repositorio.ufrn.br · Pinheiro, Antônia Jocivania. On algebras for interval-valued fuzzy logic / Antonia Jocivania Pinheiro. - 2019. 135f.: il. Tese (Doutorado)-Universidade Federal

FEDERAL UNIVERSITY OF RIO GRANDE DO NORTECENTER OF EXACT AND EARTH SCIENCES

DEPARTMENT OF INFORMATICS AND APPLIED MATHEMATICSPROGRAM OF GRADUATE STUDIES IN SYSTEMS AND COMPUTING

DOCTORATE IN COMPUTER SCIENCE

On Algebras for Interval-Valued Fuzzy Logic

Antônia Jocivania Pinheiro

Natal-RNAugust, 2019

Antônia Jocivania Pinheiro

On Algebras for Interval-Valued Fuzzy Logic

This thesis was submitted to the PostgraduateProgram in Systems and Computing of FederalUniversity of Rio Grande do Norte.Area of Concentration: Fundations for Com-puting.

Advisor: Prof. Dr. Regivan Hugo Nunes Santiago

Natal-RN

August, 2019

Pinheiro, Antônia Jocivania. On algebras for interval-valued fuzzy logic / AntoniaJocivania Pinheiro. - 2019. 135f.: il.

Tese (Doutorado)-Universidade Federal do Rio Grande do Norte,Centro de Ciências Exatas e da Terra, Pós-Graduação em Sistemase Computação, Natal, 2019. Orientador: Dr. Regivan Hugo Nunes Santiago.

1. Interval Mathematics - Tese. 2. Fuzzy Logic - Tese. 3. BCIAlgebras - Tese. 4. SBCI Algebras - Tese. 5. Fuzzy Implications- Tese. I. Santiago, Regivan Hugo Nunes. II. Título.

RN/UF/BCZM CDU 004.032.26

Universidade Federal do Rio Grande do Norte - UFRNSistema de Bibliotecas - SISBI

Catalogação de Publicação na Fonte. UFRN - Biblioteca Central Zila Mamede

Elaborado por Raimundo Muniz de Oliveira - CRB-15/429

ABSTRACT

This work aims to introduce other approaches to the interval-valued fuzzy logic. Thesenew approaches were inspired by Lodwick and Chalco’s works on constraint intervals. Theseconstraint intervals were used in this thesis to extend the fuzzy operators into two modes, namedSingle-Level Constrained Interval Operators and Constrained Interval Operators and studiedtheir properties. A new algebra, called SBCI algebra, which arises from the intervalization ofBCI-algebras, is also introduced. These algebras aims to be the algebraic model for interval-valued fuzzy logics, which take into account the notion of correctness.

A new class of fuzzy implications, called (T,N)-implications has also been studied. Theauthor investigated the behavior of the BCI/SBCI algebras and (T,N)-implications.Keywords: Interval-valued fuzzy logic, Interval Mathematics, Fuzzy Logic, BCI algebras, SBCIalgebras, Fuzzy Implications.

RESUMO

Este trabalho visa introduzir outras abordagens para a lógica fuzzy com valores intervalares.Essas novas abordagens foram inspiradas nos trabalhos de Lodwick e Chalco sobre intervalosrestritos. Esses intervalos restritos foram usados para estender os operadores fuzzy, nos quaiseles foram chamados Operadores Intervalares Restritos de Nı́vel Único (C-operador) e suaspropriedades foram estudadas. Além disso, esses operadores foram estendidos a operadores cor-retos chamados Operadores Intervalares Restritos. Uma nova álgebra, chamada SBCI álgebra,que surge da intervalização de BCI álgebras, também é introduzida. Essas álgebras têm comoobjetivo ser o modelo algébrico para lógicas fuzzy com valores intervalares que levam em contaa noção de correção.

Também foi estudada uma nova classe de implicações fuzzy, chamada (T,N)-implicações.O autor investigou o comportamento das BCI/SBCI álgebras e das (T,N)-implicações.Palavras-chave: Lógica Fuzzy com Valores Intervalares, Matemática Intervalar, Lógica Fuzzy,BCI álgebras, SBCI álgebras, Implicações Fuzzy.

Contents

1 Introduction p. 8

I Preliminaries 12

2 Interval Arithmetics and Logics p. 13

2.1 Arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 13

2.1.1 Standard Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . p. 13

2.1.1.1 Intervalization of Structures . . . . . . . . . . . . . . . . . . p. 14

2.1.2 Constrained Interval Arithmetic and Single-Level Constrained Inter-val Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 15

2.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 17

2.2.1 Fuzzy Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 18

2.2.2 Interval Fuzzy Connectives . . . . . . . . . . . . . . . . . . . . . . . . p. 22

2.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 23

3 BCI Algebras p. 24

3.1 Pseudo-BCI Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 26

3.2 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 28

II Contributions 29

4 (T,N)-Implications p. 30

4.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 30

4.2 Functional equations and (T,N)-implications . . . . . . . . . . . . . . . . . . p. 424.3 Applying (T,N)-implications to generate fuzzy subsethood measures . . . . p. 474.4 Generalization of (T,N)-implications . . . . . . . . . . . . . . . . . . . . . . . p. 52

4.4.1 Characterizations of (N ′, T,N)-Implications . . . . . . . . . . . . . . p. 644.4.2 Aggregating (N ′, T,N)-Implications . . . . . . . . . . . . . . . . . . p. 66

4.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 67

5 Intervalization of BCI algebras and Semi-BCI algebras p. 68

5.1 Intervalization of BCI algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 68

5.2 Semi-BCI algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 74

5.3 Comparing Semi-BCI and Pseudo-BCI Algebras . . . . . . . . . . . . . . . . p. 78

5.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 79

6 Single-Level Constraint Interval Fuzzy Connectives p. 80

6.1 Single-Level Constrained Interval Operators . . . . . . . . . . . . . . . . . . . p. 82

6.2 Composition of C-Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 88

6.2.1 Properties of Some Interval Fuzzy Connectives with Respect to EagerComposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 89

6.2.2 Properties of Some Interval Fuzzy Connectives with Respect to Single-Level Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 95

6.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 100

7 Constrained Interval Fuzzy Connectives p. 101

7.1 Some Properties of the Constrained Interval Fuzzy Connectives with Respectto Order ≤

KM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 111

7.1.1 (T,N)-Implications Generated from Constrained Interval Fuzzy Con-nectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 115

7.2 Some Properties of the Constrained Interval Fuzzy Connectives with Respectto Order ⊴ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 117

7.3 On the Extension of BCI and Semi-BCI Algebras via Constrained IntervalFuzzy Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 120

7.3.1 On Extension of BCI and Semi-BCI Algebras with Respect to EagerComposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 121

7.3.2 On Extension of BCI and Semi-BCI Algebras with Respect to Con-strained Punctual Composition . . . . . . . . . . . . . . . . . . . . . . p. 122

7.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 123

8 Concluding remarks p. 125

References p. 129

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1 Introduction

Interval-valued fuzzy logic was developed in order to deal with the uncertainties concerningnot only on fuzzy rules, in which fuzzy logic is very successful, but also on inputs-outputs. Incases in which the fuzzification of inputs derives from choices made by experts, fuzzy logic isnot always suitable, as the expert may be unsure when defining the membership function. Thiswas one of the motivations for the research on interval-valued fuzzy logic. Another case liesin the situation in which the inputs represent imprecise numerical data. Interval-valued fuzzylogic is a particular case of interval type-2 fuzzy logic (ZADEH, 1975; LIANG; MENDEL, 2000;BUSTINCE et al., 2015; CASTILLO et al., 2016).

Interval-valued fuzzy logic was applied in a wide variety of domains, for example: Melinand Castillo (CERVANTES; CASTILLO; MELIN, 2011) used it in the context of plant control;Figueroa et al. (FIGUEROA et al., 2005) for non-autonomous robots in the context of a robotfootball game; Lynch et al. (LYNCH; HAGRAS; CALLAGHAN, 2005) built an interval control sys-tem for large marine diesel engines; Chourasia et al. (CHOURASIA; TIWARI; GANGOPADHYAY,2014) developed a new method for assessing fetal health status based on interval type-2 fuzzylogic through fetal phonocardiography; Nguyen et al. (NGUYEN et al., 2015) used the waveletfeature in interval type-2 fuzzy logic system (IT2FLS) to reduce the computation burden andtime of IT2FLS; Leow et al. (LEOW et al., 2019) developed a hybrid of Generalized AdaptiveResonance Theory (GART) and interval type-2 fuzzy logic system algorithm; among manyothers.

The field of interval analysis has a long term development. The idea of bounding round-ing errors by computing with intervals was first given by Warmus (WARMUS, 1956), Sunaga(SUNAGA, 1958) and Moore (MOORE, 1959). However, it can be said that interval mathemat-ics and analysis began with the appearance of R. E. Moore’s book, Interval Analysis in 1966(MOORE, 1966). It deals with numerical data in the form of compact intervals in order to encodecomputational errors or inaccuracies. The interval analysis has been applied in several areas(JAULIN et al., 2001; KEARFOTT; KREINOVICH, 2013), like: Electrical power systems (BARBOZA;DIMURO; REISER, 2004), mechanical engineering (MUHANNA; ZHANG; MULLEN, 2007), chem-ical engineering (STADTHERR et al., 2007), artificial intelligence (HU et al., 2008), multi-agentsystems (DIMURO; COSTA, 2004) and geophysics (AGUIAR; DIMURO; COSTA, 2004).

Warmus (WARMUS, 1956), Teruo Sunaga (SUNAGA, 1958) and Ramon Moore (MOORE,1959, 1962) independently developed the interval arithmetic. The Moore’s arithmetic is ac-cepted as the standard approach and is called here standard interval arithmetic (SIA). There aretwo most important criteria for an interval arithmetic, namely: Correctness (accuracy) and op-timality (HICKEY; JU; EMDEN, 2001; MOORE, 1979). The first criterion establishes that the resultof an interval computation must always contain the value of the respective real function (see(MOORE, 1979, Theorem 3.1)). Although correctness is a desirable property, not every interval

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method is correct. Santiago et al. (SANTIAGO; BEDREGAL; ACIÓLY, 2006) investigates the no-tion of correctness for interval functions and its impact on some interval topological viewpoints.They call correctness as interval representations, since interval entities (algorithms and inter-vals) are seen as linguistic entities which represent real entities (functions and numbers). Aninterval function, F , is said to represent a real function, f , whenever it satisfies the followingproperty: x ∈ [a, b] ⇒ f(x) ∈ F ([a, b]) and F ([a, a]) = f(a). The second criterion estab-lishes that the resulting interval of a computation should not be greater than necessary, which iscaptured by the notion of canonical interval representation.

The process of giving the correct and optimal interval version F for a function f is calledintervalization. There are many proposals of intervalization of algebraic structures further thanthat of real numbers proposed by Moore and Sunaga, see for example the case for Łukasiewiczalgebras and MV-algebras in (BEDREGAL; SANTIAGO, 2013) and (CABRER; MUNDICI, 2014),respectively. In most cases, interval algebras fail to satisfy some properties that are satisfied bythe algebras from which they came.

In order to solve the algebraic incompatibility between the real arithmetic and Moore arith-metic, Lodwick (LODWICK, 1999) defined a new interpretation for intervals called constraintintervals. With this new approach, Lodwick proposed an alternative to Moore arithmetic onintervals in order to have X −X = [0,0], X ÷X = [1,1] if 0 ∉ X and the distributive property.This new arithmetic, called constrained interval arithmetic (CIA), is an extension of Moore’sinterval arithmetic, in the sense that they coincide in the case that there is no variable depen-dence and are distinct when there are dependencies. In this case, the CIA arithmetic presents asmaller width interval, thus improving the overestimation of Moore’s arithmetic.

This thesis presents a new approach to interval-valued fuzzy logic, in which interval oper-ators preserve some of the main algebraic properties, the overestimation problem is mitigatedand there is no loss of information. These operators were defined using the constrained intervalintroduced by Lodwick (LODWICK, 1999). In what follows, it is described how this study wasdone and also the contributions that were studied or compared with this new approach. Thereader can find a more detailed discussion in Part II.

The first contribution of this work lies on the investigation of a new class of fuzzy implica-tions, called (T,N)-implications (BEDREGAL, 2007), in which it is obtained from the compo-sition of a fuzzy negation and a t-norm. It is not difficult to find in the literature implicationsthat are obtained through other operators, among which we can mention the (S,N)-, R- andQL-implications that have been widely investigated. Many applications have already been de-veloped using fuzzy implications, such as (MAS et al., 2007; BACZYŃSKI, 2013; BACZYŃSKI et al.,2013), and they can still be applied in areas of study such as approximate reasoning, control the-ory, decision making theory, expert systems, diffuse mathematical morphology (BLOCH, 2009;YAGER, 2004; BACZYŃSKI, 2013; BANDLER; KOHOUT, 1980; BUSTINCE et al., 2013), among oth-ers. In this document, the main properties of (T,N)-implications (PINHEIRO et al., 2017, 2018a)with respect to different fuzzy negations were studied. In addition, an application to fuzzysubsethood measure was presented, in which a new subsethood measure was defined, namelyPB-subsethood measure (PINHEIRO et al., 2018b), and it was verified that it is possible to gen-erate this measure from a family of (T,N)-implications. Finally, it was defined (N ′, T,N)-implications (PINHEIRO et al., 2018), generalizes (T,N)-implications. The author also presentsa characterization of the (N ′, T,N)-implications and verify that it is possible to aggregate afamily of (N ′, T,N)-implications and this aggregation is still an (N ′, T,N)-implication.

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The second contribution lies on the application of intervalization on BCI algebras (ISÉKI,1966; HUANG, 2006). BCI algebras are the algebraic counterpart of a common fragment ofseveral important logics, like fuzzy logic, which is modeled by BL algebras (a kind of BCIalgebra). Thus, the intervalization of BCI algebras are important for the construction of analgebraic model for some interval fuzzy logics. Here, a class of BCI algebras was interval-ized and an investigation of the resulting structures was provided. Like for MV-algebras andŁukasiewicz algebras, the resulting interval structure does not belong to the same category ofits starting algebra, it is a new mathematical structure. This new structure is a generalization ofthe BCI algebras and came to be called semi-BCI algebra (SANTIAGO et al., 2019). The semi-BCI algebras were studied in detail and, in addition, the relationship between the semi-BCI andpseudo-BCI algebras has been investigated. It is verified that the only intersection between thetwo is the class of BCI algebra.

The third contribution introduces a new approach to interval-valued fuzzy logic, in whichthe notion of constraint interval, proposed by Lodwick in (LODWICK, 1999), is applied. In 2014,Chalco-Cano et al. in (CHALCO-CANO; LODWICK; BEDE, 2014) proposed a variant of constraintinterval operators that uses a single parameter (level), instead of using two parameters proposedby Lodwick. Following Lodwick and Chalco, the author extends the fuzzy operators to theso-called Single-Level Constrained Interval Operators (C-operators) and studied their mainproperties. Also, it has been shown that the fuzzy connectives are extended to their respectiveC-operators, however, not all are correct. The composition of C-operators provides two methodsof evaluation for interval compositions, called: Eager evaluation and single-level evaluation.Important properties such as the exchange principle, contraposition law (also, left and rightcontraposition law), among others, were investigated by using both methods.

The main and final contribution of this thesis presents another approach to interval-valuedfuzzy logic, in which the new operators, called Constrained Interval Operators, are very closeto Moore’s correctness, in fact, they satisfy a new correction that will be suggested, in whichit is as efficient as Moore’s correction, being, however, less demanding, which was called hereConstraint Interval Correctness. Two methods of evaluating the compositions of these opera-tors, namely: Eager and constrained punctual compositions, have been defined. In addition tothe comparative study between single-level constrained interval operators and constrained inter-val operators, the main properties of this operator have been verified in relation to two orders,namely: Kulisch-Miranker and Moore order. It has been found that this new approach guar-antees the extension of many algebraic properties and that its fuzzy operators are extended totheir respective constrained interval operators, when considering the order of Kulisch-Miranker.Also, a special class of fuzzy implications was extended, called (T,N)-implications. It hasbeen found that the (T,N)-implications generated from constrained interval operators coincidewith the best representation of the original (T,N)-implication. Finally, both algebras, BCI andsemi-BCI, are extended to their respective constrained interval algebras with respect to punctualcomposition. Here correctness is maintained from the perspective of Constrained Interval.

This thesis is organized as follows: Part I recalls some definitions and concepts usedthroughout the text in order to provide a self-contained document. It is divided into two chap-ters: Chapter 2 presents interval arithmetics and logics; and Chapter 3 introduces the BCI alge-bras. Part II presents the contributions of this work, it is divided into five chapters: Chapter 4provides a new class of fuzzy implications called (T,N)-implication. Chapter 5 proposes a newalgebra called Semi-BCI algebra, which generalizes BCI algebras. Chapter 6 proposes a newway of making interval-valued fuzzy logics, in which the operators were called Single-Level

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Constrained Interval Operators and Chapter 7 proposes another way of making interval-valuedfuzzy logics, in which the operators were called Constrained Interval Operators, which gener-alize the operators of the previous chapter. The last chapter includes some conclusions, futureworks, and the bibliography.

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Part I

Preliminaries

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2 Interval Arithmetics and Logics

2.1 Arithmetics

The limited capacity of machines to store just a finite set of finitely represented objects con-strains the automatic calculation (computation) of structures in which a machine representationof some objects exceeds such capacity. In the case of real numbers, although programs oftenprovide highly accurate results, it can happen that rounding errors built up during each step inthe computation produce results which are not even meaningful. One of the proposals to over-come this problem is due, almost simultaneously, to M. Warmus (WARMUS, 1956), T. Sunaga(SUNAGA, 1958) and R. Moore (MOORE, 1959, 1962), with the development of the so-calledinterval arithmetic, as the following section shows:

2.1.1 Standard Interval Arithmetic

Interval arithmetic is a set of operations on the set of all closed intervals I(R) = {X ∣ X =[x,x];x,x ∈ R and x ≤ x}. The operations are defined in the following way:

1. X + Y = [x + y, x + y],

2. X − Y = [x − y, x − y] ,

3. X ⋅ Y = [min{x ⋅ y, x ⋅ y, x ⋅ y, x ⋅ y},max{x ⋅ y, x ⋅ y, x ⋅ y, x ⋅ y}],

4. X/Y = [x,x] ⋅ ([1/y,1/y]), provided that 0 ∉ [y, y].

Observe that for each operation ∗ ∈ {+,−, ⋅, /}, X ⊛ Y = {x ∗ y ∈ R ∶ x ∈ X and y ∈ Y } isalways an interval.

Moore’s development of interval arithmetic is accepted as the standard approach to intervalarithmetic, which will be called standard interval arithmetic (SIA), and it is the approach tointerval arithmetic in common use. Here are some properties associated with SIA (see (MOORE,1979)), for X,Y and Z in I(R):

(1) X + (Y +Z) = (X + Y ) +Z – the associative law for addition

(2) X ⋅ (Y ⋅Z) = (X ⋅ Y ) ⋅Z – the associative law for multiplication

(3) X + Y = Y +X – the commutative law for addition

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(4) X ⋅ Y = Y ⋅X – the commutative law for multiplication

(5) [0,0] +X =X + [0,0] =X – additive identity

(6) [1,1] ⋅X =X ⋅ [1,1] =X – multiplicative identity

(7) X ⋅ (Y +Z) ⊆X ⋅ Y +X ⋅Z – the subdistributive property

Moore’s interval arithmetic presents a problem of overestimation associated with multipleoccurrences of the same variable in an expression. Also, it is clear that X −X is never 0, unlessX is a real number (a zero width interval) and X ÷X is never 1, unless X is a real number (azero width interval).

In this theory, there are two important criteria called correctness (accuracy) and optimality(HICKEY; JU; EMDEN, 2001; MOORE, 1979), which were formalized by Santiago et al. in (SAN-TIAGO; BEDREGAL; ACIÓLY, 2006), and was called interval representation and best intervalrepresentation, respectively (see, e.g., (BEDREGAL; SANTIAGO, 2013; BEDREGAL; TAKAHASHI,2005, 2006)). Both are defined in the following subsection.

2.1.1.1 Intervalization of Structures

Assuming that the set X ⊛ Y = {x ∗ y ∈ R ∶ x ∈ X and y ∈ Y } always corresponds toan interval, where ∗ ∈ {+,−, ⋅, /}, this reveals two important properties of this arithmetic (a)Correctness and (b) Optimality.

“Correctness. The criterion for correctness of a definition of interval arithmeticis that the “Fundamental Theorem of Interval Arithmetic” holds 1: when an ex-pression is evaluated using intervals, it yields an interval containing all results ofpointwise evaluations based on point values that are elements of the argument in-tervals.

[. . . ]Optimality. By optimality, it is meant that the computed floating-point interval

is not wider than necessary.”Hickey et.al(HICKEY; JU; EMDEN, 2001, p.1040)

The application of interval methods follows the following paradigm: Enclosure in intervalsthe values which are not exact by whatever reason (e.g. the value comes from an imprecisemeasurement) and applying correct and optimal operations on such intervals in order to obtainthe best interval which contains the desired output.

The property of correctness was investigated in 2006 by Santiago et al (SANTIAGO; BEDRE-GAL; ACIÓLY, 2006; BEDREGAL; SANTIAGO, 2013). Instead of correctness, they used the terminterval representation, since an interval computation could be understood not just as a machinerepresentation of real numbers, but also as a mathematical representation of real numbers (thisidea is confirmed by the Representation Theorems of Euclidean continuous functions in (SANTI-AGO; BEDREGAL; ACIÓLY, 2006; BEDREGAL; SANTIAGO, 2013)). Also, the notion of optimality

1Moore (MOORE, 1979, Theorem 3.1, p. 21): If F is an inclusion monotonic interval extension of f , then→

f (X1, ...,Xn) ⊆ F (X1, ...,Xn), where→

f (X1, ...,Xn) = {f(x1, . . . , xn) ∶ xi ∈Xi}.

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was named as best interval representation, or best representation for short. And, in what fol-lows, this notion is shown for binary operations: A binary interval operation ⊛ represents abinary real operation, ∗, whenever:

(x, y) ∈X × Y ⇒ x ∗ y ∈X ⊛ Y.

This can be easily extended to n-ary operations. The author showed that this notion is moregeneral than what is stated by the Fundamental Theorem of Interval Arithmetic, given that thereare representations which are not inclusion monotonic (see (SANTIAGO; BEDREGAL; ACIÓLY,2006, p. 238)). The formal definition follows:

Definition 2.1. An interval operation F ∶ U([0,1])n → U([0,1]) is Moore-correct or intervalrepresentation with respect to a function f ∶ [0,1]n → [0,1] whenever, for all (A1, . . . ,An) ∈U([0,1])n and ai ∈ Ai, f(a1, . . . , an) ∈ F (A1, . . . ,An). In addition, F is best interval rep-resentation with respect to function f , denoted by f̂ , if F (A1, . . . ,An) is the least intervalcontaining the set {f(a1, . . . , an) ∣ ai ∈ Ai} for all Ai ∈ U([0,1]) and i ∈ {1, . . . , n}, i.e.,

f̂(A1, . . . ,An) = [inf{f(a1, . . . , an) ∣ ai ∈ Ai}, sup{f(a1, . . . , an) ∣ ai ∈ Ai}] . (2.1)

The process of giving the correct and optimal interval version F for a function f is calledintervalization. There are many proposals of intervalization of algebraic structures further thanthat of real numbers proposed by Moore, Warmus and Sunaga. In the literature, the reader canfind proposals even for the field of Logic. For example: The Łukasiewicz implicative algebra⟨[0,1],→LK ,1⟩, where x →LK y = min(1,1 − x + y) interprets some many-valued logics andwas intervalized by Bedregal et al in (BEDREGAL; SANTIAGO, 2013). Its MV algebra counterpartwas intervalized by Cabrer et al in (CABRER; MUNDICI, 2014), also, in order to overcome thesame problems already stated for I(R). In both cases, the interval algebras did not satisfy thesame properties that are satisfied by the algebras from which they came.

In order to solve the algebraic incompatibility between the real arithmetic and Moore arith-metic, Lodwick (LODWICK, 1999) defined a new interpretation for intervals called constraintintervals, as seen below below.

2.1.2 Constrained Interval Arithmetic and Single-Level Constrained In-terval Arithmetic

In 1999, Lodwick (LODWICK, 1999) proposed an alternative to Moore arithmetic on inter-vals in order to have X −X = [0,0] and X ÷X = [1,1] if 0 ∉X . For this purpose, he defined anew way of interpreting intervals, called constrained intervals.

Definition 2.2. Given an interval X = [x,x] a constrained interval associated to X is thefunction fX ∶ [0,1]→ [0,1], s.t for 0 ≤ λx ≤ 1,

fX(λx) = (1 − λx)x + λxx= x + λxωx,

where ωx = x − x (the width of X).

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The resulting arithmetic, called constrained interval arithmetic (CIA), is defined as follows:

fX(λx) ∗ fY (λy), (2.2)

for λx, λy ∈ [0,1] and ∗ ∈ {+,−,×,÷}, where the resulting interval, X ⊛ Y , can be extractedfrom (2.2) by computing the minimum and maximum,

[ min0≤λx,λy≤1

{fX(λx) ∗ fY (λy)}, max0≤λx,λy≤1

{fX(λx) ∗ fY (λy)}] .

This arithmetic retains the desirable properties, i.e., X − X = [0,0], X ÷ X = [1,1] and thedistributive property X × (Y + Z) = (X × Y ) + (X × Z). Note that this new arithmetic isan extension of Moore’s interval arithmetic, in the sense that they coincide in the case thatthere is no dependence and are distinct when there are dependencies present. In this case, theCIA arithmetic presents a smaller width interval, thus improving the overestimation of Moore’sarithmetic. See the following example.

Example 2.1. Consider the expression X + Y −X for the intervals X = [1,2] and Y = [−1,1].Using Moore’s interval arithmetic we obtain X + Y −X = ([1,2] + [−1,1]) − [1,2] = [0,3] −[1,2] = [−2,2]. While by CIA arithmetic, fX(λx) = 1+λx and fY (λy) = −1+2λy, so (fX(λx)+fY (λy)) − fX(λx) = fY (λy) and hence X + Y −X = Y = [−1,1].

Instead of using independent lambdas for each variable, Chalco-Cano et al. in (CHALCO-CANO; LODWICK; BEDE, 2014) proposed another arithmetic, called single level constrained in-terval arithmetic (SLCIA). This arithmetic is the constrained interval arithmetic in which onlyone parameter, i.e. λy = λx. That is, if we consider two intervals X and Y , we take a levelλ ∈ [0,1] for both intervals instead for of a λx for X and λy for Y , i.e. we take

fX(λ) = (1 − λ)x + λx and fY (λ) = (1 − λ)y + λy.

After operating at all levels, the minimum and maximum are calculated, i.e.:

X ⊛ Y = [min0≤λ≤1

{fX(λ) ∗ fY (λ)},max0≤λ≤1

{fX(λ) ∗ fY (λ)}].

The SLCIA arithmetic is a restriction of the CIA, in which it maintains the desirable properties,that is, X −X = [0,0], X ÷X = [1,1] and the distributive law. However, in this approach wealso have X − Y = [0,0] when X = Y , unlike CIA.

Chalco-Cano extended the single-level interval arithmetic for expressions with intervaloperands. The evaluation of an expression is performed according to the following rule:

E(A1, . . . ,An) = [min0≤λ≤1

{E(fA1(λ), . . . , fAn(λ))},max0≤λ≤1

{E(fA1(λ), . . . , fAn(λ))}].

Considering this way of evaluating the expressions, they showed the following algebraicproperties: For all interval X,Y,Z and α,β ∈ R,

(1) X ⊕ (−Y ) =X ⊖ Y ;

(2) X ⊖ (−Y ) =X ⊕ Y ;

(3) ⊕ is associative: (X ⊕ Y )⊕Z =X ⊕ (Y ⊕Z);

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(4) ⊕ is commutative: X ⊕ Y = Y ⊕X;

(5) [0,0] is the only neutral element for ⊕: X ⊕ [0,0] =X;

(6) α⊙ (X ⊕ Y ) = (α⊙X)⊕ (α⊙ Y );

(7) (α + β)⊙X = (α⊙X)⊕ (β ⊙X);

(8) X ⊖X = [0,0];

(9) (X ⊕ Y )⊖ Y =X;

(10) (X ⊖ Y )⊖X = (−1)⊙ Y ;

(11) [0,0]⊖ (X ⊖ Y ) = Y ⊖X;

(12) X ⊖ Y = (−1)⊙ (Y ⊖X);

(13) X ⊖ Y = ((−1)⊙ Y )⊖ ((−1)⊙X);

(14) X ⊖ Y = Y ⊖X iff X ⊖ Y is symmetric;

(15) X ⊗ (Y ⊕Z) =X ⊗ Y ⊕X ⊗Z;

(16) (Y ⊕Z)⊗X = Y ⊗X ⊕Z ⊗X;

(17) X ⊗ (Y ⊖Z) =X ⊗ Y ⊖X ⊗Z;

(18) (Y ⊖Z)⊗X = Y ⊗X ⊖Z ⊗X .

Intervals not only provide a way to express a number approximations, they can also be usedas a logical value. The next section shows how intervals play this role.

2.2 Logics

In conventional or classical logic, a statement is either false or true and can not be partiallyfalse and partially true. However, in the real world, it is very common to meet complicatedproblems that are not always bivalent, nor are they always made of absolutely true or falsefacts. To model such problems, multivalent logics have emerged, such as Fuzzy Logic, inwhich it reflects the way people think, trying to shape their sense of words, decision making, orcommon sense.

Fuzzy Logic was introduced by Lofti Zadeh in 1965 in the article titled: Fuzzy Sets (ZADEH,1965). Fuzzy logic is a formalism suitable for modeling the human capacity for approximatereasoning and support for decision making in environments where there is imperfect informa-tion and gradual set belonging, allowing a sufficient variety of physical and mental tasks tobe performed without any measure or computation. It has been applied in several areas, suchas control systems (GUANRONG; TAT, 2001), decision making (CHANG; WANG, 2009), expertsystems (SILER; BUCKLEY, 2005), pattern recognition (CHOI; RHEE, 2009), etc.

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2.2.1 Fuzzy Connectives

Among the most important operators in fuzzy logic, the author highlights t-norms, t-conorms,fuzzy negations and fuzzy implications. These operators are generalizations of the classical con-junctions, disjunctions, negations and implications to fuzzy logic, respectively . The definitionsare given as follows:

Definition 2.3. (SCHWEIZER; SKLAR, 1958, 1960, 1961) A function T ∶ [0,1]2 → [0,1] is said tobe a triangular norm (t-norm, for short) if it satisfies the following conditions, for all x, y, z ∈[0,1]:

(T1) Symmetry: T (x, y) = T (y, x);

(T2) Associativity: T (x,T (y, z)) = T (T (x, y), z);

(T3) Monotonicity: If x1 ≤ x2 and y1 ≤ y2 then T (x1, y1) ≤ T (x2, y2);

(T4) 1-identity: T (x,1) = x. (boundary condition)

In fuzzy logic, the conjunction is often represented by a t-norm. The standard fuzzy con-junction TM ∶ [0,1]2 → [0,1], given by TM(x, y) = min{x, y}, called minimum t-norm, is theonly idempotent t-norm (see (KLIR; YUAN, 1995) - Theorem 3.9). Another example of t-normsis the product, denoted by TP .

Proposition 2.1. (BEDREGAL, 2007) Let T be a t-norm. Then T (0, y) = 0 for each y ∈ [0,1].

Definition 2.4. A t-norm T is called positive if, for all x, y ∈ [0,1], it satisfies the condition:T (x, y) = 0 if and only if x = 0 or y = 0.

Definition 2.5. (SCHWEIZER; SKLAR, 1961) A triangular conorm (t-conorm for short) is a binaryoperation S on the unit interval [0,1], i.e., a function S ∶ [0,1]2 → [0,1], which, for all x, y, z ∈[0,1], satisfying (T1), (T2), (T3) and

(S4) S(x,0) = x. (boundary condition)

The standard fuzzy disjunction SM ∶ [0,1]2 → [0,1] given by SM(x, y) = max{x, y},called maximum t-conorm, is the only idempotent t-conorm (see (KLIR; YUAN, 1995) - Theorem3.14).

From an axiomatical point of view, t-norms and t-conorms differ only with respect to theirboundary conditions.

In the following, the notion of fuzzy negation is recalled.

Definition 2.6. (FODOR; ROUBENS, 1994) A function N ∶ [0,1]→ [0,1] is a fuzzy negation if

(N1) N is antitonic, i.e. N(x) ≤ N(y) whenever y ≤ x;

(N2) N(0) = 1 and N(1) = 0.It is strict whenever

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(N3) N is continuous and

(N4) N(x) < N(y) whenever y < x.It is strong if

(N5) N(N(x)) = x, for each x ∈ [0,1].A fuzzy negation N is crisp if

(N6) N(x) ∈ {0,1}, for all x ∈ [0,1];A fuzzy negation N is frontier if it satisfies the property:

(N7) N(x) ∈ {0,1} if and only if x = 0 or x = 1;A fuzzy negation N is non-vanishing if

(N8) N(x) > 0 whenever x < 1.

Example 2.2. The least fuzzy negation, N⊥, and the greatest fuzzy negation, N⊺, are defined,respectively, by

N⊥(x) = {1, if x = 00, if x > 0

and

N⊺(x) = {0, if x = 11, if x < 1 .

Definition 2.7. Given a t-norm T and a fuzzy negation N , it can be said that the pair (T,N)satisfies the law of contradiction whenever

T (x,N(x)) = 0, x ∈ [0,1]. (LC)

Definition 2.8. Let T be a t-norm, S be a t-conorm and N be a fuzzy negation. Then S issaid to be N -dual to T if for all x, y ∈ [0,1] we have N(S(x, y)) = T (N(x),N(y)). It willbe denoted by ST . Analogously, T is said to be N -dual to S if for all x, y ∈ [0,1] we haveN(T (x, y)) = S(N(x),N(y)). It will be denoted by TS .

Fuzzy implication generalizes the usual material implications. In what follows they arepresented with some of their properties.

Definition 2.9. (FODOR; ROUBENS, 1994) A function I ∶ [0,1]2 → [0,1] is a fuzzy implicationif the following properties are satisfied, for all x, y, z ∈ [0,1]:

(I1) If x ≤ y then I(y, z) ≤ I(x, z);

(I2) If y ≤ z then I(x, y) ≤ I(x, z);

(I3) I(0, y) = 1;

(I4) I(x,1) = 1;

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(I5) I(1,0) = 0.

The set of all fuzzy implications will be denoted by FI .

Definition 2.10. Let I ∈ FI . The function NI ∶ [0,1]→ [0,1] defined by

NI(x) = I(x,0), x ∈ [0,1] (2.3)

is called the natural negation of I or the negation induced by I .

In the following, some of the most important properties of some fuzzy implications arepresented, which will be useful in this work (see (SMETS; MAGREZ, 1987; TRILLAS; VALVERDE,1993; FODOR; ROUBENS, 1994)).

Definition 2.11. A fuzzy implication I is said to satisfy:

(i) the identity property, if, for all x ∈ [0,1]

I(x,x) = 1; (IP)

(ii) the left neutrality property, if, for all y ∈ [0,1]

I(1, y) = y; (NP)

(iii) the exchange principle, if, for all x, y, z ∈ [0,1]

I(x, I(y, z)) = I(y, I(x, z)). (EP)

(iv) the left-ordering property, if, for all x, y ∈ [0,1]

I(x, y) = 1 whenever x ≤ y; (LOP)

(v) the right-ordering property, if, for all x, y ∈ [0,1]

I(x, y) ≠ 1 whenever x > y. (ROP)

(vi) the order property iff I satisfy (LOP) and (ROP), i.e., for all x, y ∈ [0,1]

I(x, y) = 1 iff x ≤ y; (OP)

(vii) the law of left self-distributivity2, if, for all x, y, z ∈ [0,1]

I(x, I(y, z)) = I(I(x, y), I(x, z)). (LSD)

Definition 2.12. Let I ∈ FI and let N be a fuzzy negation. I is said to satisfy the:

(i) contraposition law with respect to N , if

I(x, y) = I(N(y),N(x)), for all x, y ∈ [0,1]; (CP)

(ii) left contraposition law with respect to N , if

I(N(x), y) = I(N(y), x), for all x, y ∈ [0,1]; (L-CP)2This law was studied in (CRUZ; BEDREGAL; SANTIAGO, 2018) under the name Boolean-Like. The name given

here is more appropriate because it is closer to self left-distributive law in (FRINK, 1955).

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(iii) right contraposition law with respect to N , if

I(x,N(y)) = I(y,N(x)), for all x, y ∈ [0,1]. (R-CP)

If I satisfies the (left, right) contraposition with respect to a specific N , the following nota-tion will be used: L−CP (N), R−CP (N) and CP (N), respectively.

Proposition 2.2. (BACZYŃSKI; JAYARAM, 2008) Let I ∶ [0,1]2 → [0,1] be an any function andNI be a strong negation.

(i) If I satisfies CP (NI), then I satisfies (NP ).

(ii) If I satisfies (EP ), then I satisfies (I3), (I4), (I5), (NP ) and (CP ) only with respectto NI .

Proposition 2.3. (BACZYŃSKI; JAYARAM, 2008) If a function I ∶ [0,1]2 → [0,1] satisfies (EP )and NI is a fuzzy negation, then I satisfies R−CP (NI).

Proposition 2.4. (BACZYŃSKI; JAYARAM, 2008) If a function I ∶ [0,1]2 → [0,1] satisfies (R−CP )with respect to continuous fuzzy negation N , then I satisfies (I1) if and only if it satisfies (I2).

Definition 2.13. Let I ∈ FI and T be any t-norm and I be a fuzzy implication for T . The pair(I, T ) satisfies the T-conditionality property for T if, for each x, y ∈ [0,1],

T (x, I(x, y)) ≤ y . (TC)

Definition 2.14. Let I be a fuzzy implication and T be a t-norm. It may be said that I satisfiesthe Law of importation (LI) with respect to a t-norm T if

I(T (x, y), z) = I(x, I(y, z)), (2.4)

for all x, y, z ∈ [0,1].

It is well known that the fuzzy implications are generalizations of the implications of classi-cal logic to fuzzy logic, just as a t-norm and a t-conorm are generalizations of classical conjunc-tion and disjunction, respectively. There are some ways to generate fuzzy implications fromlogical connectives. The main fuzzy implications are generalizations of the following tautolo-gies of classical logic:

p→ q = ¬p ∨ q, p→ q = ¬p ∨ (p ∧ q) and p→ q = (¬p ∧ ¬q) ∨ q,

namely (S,N), QL and D-implications, respectively (see (FODOR, 1991; MAS; MONSERRAT;TORRENS, 2006; BACZYŃSKI; JAYARAM, 2008; BACZYŃSKI, 2004)). In addition, there is animplication that arises from the isomorphism that exists between the classical logic of twovalues and the classical set theory by the following identity

A′ ∪B = (A/B)′ =⋃{C ⊆X ∣ A ∩C ⊆ B},

where A and B are subsets of some universal set X and A′ is the complement of set A. Fuzzyimplications obtained as generalization of identity above form the family of residual implica-tions, usually called in the literature of R-implications.

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Definition 2.15. Let I be a fuzzy implication. It can be said that I is called:

(i) (S,N)-implications, if I(x, y) = S(N(x), y) for a given t-conorm S and a fuzzy negationN . If N is strong, then I is simply called S-implications.

(ii) R-implications, if I(x, y) = sup{z ∈ [0,1] ∣ T (x, z) ≤ y} for a given left-continuoust-norm T .

(iii) QL-implications, if I(x, y) = S(N(x), T (x, y)) for a given t-conorm S, a t-norm T andthe greatest fuzzy negation N⊺.

(i) D-implications, if I(x, y) = S(T (N(x),N(y)), y) for a given t-conorm S, a t-norm Tand a strong negation N .

Fuzzy connectives can be extended to take into account imprecision. The next section showshow some of the above fuzzy connectives can be extended to operate with interval values.

2.2.2 Interval Fuzzy Connectives

Let U([0,1]) be the set of closed intervals on [0,1], i.e U([0,1]) = {[x,x] ∣ 0 ≤ x ≤ x ≤ 1}.Let ⟨U([0,1]),≤⟩ be a bounded poset with [0,0] as bottom and [1,1] as top elements.

Definition 2.16. A function N ∶ U([0,1])→ U([0,1]) is an interval fuzzy negation on ⟨U([0,1]),≤⟩ if it is decreasing and satisfies N([0,0]) = [1,1] and N([1,1]) = [0,0]. If N(N(A)) = A,∀A ∈ U([0,1]), then N is called strong interval fuzzy negation.

Definition 2.17. A t-norm onU([0,1]), called interval triangular norm (it-norm) on ⟨U([0,1]),≤⟩, is a commutative, associative, increasing mapping T ∶ U([0,1])2 → U([0,1]) which satisfiesT(A, [1,1]) = A, for all A ∈ U([0,1]).

Proposition 2.5. Let T ∶ U([0,1])2 → U([0,1]) be an it-norm on the bounded poset ⟨U([0,1]),≤⟩. Then, T(A1, [0,0]) = [0,0] for all A1 ∈ U([0,1]).

Proof. Indeed, for allA1 ∈ U([0,1]), we have that [0,0] ≤ A1 ≤ [1,1]. So, since T is increasing,T([0,0],A1) ≤ T([0,0], [1,1]). From the commutativity and boundary condition of T, weobtain T(A1, [0,0]) ≤ [0,0], therefore, T(A1, [0,0]) = [0,0].

Definition 2.18. A function I ∶ U([0,1])2 → U([0,1]) is an interval fuzzy implication on⟨U([0,1]),≤⟩ if, for all A1,A2,A3 ∈ U([0,1]), I satisfies the following properties:

(I1) If A1 ≤ A2 then I(A2,A3) ≤ I(A1,A3) ;

(I2) If A2 ≤ A3 then I(A1,A2) ≤ I(A1,A3);

(I3) I([0,0],A2) = [1,1];

(I4) I(A1, [1,1]) = [1,1];

(I5) I([1,1], [0,0]) = [0,0].

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In (BEDREGAL; TAKAHASHI, 2006), Bedregal and Takahashi presented a characterizationfor the best interval representation of fuzzy implications. See the following theorem:

Theorem 2.1. (BEDREGAL; TAKAHASHI, 2006, Theorem 6.2) Let I be a fuzzy implication. Then,the best interval representation of I , denoted by Î , is given by:

Î(A1,A2) = [I(a1, a2), I(a1, a2)] ,

for all A1,A2 ∈ U([0,1]).

Following the notion of interval representation (formalized by Santiago et al. in (SANTIAGO;BEDREGAL; ACIÓLY, 2006)), Bedregal et al. (BEDREGAL; SANTIAGO, 2013) showed that theintervalization of Łukasiewicz implication does not preserve (OP), however weakening the rightside of (OP), gave rise to the pair of properties, in which the implication of Łukasiewicz satisfies:

(*) the interval r-weak order property, for all A1,A2 ∈ U([0,1]), if A1 ≤ A2 then

I(A1,A2) = [1,1]; (IR-WOP)

(**) the interval l-order property, for all A1,A2 ∈ U([0,1]), if I(A1,A2) = [1,1] then

A1 ≤ A2. (IL-OP)

Logics are usually interpreted by algebras; e.g. Classical Propositional Logics is interpretedby the usual boolean algebra {0,1}, Łukasiewicz Logics is interpreted by MV-algebras. Mostof the algebraic models for logics are BCI algebras (ISÉKI, 1966) with additional axioms. Thenext chapter introduces those algebras and an extension for that. Chapter 5 provides anothergeneralization called Semi-BCI algebras, which rises from the process of intervalization of BCIalgebras which captures the properties (IR-WOP) and (IL-OP).

2.3 Final Remarks

The definitions of some interval arithmetic in the literature and their properties were pre-sented. Besides, the definitions and main properties of fuzzy connectives and interval fuzzyconnectives were also exposed. Still in this interval context, two of the main concepts of inter-val theory were presented, namely: Correctness and optimality, where the first states that theresult of an interval computation must always contain the value of the respective real functionand the second establishes that the interval result should be as small as possible meeting thecorrectness criterion.

In the next chapter we present the BCI-algebras, which will be later intervalized (see Chap-ter 5) using the concepts of correctness and optimality.

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3 BCI Algebras

Artificial intelligence has the important task of making computers simulate humans to dealwith certainty and uncertainty in information. Certain information processing is based on theclassical logic of two values, however, it is natural and necessary to try to establish some rationallogical system as the logical basis for the uncertain information processing. This type of logicis an extension of two-valued logic. In order to construct natural and efficient inference systemsto deal with uncertainty, several types of non-classical logic systems have been developed, suchas BCI-logic.

BCI algebras were introduced by Iséki (ISÉKI, 1966) in the 60s and since then have beenextensively investigated. The term, BCI-algebra, originates from the combinatories B, C, I incombinatory logic. There are several (equivalent) definitions of a BCI algebra, differing in typeand notation. Some of them contain a ∗ symbol for the binary operation and the symbol 0 (or) for the null element. Here the BCI algebras will be used as algebras ⟨A,→,⊺⟩ and one of theconvincing arguments for this notation is that it makes obvious the connection with logic (theoriginal approach is done in signature ⟨A,∗,0⟩, see e.g. (ISÉKI, 1966) and (IMAI; ISÉKI, 1965)).

Definition 3.1. A BCI algebra is a structure C = ⟨A,→,⊺⟩, where→ is a binary operation on Aand ⊺ is an element of A, verifying, the axioms: for all x, y, z ∈X ,

(C-1) (y → z)→ ((z → x)→ (y → x)) = ⊺,

(C-2) x→ ((x→ y)→ y) = ⊺,

(C-3) x→ x = ⊺,

(C-4) if x→ y = ⊺ and y → x = ⊺ then x = y.

On any such BCI algebra it is possible to define a partial order “⪯”:

(C-5) x ⪯ y iff x→ y = ⊺.

Whenever x→ ⊺ = ⊺, i.e. x ⪯ ⊺, the BCI algebra C = ⟨A,→,⊺⟩ will be called a BCK algebra.

This relation ⪯ is called induced order ofA. It is not mandatory for ⊺ be the greatest elementof (A,⪯) (however, it is maximal), contrary to the case of BCK-algebras.

BCI algebras are the stronghold of several algebras that model important logics, includingthe fuzzy logics that are modeled by BL algebras. Thus, the intervalization of BCI algebras areimportant for the construction of an algebraic model for some interval fuzzy logics.

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Example 3.1.

(1) The Łukasiewicz implicative algebra ([0,1],→LK ,1), where x→LK y =min(1,1−x+y), is a BCI algebra.

(2) Given an abelian group (G, ⋅, e) with e as the unit element, (G,→, e) is a BCI algebra,where x→ y = y ⋅ x−1.

(3) Given a set A, consider the parts of A, denoted by P (A). The structure (P (A),⇒,∅)is a BCI algebra, with⇒ such that X ⇒ Y = Y ∩XC , where XC is the complement ofX .

Let us now recall some useful properties of BCI algebras (for more details see (HUANG,2006)):

(A-1) ⊺ ⪯ x implies x = ⊺;

(A-2) x ⪯ y implies y → z ⪯ x→ z; (First place antitonicity)

(A-3) x ⪯ y implies z → x ⪯ z → y; (Second place isotonicity)

(A-4) x ⪯ y and y ⪯ z implies x ⪯ z; (Transitivity)

(A-5) x→ (y → z) = y → (x→ z); (Exchange – EP)

(A-6) x ⪯ y → z implies y ⪯ x→ z;

(A-7) x→ y ⪯ (z → x)→ (z → y);

(A-8) ⊺→ x = x; (Left Neutrality)

(A-9) ((y → x)→ x)→ x = y → x;

(A-10) x→ y ⪯ (y → x)→ ⊺;

(A-11) (x→ y)→ ⊺ = (x→ ⊺)→ (y → ⊺);

(A-12) (x→ y)→ ⊺ = ((y → x)→ ⊺)→ ⊺;

(A-13) y → ((x→ ⊺)→ ⊺) = ((y → x)→ ⊺)→ ⊺;

(A-14) x→ ⊺ = ((x→ y)→ y)→ ⊺;

(A-15) x ⪯ y implies x→ ⊺ = y → ⊺;

(A-16) y → ⊺ ⪯ x implies x = y → ⊺.

Note that properties (A-7), (A-5) and (C-3) model the combinators B, C and I of BCI Logic(HINDLEY; SELDIN, 1986).

Proposition 3.1. Let ⟨A,→,⊺⟩ be a BCI algebra. ⟨A,→,⊺⟩ is a BCK algebra if and only if foreach x ∈ A there exists y ∈ A such that y ⪯ x and y ⪯ ⊺.

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Proof. (⇒) Straightforward because in BCK algebras ⊺ is the greatest element, i.e. x ⪯ ⊺ foreach x ∈ A.

(⇐) Suppose, by contradiction, that ⟨A,→,⊺⟩ is not a BCK algebra. Then, there existsa ∈ A such that a /⪯ ⊺. By hypothesis there exists b ∈ A such that b ⪯ a and b ⪯ ⊺. So, by (A-8)and the definition of ⪯, (b→ ⊺)→ ((⊺→ a)→ (b→ a)) = ⊺→ (a→ ⊺) = a→ ⊺ ≠ ⊺. Therefore,(C-1) fails.

BCK and BCI algebras have been extensively investigated by many researchers (see (JUN;SHIM, 2005; LIU; XU; MENG, 2007; ZHAN; LIU, 2005; LIU et al., 2000; LIU; ZHANG, 1994)). Thereare three important classes of BCI algebras: commutative BCI-algebras (MENG; XIN, 1992a),implicative BCI-algebras (MENG; XIN, 1992b) and positive implicative BCI-algebras (MENG;XIN, 1993). In addition to important concepts such as ideals/filters (these are dual concepts anddepend on the definition used, see e.g. (MENG, 1993; LIU; ZHANG, 1994; WEI; JUN, 1995)). Fora more detailed view of BCI-algebra, see e.g. (HUANG, 2006).

There are several generalizations of the BCI algebras as shown by, for example, Iorgulescuin (IORGULESCU, 2016a), in which he found thirty-one new generalizations distinct from BCI orBCK algebras and showed the hierarchies existing among these algebras (see also (IORGULESCU,2016b)). In this work the author presents a generalization (see the Pseudo-BCI algebra in thefollowing section) in which it has the signature different from the one previously mentioned,since it brings two binary operators instead of one, and propose another generalization aim-ing to capture the intervalization of point algebras that model logics (see Chapter 5). BecausePseudo-BCI has the same signature as the new generalization of BCI algebra shown in Chapter5, the author decided to show the relationship between them.

3.1 Pseudo-BCI Algebras

In (GEORGESCU; IORGULESCU, 2001a), G. Georgescu and A. Iorgulescu introduced the no-tion of pseudo-BCK algebras as an extension of BCK-algebras. The motivation was the fol-lowing: since bounded commutative BCK algebra corresponds (is categorically equivalent)to MV algebra (MUNDICI, 1986), they wanted to verify which structure corresponds to thepseudo-MV algebra, which pseudo-MV algebra is a non-commutative extension of MV al-gebras (GEORGESCU; IORGULESCU, 1999, 2001b). Years later, W. A. Dudek and Y. B. June(DUDEK; JUN, 2008) proposed a generalization of the BCI algebras, called pseudo-BCI alge-bras, as an extension of BCI-algebras.

Definition 3.2. A pseudo-BCI algebra, or PBCI algebra for short, is a structure ⟨A,≤,→,↝,⊺⟩such that “≤” is a binary relation on the setA, “→” and “↝” are binary operations onA, ⊺ ∈ Aand for all x, y, z ∈ A:

(PB-1) x→ y ≤ (y → z)↝ (x→ z),

(PB-2) x↝ y ≤ (y ↝ z)→ (x↝ z),

(PB-3) x ≤ (x→ y)↝ y,

(PB-4) x ≤ (x↝ y)→ y,

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(PB-5) x ≤ x,

(PB-6) if x ≤ y and y ≤ x, then x = y,

(PB-7) x ≤ y⇔ x→ y = ⊺⇔ x↝ y = ⊺.

Whenever x → ⊺ = ⊺, i.e. x ⪯ ⊺ for all x, the PBCI algebra ⟨A,≤,→,↝,⊺⟩ will be called aPBCK algebra.

Note that every PBCI algebra satisfying x→ y = x↝ y for all x, y ∈X is a BCI algebra.

Example 3.2.

(1) The structure A = ⟨R2,⪯,↠,→, (0,0)⟩, where (x1, y1) ↠ (x2, y2) = (x2 − x1, (y2 −y1)e−x1) and (x1, y1) → (x2, y2) = (x2 − x1, y2 − y1ex2−x1), is a PBCI algebra (proper,i.e. it is not a PBCK-algebra).

(2) The structure A = ⟨(−∞,0] ⪯,↠,→, (0,0)⟩, where

x↠ y = { 0, if x ⪯ y2yπ arctan(ln(

yx)), if y < x

and

x→ y = { 0, if x ⪯ yye−tan(

πx2y

), if y < x

is a PBCK algebra.

Proposition 3.2. LetA = ⟨A,≤,→,↝,⊺⟩ be a PBCI algebra, then the following properties holdsfor all x, y, z ∈ A:

(P-1) ⊺ ⪯ x implies x = ⊺;

(P-2) x ⪯ y implies y → z ⪯ x→ z and y ↝ z ⪯ x↝ z;

(P-3) x ⪯ y and y ⪯ z implies x ⪯ z;

(P-4) x→ (y ↝ z) = y → (x↝ z)

(P-5) x ⪯ y → z iff y ⪯ x↝ z;

(P-6) x→ y ⪯ (z → x)→ (z → y), x↝ y ⪯ (z ↝ x)↝ (z ↝ y);

(P-7) x ⪯ y implies z → x ⪯ z → y and z ↝ x ⪯ z ↝ y;

(P-8) ⊺→ x = ⊺↝ x = x;

(P-9) ((x→ y)↝ y)→ y = x→ y and ((x↝ y)→ y)↝ y = x↝ y;

(P-10) x→ y ⪯ (y → x)↝ ⊺ and x↝ y ⪯ (y ↝ x)→ ⊺;

(P-11) (x→ y)→ ⊺ = (x→ ⊺)↝ (y ↝ ⊺), (x↝ y)↝ ⊺ = (x↝ ⊺)→ (y → ⊺);

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(P-12) x→ ⊺ = x↝ ⊺.

Since its definition, PBCI algebras have been investigated by many researchers, in whichthey have defined new concepts and applications. Among these researchers worth mentioningis Xiaohong Zhang and Grzegorz Dymek, for their many published works on the subject. Formore results see for example (ZHANG, 2010; LEE; PARK, 2009; HALAŠ; KÜHR, 2009; DYMEK,2012, 2013; ZHANG; PARK; WU, 2018; ZHANG; LU; MAO, 2010; ZHANG; MA; SMARANDACHE,2017).

3.2 Final Remarks

The definition of BCI algebras and its main properties have been presented; which algebrais well known in the literature. In addition, it was exposed one of several generalizations of BCIalgebras, namely: Pseudo-BCI algebras.

The previous section closes the part preliminary of the present thesis. In what follows, thereader finds my published and submitted contributions. They can be divided into two categories:(1) Fuzzy Connectives and (2) An extension of BCI algebras, which was proposed to capturethe process of intervalization of those algebras (Semi-BCI algebras).

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Part II

Contributions

30

4 (T,N)-Implications

In this chapter, it is introduced a class of fuzzy implication called (T,N)-implication, ob-tained from the composition of a fuzzy negation and a t-norm. It has been shown under whatconstraints, (T,N)-implications preserve some main properties of fuzzy implications, such asproperty of order, the principle of exchange, the law of contraposition, among others. In ad-dition, we apply this new implication class to generate a new fuzzy subsethood measure. Andfinally the (T,N)-implications were extended to (N ′, T,N)-implications.

4.1 Definition and Basic Properties

As well as the (S,N), QL and D-implications, defined in Chapter 2, are generalizationsof implications of classical logic for fuzzy logic, the (T,N)-implications also satisfy the samefamily of implications, in which it generalizes the tautology:

p→ q = ¬(p ∧ ¬q).

This session is focused on verifying if the classical law of double negation, ¬(¬p) = p, is satis-fied thus the (T,N)-implications coincide with (S,N)-implications. See below the definitionof (T,N)-implication, initially defined by Bedregal in (BEDREGAL, 2007).

Proposition 4.1. (BEDREGAL, 2007) Let T be a t-norm and N be a fuzzy negation. Then thefunction INT ∶ [0,1]2 → [0,1] defined by

INT (x, y) = N(T (x,N(y))) (4.1)

is a fuzzy implication.

Definition 4.1. Let T be a t-norm and N be a fuzzy negation. The function INT defined by equa-tion (4.1) is called the (T,N)-implication.

Proposition 4.2. (BEDREGAL, 2007) Let N be a strong fuzzy negation and T be a t-norm. Then,

T (x, y) = N(INT (x,N(y))).

If I is a (T,N)-implication and N is a strong fuzzy negation, then by (KLEMENT; MESIAR;PAP, 2000)(p.234), we get that I is an S-implication, i.e.

I(x, y) = ST (N(x), y),

31

where, ST is as in Definition 2.8. The reciprocal is also true, since by duality we have:

ST (N(x), y) = N(T (N(N(x)),N(y))) = N(T (x,N(y))).

Remark 4.1. If N is not a strong negation and I(x, y) = S(N(x), y), then we say that I is a(S,N)-implication. It will be denoted by I(S,N).

Proposition 4.3. Let INT be a (T,N)-implication. If N is a strong fuzzy negation, then INTsatisfies (NP ), (EP ), R−CP (N) and CP (N).

Proof. (NP) Since T is a t-norm, then by the symmetry and 1-identity properties, for any y ∈[0,1], we have

INT (1, y)(4.1)= N(T (1,N(y))) T1/T4= N(N(y)) N5= y.

(EP) Since N a strong fuzzy negation and T a t-norm, we have

INT (x, INT (y, z))(4.1)= N(T (x,N(N(T (y,N(z)))))) N5= N(T (x,T (y,N(z))))T1= N(T (x,T (N(z), y))) T2= N(T (T (x,N(z)), y))T1= N(T (y, T (x,N(z)))) N5= N(T (y,N(N(T (x,N(z))))))

(4.1)= INT (y, INT (x, z)).

(R-CP) Because N is strong and from the symmetry of T, we have

INT (x,N(y))(4.1)= N(T (x,N(N(y)))) N5= N(T (x, y)) T1= N(T (y, x))N5= N(T (y,N(N(x)))) (4.1)= INT (y,N(x)).

(CP) Again, because N is strong and from the symmetry of T, we have

INT (N(y),N(x))(4.1)= N(T (N(y),N(N(x)))) N5= N(T (N(y), x))T1= N(T (x,N(y))) (4.1)= INT (x, y).

Proposition 4.4. Given a (T,N)-implication INT , the following properties are satisfied:

(i) NINT = N ;(ii) L−CP (N);

(iii) If N is strict, then R−CP (N−1).

Proof. (i) SinceN is a fuzzy negation, thenN(0) = 1, therefore from the 1-identity propertyof the t-norm T , we have, for all x ∈ [0,1],

NINT (x)(2.3)= INT (x,0)

(4.1)= N(T (x,N(0))) N2= N(T (x,1)) T4= N(x).

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(ii) From the symmetry of T , we have

INT (N(x), y)(4.1)= N(T (N(x),N(y))) T1= N(T (N(y),N(x))) (4.1)= INT (N(y), x),

for all x, y ∈ [0,1].

(iii) Again, from the symmetry property, we have

INT (x,N−1(y))(4.1)= N(T (x,N(N−1(y)))) = N(T (x, y)) T1= N(T (y, x))= N(T (y,N(N−1(x)))) (4.1)= INT (y,N−1(x)),

for all x, y ∈ [0,1].

Under some conditions, there are methods available for obtaining t-conorms and t-norms ofa (T,N)-implication and a fuzzy negation. The following propositions present these methods:

Proposition 4.5. Given a (T,N)-implication INT , define the function SINT ∶ [0,1]2 → [0,1] by

SINT (x, y) = INT (N(x), y)

for all x, y ∈ [0,1]. Then:

(i) SINT (1, x) = SINT (x,1) = 1,∀x ∈ [0,1];

(ii) SINT is increasing in both the arguments, i.e., ∀x, y, z ∈ [0,1] with y ≤ z we haveSINT (x, y) ≤ SINT (x, z) and SINT (y, x) ≤ SINT (z, x);

(iii) SINT is commutative;

(iv) If N is strong, then SINT satisfies (S4), i.e., SINT (x,0) = x;

(v) It N is strong, then SINT satisfies (S2), i.e., SINT (x,SINT (y, z)) = SINT (SINT (x, y), z).

Proof. (i) As N is a fuzzy negation and T is a t-norm, then from Proposition 2.1, we have

SINT (1, x) = INT (N(1), x)

N2= INT (0, x)(4.1)= N(T (0,N(x))) Prop.2.1= N(0) N2= 1

and

SINT (x,1) = INT (N(x),1)

(4.1)= N(T (N(x),N(1)) N2= N(T (N(x),0))T1= N(T (0,N(x))) Prop.2.1= N(0) N2= 1.

(ii) For any x, y ∈ [0,1] we get


(4.1)= N(T (N(x),N(y)))

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andSINT (x, z) = I

NT (N(x), z)

(4.1)= N(T (N(x),N(z))),hence, by the monotonicity of the t-norm T , we get that

y ≤ z N1⇒ N(z) ≤ N(y) T3⇒ T (N(x),N(z)) ≤ T (N(x),N(y)),

applying (N1), we have

N(T (N(x),N(y))) ≤ N(T (N(x),N(z)))

i.e.,SINT (x, y) ≤ SINT (x, z).

The other part follows similar.

(iii) By Proposition 4.4(ii), we have


L−CP= INT (N(y), x) = SINT (y, x).

(iv) Being N a strong fuzzy negation, we have by Proposition 4.3 that

SINT (x,0)(iii)= SINT (0, x) = I

NT (N(0), x)

N2= INT (1, x)NP= x.

(v) By Proposition 4.4(ii), we have

SINT (x,SINT (y, z)) = INT (N(x), INT (N(y), z))

L−CP= INT (N(x), INT (N(z), y)).

However, being N strong, Proposition 4.3(ii) ensures that

INT (N(x), INT (N(z), y)) = INT (N(z), INT (N(x), y)),

thus:

SINT (x,SINT (y, z)) = INT (N(z), INT (N(x), y)) = SINT (z, SINT (x, y))

(iii)= SINT (SINT (x, y), z).

From this proposition, it is possible to conclude that, if N is strong, then SINT is a t-conorm.Thus, INT is an S-implication.

Proposition 4.6. Given a (T,N)-implication IN1T , define the function TN2

IN1T

∶ [0,1]2 → [0,1] by

TN2IN1T

(x, y) = N2(IN1T (x,N2(y))), x, y ∈ [0,1],

where N2 is a fuzzy negation. Then, TN2IN1T

is a t-norm iff N2 is strict and N1 = N−12 .

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Proof. Let us assume, firstly, that TN2IN1T

is a t-norm. To prove that N2 is strict, we must show

that N2 is decreasing and continues.

(i) N2 is decreasing.In fact, suppose there are x0, y0 ∈ [0,1] com y0 < x0, such that N2(x0) = N2(y0). Thus,for (I2),

IN1T (1,N2(x0)) = IN1T (1,N2(y0))

N1⇒ TN2IN1T

(1, x0) = TN2IN1T

(1, y0)T1/T4⇒ x0 = y0.

Contradiction. Therefore, as N2 is a fuzzy negation, we have by (N1) that N2 is decreas-ing.

(ii) N2 is continuous.In fact, first there is a need to check that N2 is injective. Given x, y ∈ [0,1] with x ≠ ywe can suppose that x < y. Then, as N2 is decreasing, we have that N2(y) < N2(x), andtherefore, N2 is injective. N2 is also surjective, since given y ∈ [0,1] any, as TN2

IN1T

is a

t-norm, we have

yT4= TN2

IN1T

(y,1) = N2(IN1T (y,N2(1)))N2= N2(IN1T (y,0))

(4.1)= N2(N1(T (y,1))) T4= N2(N1(y)),

therefore, there is N1(y) ∈ [0,1] such that

N2(N1(y)) = y, (4.2)

so, N2 is surjective. Thus, N2 is bijective. It then follows that N2 is continuous be-cause otherwise there would be y ∈ [0,1] such that y ≠ N2(x), for all x ∈ [0,1], whichcontradicts the bijectivity of N2.

Finally, to ensure that N1 = N−12 , by Equation (4.2), just suffice to show that N1(N2(y)) = y.Suppose N1(N2(y)) ≠ y. Then N1(N2(y)) < y or N1(N2(y)) > y. If N1(N2(y)) < y, then by(i), we have

N2(y) < N2(N1(N2(y)))Eq.(4.2)= N2(y).

Contradiction. Analogously, we come to a contradiction when N1(N2(y)) > y. Let us assumenow that N2 is strict and N1 = N−12 . Then, for all x, y ∈ [0,1], we have

TN2IN1T

(x, y) = N2(IN1T (x,N2(y)))(4.1)= N2(N1(T (x,N1(N2(y)))))

N1=N−12= T (x, y).

Therefore, TN2IN1T

is a t-norm.

Lemma 4.1. If I ∈ FI satisfies (NP ), (EP ) and NI is a strong fuzzy negation, then

TI(x, y) = NI(I(x,NI(y)))

is a t-norm.

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Proof. By Propositions 2.2 and 2.3, we have that I satisfies CP (NI) and R−CP (NI). It willverified now that TI satisfies the conditions of the definition of t-norm:

(T1) From law of right contraposition, we get that

TI(x, y) = NI(I(x,NI(y))) R−CP= NI(I(y,NI(x))) = TI(y, x).

(T2) By virtue of NI is strong and I satisfies (EP ) and (R−CP ), we have

TI(x,TI(y, z)) N5= NI(I(x, I(y,NI(z)))) R−CP= NI(I(x, I(z,NI(y))))(EP )= NI(I(z, I(x,NI(y)))) N5= TI(z, TI(x, y)) T1= TI(TI(x, y), z).

(T3) It must be shown that, given x, y, z ∈ [0,1] with y ≤ z, then TI(x, y) ≤ TI(x, z), i.e.,

NI(I(x,NI(y))) ≤ NI(I(x,NI(z))).

Indeed,

y ≤ z N2⇒ NI(z) ≤ NI(y)I2⇒ I(x,NI(z)) ≤ I(x,NI(y))

N2⇒ NI(I(x,NI(y))) ≤ NI(I(x,NI(z))).

(T4) As I satisfies (NP ), we have for x ∈ [0,1],

TI(x,1) T1= TI(1, x) = NI(I(1,NI(x))) NP= NI(NI(x)) N5= x.

Portanto, TI uma t-norm.

Theorem 4.1. For a function I ∶ [0,1]2 → [0,1], the following statements are equivalent:

(i) I = INT is a (T,N)-implication, with N strong fuzzy negation;

(ii) I satisfies (I1), (EP ) and (NI) is a strong fuzzy negation.

Moreover, the representation of I = INT is unique in this case.

Proof. (i)⇒ (ii) Because of Proposition 4.1 we get that I = INT ∈ FI , so (I1) is satisfies. Now,by Proposition 4.3, I satisfies (EP ) and of Proposition 4.4(i) it can be concluded NI = N , andtherefore NI is strong.

(ii)⇒ (i) AsNI is strong, then by Propositions 2.2, 2.3 and 2.4 it can be concluded I ∈ FI .As I ∈ FI satisfies (NP ), (EP ) and NI is a strong fuzzy negation, we have by the Lemma 4.1that TI is a t-norm. Now, it will be demonstrated that I = INITI . In fact, for all x, y ∈ [0,1],

INITI (x, y)(4.1)= NI(TI(x,NI(y))) = NI(NI(I(x,NI(NI(y))))) N5= I(x, y).

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Finally, to prove unity assume that there are two strong fuzzy negations N1, N2 and twot-norms T1, T2 such that

I(x, y) = IN1T1 (x, y) = IN2T2

(x, y)for all x, y ∈ [0,1]. In particular, for y = 0 we obtain, for all x ∈ [0,1],

NI(x)(2.3)= I(x,0) = IN1T1 (x,0) = I

N2T2

(x,0),

so,

NI(x)(2.3)= INiTi (x,0)

(4.1)= Ni(Ti(x,Ni(0))) N2= Ni(Ti(x,1)) T4= Ni(x), i = 1,2,

thus NI = N1 = N2. Now, since NI is strong, we obtain from Proposition 4.2 that

Ti(x, y) = Ni(INiTi (x,Ni(y))), i = 1,2,

then, for all x, y ∈ [0,1],

T1(x, y) = N1(IN1T1 (x,N1(y)))hip= N1(IN2T2 (x,N1(y)))

N1=N2= N2(IN2T2 (x,N2(y))) = T2(x, y).

Therefore, the representation I = INT is unique.

Proposition 4.7. Let Nα(x) = {1, if x ≤ α0, if x > α , for some α ∈ (0,1), and T be a t-norm. Then

INαT is the implication

INαT (x, y) =⎧⎪⎪⎨⎪⎪⎩

1, if y > α or x ≤ α0, otherwise

.

Proof. Indeed,

INαT (x, y) = Nα(T (x,Nα(y))) = {1, if y > α or x = 0

Nα(x), if y ≤ α e x > 0= { 1, if y > α or x ≤ α

0, otherwise.

Remark 4.2.

1. The t-norm T is irrelevant for INαT .

2. There is no (S,N)-implication such that INαT = I(S,N), since if it exists and N is a fuzzynegation, then N(1) = 0 and S(N(1), y) = y ∉ {0,1}, therefore S(N(x), y) ≠ INαT .

3. There is no t-norm T such that INαT = IT , since by definition, for x = 1:

IT (1, y) = sup{z ∈ [0,1] ∣ T (1, z) ≤ y}= sup{z ∈ [0,1] ∣ z ≤ y}= y ≠ INαT (1, y), ∀y ∈ (0,1).

Therefore, IT ≠ INαT .

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Theorem 4.2. Let T be a t-norm and N be a fuzzy negation. Then N is strong iff INT is anS-implication with N as underlying negation.

Proof. Assume that N is a strong fuzzy negation, then by (KLEMENT; MESIAR; PAP, 2000)(p.234), we get that INT is an S-implication. Conversely, let I

NT be an S-implication. Then

there exist a fuzzy negation N ′ and a t-conorm S such that INT (x, y) = S(N ′(x), y), for allx, y ∈ [0,1]. Therefore,

x = S(N ′(1), x) = INT (1, x) = N(T (1,N(x))) = N(N(x)), ∀x ∈ [0,1],

i.e., N(N(x)) = x for all x ∈ [0,1]. Moreover,

N ′(x) (S4)= S(N ′(x),0) = INT (x,0) = N(T (x,N(0)))(N2)= N(T (x,1)) (T4)= N(x) ∀x ∈ [0,1],

so, N ′ = N .

From Theorem 4.2, it is implied that every result which holds for S-implications, where Nis a strong fuzzy negation, also holds for INT . Therefore, for the purposes of this work, there isno interest in this kind of outcomes (results with a strong N ). The study keeps on focusing onproving results for non-strong fuzzy negations.

Proposition 4.8. Let INT be a (T,N)-implication and let N be a non-strong fuzzy negation:

(i) INT does not satisfy (NP );

(ii) If N is strict, then INT does not satisfy (EP );

(iii) If N is strict, then INT does not satisfy R−CP (N);

(iv) If N is strict, then INT does not satisfy CP (N).

Proof. (i) In fact, since N is not a strong fuzzy negation, then there exists x ∈ [0,1] suchthat N(N(x)) ≠ x, so

INT (1, x) = N(T (1,N(x))) = N(N(x)) ≠ x.

(ii) Since N is not strong, then there exists x ∈ [0,1] such that N(N(x)) ≠ x, so

INT (1, INT (x,0)) = N(T (1,N(N(T (x,N(0))))))N2= N(T (1,N(N(T (x,1)))))

T4= N(T (1,N(N(x)))) T4= N(N(N(x)))

and

INT (x, INT (1,0)) = N(T (x,N(N(T (1,N(0))))))N2= N(T (x,N(N(T (1,1)))))

T4= N(T (x,N(N(1)))) N2= N(T (x,1)) T4= N(x).

Therefore, by N being strict, INT (1, INT (x,0)) ≠ INT (x, INT (1,0)).

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(iii) Again, by hypothesis, there exists x ∈ [0,1] such that N(N(x)) ≠ x, then

INT (x,N(1)) = N(T (x,1))T4= N(x)

and

INT (1,N(x)) = N(T (1,N(N(x))))T4= N(N(N(x))).

So, since N is strict, INT (x,N(1)) ≠ INT (1,N(x)).

(iv) Again, there exists x ∈ [0,1] such that N(N(x)) ≠ x, then

INT (x,0) = N(T (x,1))T4= N(x)

and

INT (N(0),N(x)) = INT (1,N(x)) = N(T (1,N(N(x))))T4= N(N(N(x))).

Therefore, once N is strict, INT (x,0) ≠ INT (N(0),N(x)).

Remark 4.3. As any QL-implication (R-implication, D-implication) (see Definition 2.15) sat-isfies (NP), we have by Proposition 4.8(i) that if a (T,N)-implication is a QL-implication (R-implication, D-implication), then N is strong and therefore it is an S-implication.

Remark 4.4. By (DIMURO et al., 2017), a fuzzy negation N ∶ [0,1] → [0,1] is crisp if and onlyif there exists α ∈ [0,1[ such that N = Nα or there exists α ∈ ]0,1] such that N = Nα, where

Nα(x) =⎧⎪⎪⎨⎪⎪⎩

0, if x > α1, if x ≤ α

and

Nα(x) =⎧⎪⎪⎨⎪⎪⎩

0, if x ≥ α1, if x < α.

Theorem 4.3. Let INT be a (T,N)-implication and let N be a crisp fuzzy negation. Then:

(i) INT satisfies (EP );

(ii) INT satisfies R −CP (N);

(iii) INT satisfies CP (N);

(iv) INT does not satisfy (NP );

(v) INT does not satisfy (ROP );

(vi) INT satisfies (LOP ).

Proof. Consider N = Nα, for some α ∈ [0,1[, then:

39

(i) Given x, y, z ∈ [0,1]: (1) If z ≤ α, then Nα(z) = 1, so

INαT (x, INαT (y, z)) = Nα(T (x,Nα(Nα(T (y,Nα(z)))))) = Nα(T (x,Nα(Nα(y))))

=⎧⎪⎪⎨⎪⎪⎩

Nα(T (x,Nα(0))), if y > αNα(T (x,Nα(1))), if y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

Nα(x), if y > α1, if y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and y > α1, otherwise

and

INαT (y, INαT (x, z)) = Nα(T (y,Nα(Nα(T (x,Nα(z)))))) = Nα(T (y,Nα(Nα(x))))

=⎧⎪⎪⎨⎪⎪⎩

Nα(y), if x > α1, if x ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and y > α1, otherwise.

Therefore, for z ≤ α, INαT (x, INαT (y, z)) = INαT (y, INαT (x, z)). (2) If z > α, thenNα(z) = 0,so:

INαT (x, INαT (y, z)) = Nα(T (x,Nα(Nα(T (y,Nα(z)))))) = Nα(T (x,Nα(Nα(0))))= Nα(T (x,0)) = Nα(0) = 1

and

INαT (y, INαT (x, z)) = Nα(T (y,Nα(Nα(T (x,Nα(z)))))) = Nα(T (y,Nα(Nα(0))))= Nα(T (y,0)) = Nα(0) = 1.

In any case, INαT (x, INαT (y, z)) = INαT (y, INαT (x, z)).

(ii) Given x, y ∈ [0,1]:

INαT (x,Nα(y)) = Nα(T (x,Nα(Nα(y)))) =⎧⎪⎪⎨⎪⎪⎩

Nα(x), if y > αNα(0), if y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and y > α1, otherwise

and

INαT (y,Nα(x)) = Nα(T (y,Nα(Nα(x)))) =⎧⎪⎪⎨⎪⎪⎩

Nα(y), if x > αNα(0), if x ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if y > α and x > α1, otherwise.

Therefore, INαT (x,Nα(y)) = INαT (y,Nα(x)).

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(iii) Given x, y ∈ [0,1]:

INαT (x, y) = Nα(T (x,Nα(y))) =⎧⎪⎪⎨⎪⎪⎩

Nα(x), if y ≤ αNα(0), if y > α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and y ≤ α1, otherwise

and

INαT (Nα(y),Nα(x)) = Nα(T (Nα(y),Nα(Nα(x)))) =⎧⎪⎪⎨⎪⎪⎩

Nα(Nα(y)), if x > αNα(0), if x ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and y ≤ α1, otherwise.

Therefore, INαT (x,Nα(y)) = INαT (y,Nα(x)).

(iv) Indeed,

INαT (1, y) = Nα(T (1,Nα(y))) =⎧⎪⎪⎨⎪⎪⎩

Nα(0), if y > αNα(1), if y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

1, if y > α0, if y ≤ α.

Therefore, INαT (1, y) ≠ y for all y ∈ (0,1).

(v) Indeed, if y < x ≤ α, we have that INαT (x, y) = Nα(T (x,Nα(y))) = Nα(T (x,1)) =Nα(x) = 1.

(vi) If x ≤ y, we have the following cases: (1) If x ≤ y ≤ α, then Nα(x) = Nα(y) = 1 andNα(T (x,Nα(y))) = Nα(x) = 1; (2) If x ≤ α < y, then Nα(x) = 1 and Nα(y) = 0,so Nα(T (x,Nα(y))) = Nα(0) = 1; (3) If α < x ≤ y, then Nα(x) = Nα(y) = 0, soNα(T (x,Nα(y))) = Nα(0) = 1. In any case, INαT (x, y) = 1.

The proof follows analogously for N = Nα, for some α ∈ ]0,1].

Remark 4.5. By item (v) of Theorem 4.3, it can be concluded INT does not satisfy (OP) when Nis crisp.

Proposition 4.9. Let T be a positive t-norm and let N be a frontier fuzzy negation. ThenINT (x, y) = 1, if and only if x = 0 or y = 1.

Proof. Assume INT (x, y) = 1, then N(T (x,N(y))) = 1 and, therefore, T (x,N(y)) = 0, sinceN is a frontier fuzzy negation. Now, take a positive T , so we have x = 0 or N(y) = 0, andtherefore x = 0 or y = 1. Conversely, if x = 0 or y = 1 then by (vi) of Theorem 4.3, it followsstraightforward.

Proposition 4.10. LetN be a fuzzy negation and let T be the minimum t-norm. IfN(N(x)) ≤ x,then INT (x, INT (x, y)) = INT (x, y).

41

Proof. By definition, INT (x, INT (x, y)) = N(T (x,N(N(T (x,N(y)))))). Given a t-norm T ,T (x,N(y)) ≤ x, and so

N(N(T (x,N(y)))) ≤ N(N(x))hip.≤ x.

As T is the minimum t-norm, T (x,N(N(T (x,N(y))))) = N(N(T (x,N(y)))), so we haveINT (x, INT (x, y)) = N(N(N(T (x,N(y))))). But, since N is a fuzzy negation, we have thatN(N(x)) ≤ x implies N(x) ≤ N(N(N(x))). On the other hand, taking y = N(x) andsubstituting it in N(N(y)) ≤ y, we obtain N(N(N(x))) ≤ N(x). And so, N(N(N(x))) =N(x). Therefore,

INT (x, INT (x, y)) = N(N(N(T (x,N(y))))) = N(T (x,N(y))) = INT (x, y).

Proposition 4.11. Let I = INT be a (T,N)-implication. If N is a crisp fuzzy negation, thenINT (x, INT (x, y)) = INT (x, y).

Proof. Suppose N = Nα, then:

INαT (x, INαT (x, y)) = Nα(T (x,Nα(Nα(T (x,Nα(y))))))

=⎧⎪⎪⎨⎪⎪⎩

Nα(T (x,Nα(Nα(x)))), if y ≤ αNα(T (x,0)) = 1, if y > α

=⎧⎪⎪⎨⎪⎪⎩

1, if y > α or (x ≤ α and y ≤ α)0, if x > α

and

INαT (x, y) = Nα(T (x,Nα(y))) =⎧⎪⎪⎨⎪⎪⎩

Nα(x), if y ≤ α1, if y > α

=⎧⎪⎪⎨⎪⎪⎩

1, if y > α or (x ≤ α and y ≤ α)0, if x > α.

Therefore, INαT (x, INαT (x, y)) = INαT (x, y). The result follows analogously for N = Nα.

Proposition 4.12. Let I = INT be a (T,N)-implication. If N is a strict fuzzy negation andINT (x, INT (x, y)) = INT (x, y), then N is strong.

Proof. As INT (x, INT (x, y)) = INT (x, y), in particular we have INT (1, INT (1, y)) = INT (1, y), i.e.N(T (1,N(N(T (1,N(y)))))) = N(T (1,N(y))). Since T is a t-norm, N(N(N(N(y)))) =N(N(y)), and therefore, as N is strict, N(N(y)) = y,∀y ∈ [0,1].

Proposition 4.13. Let INT be a (T,N)-implication and let N be a strong fuzzy negation. ThenINT (x, INT (x, y)) = INT (x, y) if and only if T is idempotent.

Proof. Suppose INT (x, INT (x, y)) = INT (x, y), for y = 0 we have:

INT (x,0) = N(T (x,N(0))) = N(T (x,1)) = N(x)

42

and

INT (x, INT (x,0)) = INT (x,N(x)) = N(T (x,N(N(x))))Nstrong= N(T (x,x)),

so N(x) = N(T (x,x)). Since N is strong, T (x,x) = x. Conversely, as T is idempotent, thenT is minimum, and therefore, by Proposition 4.10, the result follows.

4.2 Functional equations and (T,N)-implications

As already mentioned, functional equations are the ones in which the unknowns are func-tions instead of being a traditional variable. This section investigates the validity of some func-tional equations by the (T,N)-implication. In (BACZYŃSKI; JAYARAM, 2008), Baczyński statesthat functional equations come up as generalizations of the corresponding tautologies in classi-cal logic involving boolean implications. The results presented in the sequel consider the law ofimportation (LI), Equation 2.4, and four basic distributive equations involving an implication,which will be discussed later.

The exchange principle (EP) is one of the crucial properties of fuzzy implications. Dueto the commutativity property of t-norm T , one of the conditions for an implication to satisfyit is that (LI) is also satisfied. The well-known fuzzy implications called (S, N), R, QL andD-implications satisfy (LI) under some conditions (see (JAYARAM, 2008; MAS; MONSERRAT;TORRENS, 2009)). In addition, some possible applications were pointed out in (JAYARAM, 2008).As follows, the conditions under which (T,N)-implications satisfy (LI).

Proposition 4.14. Let INT be a (T,N)-implication. Then:

(i) If N is strong then INT satisfies (LI) with respect to the t-norm T ;

(ii) If N is continuous and INT satisfies (LI) with respect to the t-norm T , then N is strong.

Proof. (i) Indeed, for all x, y, z ∈ [0,1], by the associativity of T ,

INT (x, INT (y, z)) = N(T (x,N(N(T (y,N(z)))))) = N(T (x,T (y,N(z))))= N(T (T (x, y),N(z))) = INT (T (x, y), z).

(ii) As INT satisfies (LI) with respect to the t-norm T , then, for x = y = 1, INT (1, INT (1, z)) =INT (T (1,1), z) ⇒ N(T (1,N(N(T (1,N(z)))))) = N(T (1,N(z))) for all z ∈ [0,1],still by the boundary condition of T ,

N(N(N(N(z)))) = N(N(z)). (4.3)

Given that N is continuous, for all y ∈ [0,1] there exists x′ ∈ [0,1] such that N(x′) = y.Only for this x′ there exists x ∈ [0,1] such thatN(x) = x′. Thus, for all y ∈ [0,1] there ex-ists x ∈ [0,1] such thatN(N(x)) = y. Therefore, by Equation (4.3),N(N(N(N(x)))) =N(N(x))⇒ N(N(y)) = y, for all y ∈ [0,1].

43

Note that if N is continuous and non-strong then INT does not satisfy (LI). However, thereare non-continuous negationsN such that INT satisfies (LI) for some t-norm T . See the followingexample:

Example 4.1. Take a crisp negation N given by N = Nα and the minimum t-norm T , so

INαT (x, INαT (y, z)) = Nα(T (x,Nα(Nα(T (y,Nα(z))))))

=⎧⎪⎪⎨⎪⎪⎩

Nα(T (x,Nα(Nα(y)))), if z ≤ α1, if z > α

=⎧⎪⎪⎨⎪⎪⎩

Nα(x), if z ≤ α and y > α1, if z > α or y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if z ≤ α and y > α and x > α1, otherwise

and

INαT (T (x, y), z) = Nα(T (T (x, y),Nα(z))) =⎧⎪⎪⎨⎪⎪⎩

Nα(T (T (x, y),1)), if z ≤ α1, if z > α

=⎧⎪⎪⎨⎪⎪⎩

Nα(T (x, y)), if z ≤ α1, if z > α

=⎧⎪⎪⎨⎪⎪⎩

0, if z ≤ α and T (x, y) > α1, if z > α or T (x, y) ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if z ≤ α and x > α and y > α1, otherwise

,

thus, INαT satisfies (LI).

Another example can be given by taking the crisp fuzzy negation N = Nα with α = 0 andany t-norm T . In this case, by Proposition 2.1 we also have that INαT satisfies (LI).

In classical logic, the distributivity of binary operators over one another can somehow definethe framework of the algebra imposed by these operators. In fuzzy logic, one can find a varietyof studies on the distributivity of t-norms over t-conorms (BERTOLUZZA, 1993; BERTOLUZZA;DOLDI, 2004; CARBONELL et al., 1996; KLEMENT; MESIAR; PAP, 2000). In this sense, taking intoaccount the four basic distributive equations involving an implication, Equations 4.4, 4.5, 4.6,4.7, in the next propositions, some generalizations of them - which yield the distributivity of(T,N)-implications over t-norms and t-conorms - are presented.

I(T (x, y), z) = S(I(x, z), I(y, z)) (4.4)I(S(x, y), z) = T (I(x, z), I(y, z)) (4.5)

I(x,S1(y, z)) = S2(I(x, y), I(x, z)) (4.6)I(x,T1(y, z)) = T2(I(x, y), I(x, z)) (4.7)

Proposition 4.15. Let INT be a (T,N)-implication and S be a t-conorm. Then:

(i) If T is N -dual of S and the range of N is a subset of the idempotent elements of T thenINT satisfies Equation (4.4) with respect to the t-norm T and to the t-conorm S;

44

(ii) If INT satisfies Equation (4.4) with respect to the t-norm T and to the t-conorm S, then(1) T is N -dual of S and,(2) If N is strict then the range of N is a subset of the idempotent elements of T .

Proof. (i) As T is N -dual of S and the range of N is a subset of the idempotent elements ofT , i.e., T (N(x),N(x)) = N(x) for all x ∈ [0,1], then, for all x, y, z ∈ [0,1] :

S(INT (x, z), INT (y, z)) = S(N(T (x,N(z))),N(T (y,N(z))))= N(T (T (x,N(z)), T (y,N(z))))

(T1) (T2)= N(T (T (x, y), T (N(z),N(z))))= N(T (T (x, y),N(z)))= INT (T (x, y), z).

(ii) (1) As INT satisfies Equation (4.4) with respect to the t-norm T and to the t-conorm S,then, for z = 0, N(T (T (x, y),N(0))) = S(N(T (x,N(0))),N(T (y,N(0)))), so by(T4), N(T (x, y)) = S(N(x),N(y)) for all x, y ∈ [0,1] and(2) In particular, for x = y = 1, S(INT (1, z), INT (1, z)) = N(T (T (1,1),N(z))), so by(T4), S(N(N(z)),N(N(z))) = N(N(z)) for all z ∈ [0,1], since T is N -dual of S wehave N(T (N(z),N(z))) = N(N(z)) N strict⇒ T (N(z),N(z)) = N(z), for all z ∈ [0,1].

Corollary 4.1. Let N be a strict negation and T be a t-norm. Then, INT satisfies Equation (4.4)if and only if T = TM and S = SM .

In the previous corollary, the continuity ofN ensures that if INT satisfies Equation (4.4) thenT is minimum. However, there are non-continuous negations such that INT satisfies Equation(4.4) for some t-norms. See the following example:

Example 4.2. Take a crisp negation N given by N = Nα and take T as the minimum t-norm, so

S(INαT (x, z), INαT (y, z)) = S(Nα(T (x,Nα(z))),Nα(T (y,Nα(z))))

=⎧⎪⎪⎨⎪⎪⎩

S(Nα(x),Nα(y)), if z ≤ α1, if z > α

=⎧⎪⎪⎨⎪⎪⎩


and, by Example 4.1

INαT (T (x, y), z) =⎧⎪⎪⎨⎪⎪⎩


thus, INαT satisfies Equation (4.4).

Another example can be given for any t-norm T . Just take the crisp fuzzy negation N = Nαwith α = 0. Then, by Proposition 2.1 we also have that IN0T satisfies Equation (4.4).

45

Proposition 4.16. Let INT be a (T,N)-implication. Then,

(i) INT satisfies Equation (4.5) for TM and SM , i.e., considering TM as T and SM as S inEquation (4.5);

(ii) If INT satisfies Equation (4.5) with respect to the t-norm T and to the t-conorm S, then(1) S is N -dual of T and(2) If N is strict then the range of N is a subset of the idempotent elements of S.

Proof. (i) For all x, y, z ∈ [0,1], if x ≤ y then SM(x, y) = y and, by (T3) and (N1), INT (y, z) ≤INT (x, z), so

TM(INT (x, z), INT (y, z)) = INT (y, z) = INT (SM(x, y), z).

Therefore, INT satisfies Equation (4.5). Similarly, if x > y the result follows.

(ii) (1) As INT satisfies Equation (4.5) with respect to the t-norm T and to the t-conorm S,then, for z = 0, N(T (S(x, y),N(0))) = T (N(T (x,N(0))),N(T (y,N(0)))), so by(T4), N(S(x, y)) = T (N(x),N(y)) for all x, y ∈ [0,1].(2) In particular, for x = y = 1, T (INT (1, z), INT (1, z)) = INT (S(1,1), z), so by (T4),T (N(N(z)),N(N(z))) = N(N(z)) for all z ∈ [0,1], since S is N -dual of T we haveN(S(N(z),N(z))) = N(N(z)) N strict⇒ S(N(z),N(z)) = N(z), for all z ∈ [0,1].

Corollary 4.2. Let N be a strict negation and T be a t-norm. Then, INT satisfies Equation (4.5)if and only if T = TM and S = SM .

Proposition 4.17. Let INT be a (T,N)-implication and S1 and S2 be t-conorms. Then:

(i) If S1 = S2 = SM then, for any t-norm T and any negation N , INT satisfies Equation (4.6);

(ii) If INT satisfies Equation (4.6) with respect to t-conorms S1 and S2, then:(1) The range of N is a subset of the idempotent elements of S2 and(2) If N is strict then S1 = S2 = SM .

Proof. (i) For all x, y, z ∈ [0,1], if y ≤ z then SM(y, z) = z and, by (N1) and (T3), INT (x, y) ≤INT (x, z), so

SM(INT (x, y), INT (x, z)) = INT (x, z) = INT (x,SM(y, z)).

Therefore, INT satisfies Equation (4.6). Similarly, if y > z the result follows.

(ii) (1) As INT satisfies Equation (4.6) then, in particular for y = z = 0,

N(T (x,N(S1(0,0)))) = S2(N(T (x,N(0))),N(T (x,N(0)))),

so by (T4), N(x) = S2(N(x),N(x)), for all x ∈ [0,1]. (2) Since N is strict andS2(N(x),N(x)) = N(x) for all x ∈ [0,1], then

S2(y, y) = S2(N(N−1(y)),N(N−1(y))) = N(N−1(y)) = y

46

for all y ∈ [0,1], so S2 = SM . On the other hand, doing x = 1 and z = y, we haveN(T (1,N(S1(y, y)))) = S2(N(T (1,N(y))),N(T (1,N(y)))) for all y ∈ [0,1], so by(T4),

N(N(S1(y, y))) = S2(N(N(y)),N(N(y))) S2=SM= N(N(y)),

for all y ∈ [0,1]. Thus, S1(y, y) = y for all y ∈ [0,1], since N is strict. Therefore,S1 = SM .

Corollary 4.3. Let N be a strict negation and T be a t-norm. Then, INT satisfies Equation (4.6)if and only if S1 = S2 = SM .

Proposition 4.18. Let INT be a (T,N)-implication and T1 and T2 be t-norms. Then:

(i) If T1 = T2 = TM then, for any t-norm T and any negation N , INT satisfies Equation (4.7);

(ii) If INT satisfies Equation (4.7) with respect to t-norms T1 and T2, then:(1) The range of N is a subset of the idempotent elements of T2 and(2) If N is strict then T1 = T2 = TM .

Proof. (i) For all x, y, z ∈ [0,1], if y ≤ z then TM(y, z) = y and, by (N1) and (T3), INT (x, y) ≤INT (x, z), so

TM(INT (x, y), INT (x, z)) = INT (x, y) = INT (x,TM(y, z)).

Therefore, INT satisfies Equation (4.7). Similarly, if y > z the result follows.

(ii) (1) As INT satisfies Equation (4.7) then, in particular for y = z = 0,

N(T (x,N(T1(0,0)))) = T2(N(T (x,N(0))),N(T (x,N(0)))),

so by (T4), N(x) = T2(N(x),N(x)), for all x ∈ [0,1]. (2) Since N is strict andthe range of N a subset of the idempotent elements of T2, we have that T2(x,x) =T2(N(N−1(x)),N(N−1(x))) = N(N−1(x)) = x. On the other hand, for x = 1 andz = y, N(T (1,N(T1(y, y)))) = T2(N(T (1,N(y))),N(T (1,N(y)))), so by (T4),

N(N(T1(y, y))) = T2(N(N(y)),N(N(y))) T2=TM= N(N(y)),

for all y ∈ [0,1]. Thus, T1(y, y) = y for all y ∈ [0,1], since N is strict. Therefore,T1 = TM .

There are other conditions for t-norms and negations that imply that a (T,N)-implicationsatisfies equation (4.7). The following example ensures that if we take T1 = TM and the crispnegation N , given by N = Nα with α ∈ [0,1), then, independently from t-norms T and T2, INTsatisfies Equation (4.7).

47

Example 4.3. Take the crisp negation N given by N = Nα and take T1 as the minimum t-norm,so

T2(INαT (x, y), INαT (x, z)) = T2(Nα(T (x,Nα(y))),Nα(T (x,Nα(z))))

=⎧⎪⎪⎨⎪⎪⎩

Nα(T (x,Nα(y))), if z > αT2(Nα(T (x,Nα(y))),Nα(x)), if z ≤ α

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1, if z > α and y > αNα(x), if z > α and y ≤ αNα(x), if z ≤ α and y > αT2(Nα(x),Nα(x)), if z ≤ α and y ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if x > α and T1(y, z) ≤ α1, otherwise

and

INαT (x,T1(y, z)) = Nα(T (x,Nα(T1(y, z)))) =⎧⎪⎪⎨⎪⎪⎩

1, if T1(y, z) > αNα(x), if T1(y, z) ≤ α

=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1, if T1(y, z) > α0, if T1(y, z) ≤ α and x > α1, if T1(y, z) ≤ α and x ≤ α

=⎧⎪⎪⎨⎪⎪⎩

0, if T1(y, z) ≤ α and x > α1, otherwise

thus, INT satisfies Equation (4.7).

4.3 Applying (T,N)-implications to generate fuzzy subset-hood measures

Fuzzy subsethood measures determine up to what extent a fuzzy set is included into anotherfuzzy set. Formally, given two fuzzy sets A,B ∈ F (X), it can be said that A is included in B,written A ≤ B, if the inequality: A(x) ≤ B(x) holds for every x ∈ X (ZADEH, 1965), where ≤defines a partial order in F (X) which extends the linear order between real numbers.

Clearly, this form

repositorio.ufrn.br · Pinheiro, Antônia Jocivania. On algebras for interval-valued fuzzy logic / Antonia Jocivania Pinheiro. - 2019. 135f.: il. Tese (Doutorado)-Universidade Federal

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