . . . ?disjunction property of Wallman . . . . . . . . . . ?distance . . . . . . . . . . . . . . . . . . . . . . . . ?distributive

in�nite . . . . . . . . . . . . . . . . . . . . . . ?domain

of funcoid . . . . . . . . . . . . . . . . . . . . . ?downgrading . . . . . . . . . . . . . . . . . . . . . ?

staroid . . . . . . . . . . . . . . . . . . . . ?, ?dual

order . . . . . . . . . . . . . . . . . . . . . . . ?poset . . . . . . . . . . . . . . . . . . . . . . . ?

257

dualitypartial order . . . . . . . . . . . . . . . . . . . ?

edge . . . . . . . . . . . . . . . . . . . . . . . . . . ?edge part . . . . . . . . . . . . . . . . . . . . . . . ?element

of �ltrator . . . . . . . . . . . . . . . . . . . . . ?embedding

reloids into funcoids . . . . . . . . . . . . . ?, ?endo-funcoid . . . . . . . . . . . . . . . . . . . . . ?endomorphism . . . . . . . . . . . . . . . . . . . . ?

symmetric . . . . . . . . . . . . . . . . . . . . . ?transitive . . . . . . . . . . . . . . . . . . . . . ?

endomorphism series . . . . . . . . . . . . . . . . . ?endo-reloid . . . . . . . . . . . . . . . . . . . . . . ?entirely de�ned

morphism . . . . . . . . . . . . . . . . . . . . . ?equivalent

�lters . . . . . . . . . . . . . . . . . . . . . . . ?�lter

closed . . . . . . . . . . . . . . . . . . . . . . . ?co�nite . . . . . . . . . . . . . . . . . . . . . . ?directly isomorphic . . . . . . . . . . . . . . . . ?Fréchet . . . . . . . . . . . . . . . . . . . . . . ?on a meet-semilattice . . . . . . . . . . . . . . . ?on a poset . . . . . . . . . . . . . . . . . . . . . ?on a set . . . . . . . . . . . . . . . . . . . . . . ?on meet-semilattice . . . . . . . . . . . . . . . . ?on poset . . . . . . . . . . . . . . . . . . . . . . ?on powerset . . . . . . . . . . . . . . . . . . . . ?on set . . . . . . . . . . . . . . . . . . . . . . . ?principal . . . . . . . . . . . . . . . . . . . ?, ?proper . . . . . . . . . . . . . . . . . . . . . . . ?rebase . . . . . . . . . . . . . . . . . . . . . . . ?

�lter base . . . . . . . . . . . . . . . . . . . . . . . ?generalized . . . . . . . . . . . . . . . . . . . . ?

of a �lter . . . . . . . . . . . . . . . . . . . ?generated by . . . . . . . . . . . . . . . . . . . ?

�lter-closed . . . . . . . . . . . . . . . . . . . . . . ?�ltrator . . . . . . . . . . . . . . . . . . . . . . . . ?

central . . . . . . . . . . . . . . . . . . . . . . ?complete lattice . . . . . . . . . . . . . . . . . . ?down-aligned . . . . . . . . . . . . . . . . . . . ?�ltered . . . . . . . . . . . . . . . . . . . . . . ?lattice . . . . . . . . . . . . . . . . . . . . . . . ?of funcoids . . . . . . . . . . . . . . . . . . . . ?powerset . . . . . . . . . . . . . . . . . . . . . ?pre�ltered . . . . . . . . . . . . . . . . . . . . . ?semi�ltered . . . . . . . . . . . . . . . . . . . . ?star-separable . . . . . . . . . . . . . . . . . . . ?up-aligned . . . . . . . . . . . . . . . . . . . . . ?with co-separable core . . . . . . . . . . . . . . ?with �nitely join-closed core . . . . . . . . . . . ?with �nitely meet-closed core . . . . . . . . . . ?with join-closed core . . . . . . . . . . . . . . . ?with meet-closed core . . . . . . . . . . . . . . ?with separable core . . . . . . . . . . . . . . . . ?

�nite intersection property . . . . . . . . . . . . . ?form

of star-morphism . . . . . . . . . . . . . . . . . ?free star

complete . . . . . . . . . . . . . . . . . . . . . ?funcoid . . . . . . . . . . . . . . . . . . . . . ?, ?, ?

co-complete . . . . . . . . . . . . . . . . . . . . ?complete . . . . . . . . . . . . . . . . . . . . . ?destination . . . . . . . . . . . . . . . . . . . . ?identity . . . . . . . . . . . . . . . . . . . . . . ?

induced by reloid . . . . . . . . . . . . . . . . . ?injective . . . . . . . . . . . . . . . . . . . . . . ?monovalued . . . . . . . . . . . . . . . . . . . . ?open . . . . . . . . . . . . . . . . . . . . . . . . ?pointfree . . . . . . . . . . . . . . . . . . . . . ?

co-complete . . . . . . . . . . . . . . . . . . ?co-completion . . . . . . . . . . . . . . . . . ?complete . . . . . . . . . . . . . . . . . . . ?composable . . . . . . . . . . . . . . . . . . ?composition . . . . . . . . . . . . . . . . . . ?destination . . . . . . . . . . . . . . . . . . ?domain . . . . . . . . . . . . . . . . . . . . ?identity . . . . . . . . . . . . . . . . . . . . ?image . . . . . . . . . . . . . . . . . . . . . ?injective . . . . . . . . . . . . . . . . . . . . ?monovalued . . . . . . . . . . . . . . . . . . ?order . . . . . . . . . . . . . . . . . . . . . . ?restricted identity . . . . . . . . . . . . . . . ?restricting . . . . . . . . . . . . . . . . . . . ?source . . . . . . . . . . . . . . . . . . . . . ?zero . . . . . . . . . . . . . . . . . . . . . . ?

principal . . . . . . . . . . . . . . . . . . . . . ?restricted identity . . . . . . . . . . . . . . . . ?reverse . . . . . . . . . . . . . . . . . . . . . . ?separable . . . . . . . . . . . . . . . . . . . . . ?source . . . . . . . . . . . . . . . . . . . . . . . ?

funcoidal reloid . . . . . . . . . . . . . . . . . . . . ?function

variadic . . . . . . . . . . . . . . . . . . . . . . ?function space of posets . . . . . . . . . . . . . . . ?Galois

connection . . . . . . . . . . . . . . . . . . . . ?Galois connection

between funcoids and reloids . . . . . . . . ?, ?generalized closure . . . . . . . . . . . . . . . . . . ?

pointfree . . . . . . . . . . . . . . . . . . . . . ?graph

of anchored relation . . . . . . . . . . . . . ?, ?greatest element . . . . . . . . . . . . . . . . . ?, ?Grothendieck universe . . . . . . . . . . . . . . ?, ?group . . . . . . . . . . . . . . . . . . . . . . . . . ?

permutation . . . . . . . . . . . . . . . . . . . ?transitive . . . . . . . . . . . . . . . . . . . ?

group theory . . . . . . . . . . . . . . . . . . . ?, ?groupoid . . . . . . . . . . . . . . . . . . . . . . . ?ideal . . . . . . . . . . . . . . . . . . . . . . . . . . ?idempotent . . . . . . . . . . . . . . . . . . . . . . ?identity . . . . . . . . . . . . . . . . . . . . . . . . ?identity relation . . . . . . . . . . . . . . . . . ?, 17image

of funcoid . . . . . . . . . . . . . . . . . . . . . ?independent family . . . . . . . . . . . . . . . . . . ?inequality

triangle . . . . . . . . . . . . . . . . . . . . . . ?in�mum . . . . . . . . . . . . . . . . . . . . . . . . ?in�nite distributive . . . . . . . . . . . . . . . . . . ?injective

funcoid . . . . . . . . . . . . . . . . . . . . . . ?morphism . . . . . . . . . . . . . . . . . . . . . ?reloid . . . . . . . . . . . . . . . . . . . . . . . ?

intersecting elements . . . . . . . . . . . . . . . . . ?inverse . . . . . . . . . . . . . . . . . . . . . . . . ?isomorphic

�lters . . . . . . . . . . . . . . . . . . . . . . . ?isomorphism . . . . . . . . . . . . . . . . . . . . . ?join . . . . . . . . . . . . . . . . . . . . . . . . . . ?

258 Index

binary . . . . . . . . . . . . . . . . . . . . . . . ?join in�nite distributive . . . . . . . . . . . . . . . ?join semilattice

homomorphism . . . . . . . . . . . . . . . . . . ?joining elements . . . . . . . . . . . . . . . . . . . ?Kuratowski's lemma . . . . . . . . . . . . . . . ?, ?lattice . . . . . . . . . . . . . . . . . . . . . . . . . ?

boolean . . . . . . . . . . . . . . . . . . . . . . ?center . . . . . . . . . . . . . . . . . . . . . . . ?co-brouwerian . . . . . . . . . . . . . . . . . . . ?co-Heyting . . . . . . . . . . . . . . . . . . . . ?complete . . . . . . . . . . . . . . . . . . . . . ?distributive . . . . . . . . . . . . . . . . . . . . ?homomorphism . . . . . . . . . . . . . . . . . . ?

least element . . . . . . . . . . . . . . . . . . . ?, ?limit . . . . . . . . . . . . . . . . . . . . . . . . . . ?

generalized . . . . . . . . . . . . . . . . . ?, ?, ?linearly ordered set . . . . . . . . . . . . . . . . . . ?lower bound

nontrivial . . . . . . . . . . . . . . . . . . . . 17maximal element . . . . . . . . . . . . . . . . . . . ?maximum . . . . . . . . . . . . . . . . . . . . . . . ?meet . . . . . . . . . . . . . . . . . . . . . . . . . ?

binary . . . . . . . . . . . . . . . . . . . . . . . ?meet in�nite distributive . . . . . . . . . . . . . . ?meet semilattice

homomorphism . . . . . . . . . . . . . . . . . . ?metacomplete . . . . . . . . . . . . . . . . . . . . ?metainjective . . . . . . . . . . . . . . . . . . . . . ?metamonovalued . . . . . . . . . . . . . . . . . . . ?minimal element . . . . . . . . . . . . . . . . . . . ?minimum . . . . . . . . . . . . . . . . . . . . . . . ?monotone . . . . . . . . . . . . . . . . . . . . . . . ?monovalued

funcoid . . . . . . . . . . . . . . . . . . . . . . ?morphism . . . . . . . . . . . . . . . . . . . . . ?reloid . . . . . . . . . . . . . . . . . . . . . . . ?

morphism . . . . . . . . . . . . . . . . . . . . . . . ?bijective . . . . . . . . . . . . . . . . . . . . . . ?entirely de�ned . . . . . . . . . . . . . . . . . . ?identity . . . . . . . . . . . . . . . . . . . . . . ?injective . . . . . . . . . . . . . . . . . . . . . . ?monovalued . . . . . . . . . . . . . . . . . . . . ?surjective . . . . . . . . . . . . . . . . . . . . . ?symmetric . . . . . . . . . . . . . . . . . . . . . ?transitive . . . . . . . . . . . . . . . . . . . . . ?unitary . . . . . . . . . . . . . . . . . . . . . . ?

multifuncoid . . . . . . . . . . . . . . . . . . . . . ?completary . . . . . . . . . . . . . . . . . . . . ?

multigraphdirected . . . . . . . . . . . . . . . . . . . . . . ?

multireloid . . . . . . . . . . . . . . . . . . . . . . ?convex . . . . . . . . . . . . . . . . . . . . . . . ?principal . . . . . . . . . . . . . . . . . . . . . ?

object . . . . . . . . . . . . . . . . . . . . . . . . . ?open map . . . . . . . . . . . . . . . . . . . . . . . ?order

dual . . . . . . . . . . . . . . . . . . . . . . . . ?of pre-multifuncoid sketches . . . . . . . . . . . ?Rudin-Keisler . . . . . . . . . . . . . . . . . . . ?

order embedding . . . . . . . . . . . . . . . . . . . ?order homomorphism . . . . . . . . . . . . . . . . ?order isomorphism . . . . . . . . . . . . . . . . . . ?ordinal . . . . . . . . . . . . . . . . . . . . . . . . ?ordinal number . . . . . . . . . . . . . . . . . . . . ?ordinal variadic . . . . . . . . . . . . . . . . . . . . ?

partial order . . . . . . . . . . . . . . . . . . . . . ?restricted . . . . . . . . . . . . . . . . . . . . . ?strict . . . . . . . . . . . . . . . . . . . . . . . ?

partially ordered setatomic . . . . . . . . . . . . . . . . . . . . . . . ?atomistic . . . . . . . . . . . . . . . . . . . . . ?bounded . . . . . . . . . . . . . . . . . . . . . . ?

partitionstrong . . . . . . . . . . . . . . . . . . . . . . . ?weak . . . . . . . . . . . . . . . . . . . . . . . . ?

path . . . . . . . . . . . . . . . . . . . . . . . . . . ?poset . . . . . . . . . . . . . . . . . . . . . . . . . ?

atomic . . . . . . . . . . . . . . . . . . . . . . . ?atomistic . . . . . . . . . . . . . . . . . . . . . ?bounded . . . . . . . . . . . . . . . . . . . . . . ?dual . . . . . . . . . . . . . . . . . . . . . . . . ?separable . . . . . . . . . . . . . . . . . . . . . ?starrish . . . . . . . . . . . . . . . . . . . . . . ?

precategory . . . . . . . . . . . . . . . . . . . . . . ?dagger . . . . . . . . . . . . . . . . . . . . . . . ?partially ordered . . . . . . . . . . . . . . . . . ?

pre-category . . . . . . . . . . . . . . . . . . . . . ?partially ordered . . . . . . . . . . . . . . . . . ?

with star-morphisms . . . . . . . . . . . . . ?with star morphisms

induced by dagger pre-category . . . . . . . ?with star-morphisms . . . . . . . . . . . . . . . ?

quasi-invertible . . . . . . . . . . . . . . . . ?preclosure . . . . . . . . . . . . . . . . . . . . . . . ?pre-multifuncoid . . . . . . . . . . . . . . . . . . . ?pre-multifuncoid sketch . . . . . . . . . . . . . . . ?

on powersets . . . . . . . . . . . . . . . . . . . ?preorder . . . . . . . . . . . . . . . . . . . . . . . ?pre-staroid . . . . . . . . . . . . . . . . . . . . . . ?

corresponding to pre-multifuncoid . . . . . . . . ?pre-topology

induced by metric . . . . . . . . . . . . . . . . ?prime element . . . . . . . . . . . . . . . . . . . . ?principal

funcoid . . . . . . . . . . . . . . . . . . . . . . ?product

cartesian . . . . . . . . . . . . . . . . . . . . . ?cross-composition . . . . . . . . . . . . . . . ?, ?displaced . . . . . . . . . . . . . . . . . . . . . ?funcoidal . . . . . . . . . . . . . . . . . . ?, ?, ?oblique . . . . . . . . . . . . . . . . . . . . . . ?ordinated . . . . . . . . . . . . . . . . . . . . . ?reindexation . . . . . . . . . . . . . . . . . ?, ?reloidal . . . . . . . . . . . . . . . . . . . . . . ?

starred . . . . . . . . . . . . . . . . . . . . . ?second . . . . . . . . . . . . . . . . . . . . . . . ?simple . . . . . . . . . . . . . . . . . . . . . . . ?staroidal . . . . . . . . . . . . . . . . . . . . . ?subatomic . . . . . . . . . . . . . . . . . . . . . ?

projectionstaroidal . . . . . . . . . . . . . . . . . . . . . ?

proximity . . . . . . . . . . . . . . . . . . . . . . . ?pseudocomplement . . . . . . . . . . . . . . . . . . ?

dual . . . . . . . . . . . . . . . . . . . . . . . . ?pseudodi�erence . . . . . . . . . . . . . . . . . . . ?pseudofuncoid . . . . . . . . . . . . . . . . . . . 13quasicomplement . . . . . . . . . . . . . . . . . . . ?

dual . . . . . . . . . . . . . . . . . . . . . . . . ?quasidi�erence . . . . . . . . . . . . . . . . . . ?, ?

second . . . . . . . . . . . . . . . . . . . . . . . ?quasi-proximity . . . . . . . . . . . . . . . . . . . . ?

Index 259

relationanchored . . . . . . . . . . . . . . . . . . . ?, ?

between posets . . . . . . . . . . . . . . . . ?�nitary . . . . . . . . . . . . . . . . . . . . ?in�nitary . . . . . . . . . . . . . . . . . . . ?on powersets . . . . . . . . . . . . . . . . . ?

reloid . . . . . . . . . . . . . . . . . . . . . . . . . ?co-complete . . . . . . . . . . . . . . . . . . . . ?complete . . . . . . . . . . . . . . . . . . . . . ?convex . . . . . . . . . . . . . . . . . . . . . . . ?destination . . . . . . . . . . . . . . . . . . . . ?domain . . . . . . . . . . . . . . . . . . . . . . ?identity . . . . . . . . . . . . . . . . . . . . . . ?image . . . . . . . . . . . . . . . . . . . . . . . ?induced by funcoid . . . . . . . . . . . . . . . . ?injective . . . . . . . . . . . . . . . . . . . . . . ?inward . . . . . . . . . . . . . . . . . . . . . . . ?monovalued . . . . . . . . . . . . . . . . . . . . ?outward . . . . . . . . . . . . . . . . . . . . . . ?principal . . . . . . . . . . . . . . . . . . . . . ?reverse . . . . . . . . . . . . . . . . . . . . . . ?source . . . . . . . . . . . . . . . . . . . . . . . ?

restricted identity reloid . . . . . . . . . . . . . . . ?restricting

funcoid . . . . . . . . . . . . . . . . . . . . . . ?rectangular . . . . . . . . . . . . . . . . . . . . ?reloid . . . . . . . . . . . . . . . . . . . . . . . ?square . . . . . . . . . . . . . . . . . . . . . . . ?

Rudin-Keisler equivalence . . . . . . . . . . . . . . ?semigroup . . . . . . . . . . . . . . . . . . . . . . . ?semilattice

join-semilattice . . . . . . . . . . . . . . . . . . ?meet-semilattice . . . . . . . . . . . . . . . . . ?

separable . . . . . . . . . . . . . . . . . . . . . . . ?atomically . . . . . . . . . . . . . . . . . . . . . ?

separation subset . . . . . . . . . . . . . . . . . . . ?set

closed . . . . . . . . . . . . . . . . . . . . . . . ?in metric space . . . . . . . . . . . . . . . . ?

open . . . . . . . . . . . . . . . . . . . . . . . . ?in metric space . . . . . . . . . . . . . . . . ?

partially ordered . . . . . . . . . . . . . . . . . ?small set . . . . . . . . . . . . . . . . . . . . . . . ?small staroid

generated . . . . . . . . . . . . . . . . . . . . . ?source . . . . . . . . . . . . . . . . . . . . . . . . . ?space

metric . . . . . . . . . . . . . . . . . . . . . . . ?

preclosure . . . . . . . . . . . . . . . . . . . . . ?induced by topology . . . . . . . . . . . . . ?

pre-topological . . . . . . . . . . . . . . . . . . ?proximity . . . . . . . . . . . . . . . . . . . . . ?topological . . . . . . . . . . . . . . . . . . . . ?

induced by preclosure . . . . . . . . . . . . ?uniform . . . . . . . . . . . . . . . . . . . . . . ?

starcore . . . . . . . . . . . . . . . . . . . . . . . . ?free . . . . . . . . . . . . . . . . . . . . . . . . ?full . . . . . . . . . . . . . . . . . . . . . . . . ?

star composition . . . . . . . . . . . . . . . . . . . ?star-morphism . . . . . . . . . . . . . . . . . . . . ?staroid . . . . . . . . . . . . . . . . . . . . . . . . ?

completary . . . . . . . . . . . . . . . . . . ?, ?identity

big . . . . . . . . . . . . . . . . . . . . . . 17small . . . . . . . . . . . . . . . . . . . . . 17

principal . . . . . . . . . . . . . . . . . . . . . ?starrish . . . . . . . . . . . . . . . . . . . . . . . . ?straight map . . . . . . . . . . . . . . . . . . . . . ?subcategory

wide . . . . . . . . . . . . . . . . . . . . . . . . ?subelement . . . . . . . . . . . . . . . . . . . . . . ?sublattice

closed . . . . . . . . . . . . . . . . . . . . . . . ?substractive . . . . . . . . . . . . . . . . . . . . . . ?sum

structured . . . . . . . . . . . . . . . . . . . . . ?supremum . . . . . . . . . . . . . . . . . . . . . . ?surjective

morphism . . . . . . . . . . . . . . . . . . . . . ?�-thick . . . . . . . . . . . . . . . . . . . . . . . . ?�-thick . . . . . . . . . . . . . . . . . . . . . . . . ?topology . . . . . . . . . . . . . . . . . . . . . . . ?torning . . . . . . . . . . . . . . . . . . . . . . . . ?�-totally bounded . . . . . . . . . . . . . . . . . . ?�-totally bounded . . . . . . . . . . . . . . . . . . ?totally ordered set . . . . . . . . . . . . . . . . . . ?ultra�lter . . . . . . . . . . . . . . . . . . . . . . . ?

trivial . . . . . . . . . . . . . . . . . . . . . . . ?uncurrying . . . . . . . . . . . . . . . . . . . . ?, ?uniformity . . . . . . . . . . . . . . . . . . . . . . ?upgrading . . . . . . . . . . . . . . . . . . . . . . . ?

staroid . . . . . . . . . . . . . . . . . . . . ?, ?upper set . . . . . . . . . . . . . . . . . . . . . . . ?vertex . . . . . . . . . . . . . . . . . . . . . . . . . ?Wallman's disjunction property . . . . . . . . . . . ?

260 Index

Algebraic General Topology · Before going to topology, this book studies properties of co-brouwerian lattices and lters. Keywords:algebraic general topology, quasi-uniform spaces,

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