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Research ArticlePawlak Algebra and Approximate Structure on Fuzzy Lattice
Ying Zhuang1 Wenqi Liu1 Chin-Chia Wu2 and Jinhai Li1
1 Faculty of Science Kunming University of Science and Technology Kunming 650500 China2Department of Statistics Feng Chia University Taichung 40724 Taiwan
Correspondence should be addressed to Ying Zhuang shadowzhuang163com
Received 26 June 2014 Revised 13 July 2014 Accepted 13 July 2014 Published 23 July 2014
Academic Editor Yunqiang Yin
Copyright copy 2014 Ying Zhuang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The aim of this paper is to investigate the general approximation structure weak approximation operators and Pawlak algebra inthe framework of fuzzy lattice lattice topology and auxiliary ordering First we prove that the weak approximation operator spaceforms a complete distributive lattice Then we study the properties of transitive closure of approximation operators and apply themto rough set theory We also investigate molecule Pawlak algebra and obtain some related properties
1 Introduction
The theory of rough sets was originally proposed by Pawlak[1] in 1982 as amathematical approach to handle imprecisionvagueness and uncertainty in data analysis which has beenapplied successfully to many areas such as knowledge rep-resentation data mining pattern recognition and decisionmaking (Sun et al [2] Wang et al [3]) This theory takesinto consideration the indiscernibility between objects Theindiscernibility is typically characterized by an equivalencerelation Rough sets are the results of approximating crisp setsusing equivalence classes However the requirement of anequivalence relation seems to be a very restrictive conditionwhich may limit the applications of rough set theory sincethis requirement can deal only with complete informationsystems Therefore some interesting and meaningful exten-sions of Pawlakrsquos rough set models have been proposed inthe literature For example some interesting extensions toequivalence relations have been proposed such as tolerancerelations or similarity relations general binary relations onthe discourse partitions and general binary relations on theneighborhood system from topological space and generalapproximation spaces (examples of this approach can befound in Chen et al [4] Wang and Hu [5] and Yin et al [6ndash8]) some general notions of rough sets such as rough fuzzysets fuzzy rough sets and soft rough sets have been proposedand discussed (examples of this approach can be found in Aliet al [9] Feng et al [10] Li and Yin [11] Yao et al [12] and
Zhang et al [13]) and rough set models on two universes ofdiscourse which can be interpreted by the notions of intervalstructure and generalized approximation space have beenextensively studied by Li and Zhang [14] Ma and Sun [15]and Yao and Lin [16]
The relationships between lattice theory and rough setsare another topic receiving much attention in recent yearsCattaneo and Ciucci [17] focus on the study on latticeswith interior and closure operators and abstract approxima-tion spaces in which the nonequational notion of abstractapproximation space for roughness theory is introducedand its relationship with the equational definition of latticewith Tarski interior and closure operations is studied Theircategorical isomorphism is also proved and the role of theTarski interior and closure with an algebraic semantic of aS4-like model of modal logic is widely investigated Jarvinen[18] investigates lattice-theoretical foundations of rough settheory in which closure operators in a more general settingof ordered sets fixpoints of Galois connections rough setapproximations and definable sets and the lattice structuresof the ordered set of all rough sets determined by differentkinds of indiscernibility relations are studied in detail Thepurpose of this paper is to investigate the general approxi-mation structure weak approximation operators and Pawlakalgebra in the framework of fuzzy lattice lattice topologyand auxiliary orderingThe relationships between the Pawlakapproximation structures and these mathematic structuresare established
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 697107 9 pageshttpdxdoiorg1011552014697107
2 The Scientific World Journal
The remaining part of the paper is organized as followsSection 2 introduces the relevant definitions which will beused throughout the paper Section 3 investigates the Pawlakalgebra and weak Pawlak algebra on fuzzy lattice Section 4discusses the relationships between Pawlak algebra and aux-iliary ordering Section 5 focuses on the study of molecu-lar Pawlak algebra Section 6 investigates the properties ofPawlak algebra based on binary relation Finally Section 7concludes the paper and suggests some future research topics
2 Preliminaries
In this section we introduce some basic definitions (see [319 20]) which will be used throughout the paper
Definition 1 Apartially ordered set (119871 le) is said to be a latticeif inf119909 119910 and sup119909 119910 denoted by and and or respectivelyexist for all 119909 119910 isin 119871 A lattice 119871 is said to be complete if forevery 119860 sube 119871⋀
120572isin119860120572 and⋁
120572isin119860120572 exist
Definition 2 Let (119871 le) be a complete lattice with the max-imum element 1 and minimum element 0 and ldquo≺rdquo a binaryrelation on 119871 If the following conditions hold for all 120572 120573 120583120578 120574 isin 119871
(i) 120572 ≺ 120573 rArr 120572 le 120573(ii) 120583 le 120572 ≺ 120573 le 120578 rArr 120583 ≺ 120578(iii) 120572 ≺ 120574 120573 ≺ 120574 rArr 120572 or 120573 ≺ 120574(iv) forall120572 isin 119871 0 ≺ 120572 ≺ 1
then the relation ldquo≺rdquo is called an auxiliary ordering on 119871 If therelation ldquo≺rdquo satisfies conditions (i) (ii) and (iv) it is called aweak auxiliary ordering on119871Theweak auxiliary ordering ldquo≺rdquois called completely approximate if
then ldquo≺rdquo is called a strong auxiliary ordering on 119871
Definition 3 Let (119871 or and 0 1) be a completely distributivelattice If the mapping 119888 119871 rarr 119871 satisfies the followingconditions
(i) reverse law forall120572 120573 isin 119871 if 120572 le 120573 then 120573119888 le 120572119888(ii) recovery law forall120572 isin 119871 (120572119888)119888 = 120572
then (119871 or and119888 0 1) is called a fuzzy lattice
Remark 4 The notion of fuzzy lattice just given is first intro-duced in Wang [20] and is also called de Morgan completelydistributive lattice in the literature To keep consistency weadopt the term ldquofuzzy latticerdquo in this paper
Definition 5 Let (119871 or and119888 0 1) be a fuzzy lattice and 120587 sube 119871If the subset 120587 satisfies the following conditions
(i) 0 notin 120587(ii) 120572 120573 isin 120587 120572 and 120573 = 0 rArr 120572 le 120573 or 120573 le 120572(iii) 120572 120573 120574 isin 120587 120572 le 120574 120573 le 120574 rArr 120572 and 120573 = 0(iv) 120578 = or120572 | 120572 isin 120587 120572 le 120578 for all 120578 isin 119871(v) if 120587
0sube 120587 and 120587
0is linear order subset then or120587
0isin 120587
then 120587 is called amolecular set and the element of 120587 is calleda molecule The fuzzy lattice with molecule (119871(120587) or and119888 0 1)
is called amolecular lattice
Definition 6 Let (119871 or and119888 0 1) be a fuzzy lattice and 120575 sube 119871If the subset 120575 satisfies the following conditions
then (119871 120575) is called a lattice topology space
3 The Pawlak Algebra and Weak PawlakAlgebra on Fuzzy Lattices
In this section we investigate the structural properties ofPawlak rough approximations on fuzzy lattices Let us beginwith introducing the following concepts
Definition 7 (see [20])) Let (119871 or and119888 0 1) be a fuzzy latticeIf the dual mappings 119886119901119903 119871 rarr 119871 and 119886119901119903 119871 rarr 119871 satisfythe following conditions
(P3) 119886119901119903(120572 and 120573) = (119886119901119903120572) and (119886119901119903120573) (forall120572 120573 isin 119871)
(P4) 120572 le 119886119901119903(120572) forall120572 isin 119871
then (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a Pawlak algebra 119886119901119903(resp 119886119901119903 pair (119886119901119903 119886119901119903) ) is called upper approximationoperator (resp lower approximation operator dual approx-imation operator) on 119871 If 119886119901119903(120572) = 119886119901119903(120572) then 120572 is calleda definable element If 119886119901119903120572 = 119886119901119903120572 then 120572 is called a roughelement
If (P3) is replaced by the following condition (P3)lowast
(P3)lowast 120572 le 120573 rArr 119886119901119903(120572) le 119886119901119903(120573)
then 119886119901119903 (resp 119886119901119903 pair (119886119901119903 119886119901119903)) is called a weak upperapproximation operator (resp weak lower approximationoperator weak dual approximation operator) on 119871 andthe system (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a weak Pawlakalgebra
Let (119871 or and119888 0 1) be a fuzzy lattice Denote by 119860119875119877(119871)
(resp 119860119875119877119882(119871)) the set of all dual approximation operators
(resp dual weak approximation operators) on 119871 respectivelyand by 120590
119886119901119903
0(119871) the set of all definable elements in the Pawlak
algebra (119871 or and119888 0 1 119886119901119903 119886119901119903)
The Scientific World Journal 3
Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903
1
1198861199011199031) (119886119901119903
2
1198861199011199032) isin 119860119875119877
119908(119871) Then the dual weak
approximation operator (1198861199011199032
1198861199011199032) is called rougher than
(1198861199011199031
1198861199011199031) denoted by (119886119901119903
1
1198861199011199031) ≺ (119886119901119903
2
1198861199011199032) if the
following inequalities hold
forall120572 isin 119871 1198861199011199032
(120572) le 1198861199011199031
(120572) le 1198861199011199031(120572) le 119886119901119903
2(120572) (2)
Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877
119908(119871)
Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so
119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)
Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast
holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required
Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows
(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903
1(120572) le
1198861199011199031(119886119901119903
2(120572)) = (119886119901119903
1∘ 119886119901119903
2)(120572)
(3) From the proof of Theorem 10 we have
cl (119886119901119903 119886119901119903) = (
infin
⋀119896=1
119886119901119903(119896)
infin
⋁119896=1
119886119901119903(119896)
) (10)
And it is easy to prove by mathematical induction that
119886119901119903(119896)
(120572 and 120573)
= 119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573) for all positive integer 119896
(11)
Therefore
(
infin
⋀119896=1
119886119901119903(119896)
)(120572 and 120573) =
infin
⋀119896=1
119886119901119903(119896)
(120572 and 120573)
=
infin
⋀119896=1
(119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573))
= (
infin
⋀119896=1
119886119901119903(119896)
(120572)) and (
infin
⋀119896=1
119886119901119903(119896)
(120573))
(12)
It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator
Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871
Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has
(
infin
⋀119896=1
119886119901119903(119896)
) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)
(
infin
⋁119896=1
119886119901119903(119896)
) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)
Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879
119886119901119903 for any 120572 isin 119871 we have
le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds
As a consequence ofTheorem 15 we obtain the followingresult
Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903
lowast= ⋁
infin
119896=1119886119901119903
(119896) is a closure operator on 119871
4 The Relationships between Pawlak Algebraand Auxiliary Ordering
Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately
The following results present the relationships betweenPawlak algebra and strong auxiliary ordering
Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903
≺
119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows
(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺
120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
The remaining part of the paper is organized as followsSection 2 introduces the relevant definitions which will beused throughout the paper Section 3 investigates the Pawlakalgebra and weak Pawlak algebra on fuzzy lattice Section 4discusses the relationships between Pawlak algebra and aux-iliary ordering Section 5 focuses on the study of molecu-lar Pawlak algebra Section 6 investigates the properties ofPawlak algebra based on binary relation Finally Section 7concludes the paper and suggests some future research topics
2 Preliminaries
In this section we introduce some basic definitions (see [319 20]) which will be used throughout the paper
Definition 1 Apartially ordered set (119871 le) is said to be a latticeif inf119909 119910 and sup119909 119910 denoted by and and or respectivelyexist for all 119909 119910 isin 119871 A lattice 119871 is said to be complete if forevery 119860 sube 119871⋀
120572isin119860120572 and⋁
120572isin119860120572 exist
Definition 2 Let (119871 le) be a complete lattice with the max-imum element 1 and minimum element 0 and ldquo≺rdquo a binaryrelation on 119871 If the following conditions hold for all 120572 120573 120583120578 120574 isin 119871
(i) 120572 ≺ 120573 rArr 120572 le 120573(ii) 120583 le 120572 ≺ 120573 le 120578 rArr 120583 ≺ 120578(iii) 120572 ≺ 120574 120573 ≺ 120574 rArr 120572 or 120573 ≺ 120574(iv) forall120572 isin 119871 0 ≺ 120572 ≺ 1
then the relation ldquo≺rdquo is called an auxiliary ordering on 119871 If therelation ldquo≺rdquo satisfies conditions (i) (ii) and (iv) it is called aweak auxiliary ordering on119871Theweak auxiliary ordering ldquo≺rdquois called completely approximate if
then ldquo≺rdquo is called a strong auxiliary ordering on 119871
Definition 3 Let (119871 or and 0 1) be a completely distributivelattice If the mapping 119888 119871 rarr 119871 satisfies the followingconditions
(i) reverse law forall120572 120573 isin 119871 if 120572 le 120573 then 120573119888 le 120572119888(ii) recovery law forall120572 isin 119871 (120572119888)119888 = 120572
then (119871 or and119888 0 1) is called a fuzzy lattice
Remark 4 The notion of fuzzy lattice just given is first intro-duced in Wang [20] and is also called de Morgan completelydistributive lattice in the literature To keep consistency weadopt the term ldquofuzzy latticerdquo in this paper
Definition 5 Let (119871 or and119888 0 1) be a fuzzy lattice and 120587 sube 119871If the subset 120587 satisfies the following conditions
(i) 0 notin 120587(ii) 120572 120573 isin 120587 120572 and 120573 = 0 rArr 120572 le 120573 or 120573 le 120572(iii) 120572 120573 120574 isin 120587 120572 le 120574 120573 le 120574 rArr 120572 and 120573 = 0(iv) 120578 = or120572 | 120572 isin 120587 120572 le 120578 for all 120578 isin 119871(v) if 120587
0sube 120587 and 120587
0is linear order subset then or120587
0isin 120587
then 120587 is called amolecular set and the element of 120587 is calleda molecule The fuzzy lattice with molecule (119871(120587) or and119888 0 1)
is called amolecular lattice
Definition 6 Let (119871 or and119888 0 1) be a fuzzy lattice and 120575 sube 119871If the subset 120575 satisfies the following conditions
then (119871 120575) is called a lattice topology space
3 The Pawlak Algebra and Weak PawlakAlgebra on Fuzzy Lattices
In this section we investigate the structural properties ofPawlak rough approximations on fuzzy lattices Let us beginwith introducing the following concepts
Definition 7 (see [20])) Let (119871 or and119888 0 1) be a fuzzy latticeIf the dual mappings 119886119901119903 119871 rarr 119871 and 119886119901119903 119871 rarr 119871 satisfythe following conditions
(P3) 119886119901119903(120572 and 120573) = (119886119901119903120572) and (119886119901119903120573) (forall120572 120573 isin 119871)
(P4) 120572 le 119886119901119903(120572) forall120572 isin 119871
then (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a Pawlak algebra 119886119901119903(resp 119886119901119903 pair (119886119901119903 119886119901119903) ) is called upper approximationoperator (resp lower approximation operator dual approx-imation operator) on 119871 If 119886119901119903(120572) = 119886119901119903(120572) then 120572 is calleda definable element If 119886119901119903120572 = 119886119901119903120572 then 120572 is called a roughelement
If (P3) is replaced by the following condition (P3)lowast
(P3)lowast 120572 le 120573 rArr 119886119901119903(120572) le 119886119901119903(120573)
then 119886119901119903 (resp 119886119901119903 pair (119886119901119903 119886119901119903)) is called a weak upperapproximation operator (resp weak lower approximationoperator weak dual approximation operator) on 119871 andthe system (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a weak Pawlakalgebra
Let (119871 or and119888 0 1) be a fuzzy lattice Denote by 119860119875119877(119871)
(resp 119860119875119877119882(119871)) the set of all dual approximation operators
(resp dual weak approximation operators) on 119871 respectivelyand by 120590
119886119901119903
0(119871) the set of all definable elements in the Pawlak
algebra (119871 or and119888 0 1 119886119901119903 119886119901119903)
The Scientific World Journal 3
Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903
1
1198861199011199031) (119886119901119903
2
1198861199011199032) isin 119860119875119877
119908(119871) Then the dual weak
approximation operator (1198861199011199032
1198861199011199032) is called rougher than
(1198861199011199031
1198861199011199031) denoted by (119886119901119903
1
1198861199011199031) ≺ (119886119901119903
2
1198861199011199032) if the
following inequalities hold
forall120572 isin 119871 1198861199011199032
(120572) le 1198861199011199031
(120572) le 1198861199011199031(120572) le 119886119901119903
2(120572) (2)
Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877
119908(119871)
Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so
119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)
Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast
holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required
Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows
(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903
1(120572) le
1198861199011199031(119886119901119903
2(120572)) = (119886119901119903
1∘ 119886119901119903
2)(120572)
(3) From the proof of Theorem 10 we have
cl (119886119901119903 119886119901119903) = (
infin
⋀119896=1
119886119901119903(119896)
infin
⋁119896=1
119886119901119903(119896)
) (10)
And it is easy to prove by mathematical induction that
119886119901119903(119896)
(120572 and 120573)
= 119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573) for all positive integer 119896
(11)
Therefore
(
infin
⋀119896=1
119886119901119903(119896)
)(120572 and 120573) =
infin
⋀119896=1
119886119901119903(119896)
(120572 and 120573)
=
infin
⋀119896=1
(119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573))
= (
infin
⋀119896=1
119886119901119903(119896)
(120572)) and (
infin
⋀119896=1
119886119901119903(119896)
(120573))
(12)
It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator
Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871
Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has
(
infin
⋀119896=1
119886119901119903(119896)
) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)
(
infin
⋁119896=1
119886119901119903(119896)
) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)
Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879
119886119901119903 for any 120572 isin 119871 we have
le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds
As a consequence ofTheorem 15 we obtain the followingresult
Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903
lowast= ⋁
infin
119896=1119886119901119903
(119896) is a closure operator on 119871
4 The Relationships between Pawlak Algebraand Auxiliary Ordering
Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately
The following results present the relationships betweenPawlak algebra and strong auxiliary ordering
Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903
≺
119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows
(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺
120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903
1
1198861199011199031) (119886119901119903
2
1198861199011199032) isin 119860119875119877
119908(119871) Then the dual weak
approximation operator (1198861199011199032
1198861199011199032) is called rougher than
(1198861199011199031
1198861199011199031) denoted by (119886119901119903
1
1198861199011199031) ≺ (119886119901119903
2
1198861199011199032) if the
following inequalities hold
forall120572 isin 119871 1198861199011199032
(120572) le 1198861199011199031
(120572) le 1198861199011199031(120572) le 119886119901119903
2(120572) (2)
Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877
119908(119871)
Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so
119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)
Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast
holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required
Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows
(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903
1(120572) le
1198861199011199031(119886119901119903
2(120572)) = (119886119901119903
1∘ 119886119901119903
2)(120572)
(3) From the proof of Theorem 10 we have
cl (119886119901119903 119886119901119903) = (
infin
⋀119896=1
119886119901119903(119896)
infin
⋁119896=1
119886119901119903(119896)
) (10)
And it is easy to prove by mathematical induction that
119886119901119903(119896)
(120572 and 120573)
= 119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573) for all positive integer 119896
(11)
Therefore
(
infin
⋀119896=1
119886119901119903(119896)
)(120572 and 120573) =
infin
⋀119896=1
119886119901119903(119896)
(120572 and 120573)
=
infin
⋀119896=1
(119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573))
= (
infin
⋀119896=1
119886119901119903(119896)
(120572)) and (
infin
⋀119896=1
119886119901119903(119896)
(120573))
(12)
It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator
Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871
Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has
(
infin
⋀119896=1
119886119901119903(119896)
) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)
(
infin
⋁119896=1
119886119901119903(119896)
) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)
Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879
119886119901119903 for any 120572 isin 119871 we have
le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds
As a consequence ofTheorem 15 we obtain the followingresult
Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903
lowast= ⋁
infin
119896=1119886119901119903
(119896) is a closure operator on 119871
4 The Relationships between Pawlak Algebraand Auxiliary Ordering
Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately
The following results present the relationships betweenPawlak algebra and strong auxiliary ordering
Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903
≺
119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows
(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺
120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903
1(120572) le
1198861199011199031(119886119901119903
2(120572)) = (119886119901119903
1∘ 119886119901119903
2)(120572)
(3) From the proof of Theorem 10 we have
cl (119886119901119903 119886119901119903) = (
infin
⋀119896=1
119886119901119903(119896)
infin
⋁119896=1
119886119901119903(119896)
) (10)
And it is easy to prove by mathematical induction that
119886119901119903(119896)
(120572 and 120573)
= 119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573) for all positive integer 119896
(11)
Therefore
(
infin
⋀119896=1
119886119901119903(119896)
)(120572 and 120573) =
infin
⋀119896=1
119886119901119903(119896)
(120572 and 120573)
=
infin
⋀119896=1
(119886119901119903(119896)
(120572) and 119886119901119903(119896)
(120573))
= (
infin
⋀119896=1
119886119901119903(119896)
(120572)) and (
infin
⋀119896=1
119886119901119903(119896)
(120573))
(12)
It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator
Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871
Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has
(
infin
⋀119896=1
119886119901119903(119896)
) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)
(
infin
⋁119896=1
119886119901119903(119896)
) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)
Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879
119886119901119903 for any 120572 isin 119871 we have
le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds
As a consequence ofTheorem 15 we obtain the followingresult
Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903
lowast= ⋁
infin
119896=1119886119901119903
(119896) is a closure operator on 119871
4 The Relationships between Pawlak Algebraand Auxiliary Ordering
Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately
The following results present the relationships betweenPawlak algebra and strong auxiliary ordering
Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903
≺
119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows
(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺
120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds
As a consequence ofTheorem 15 we obtain the followingresult
Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903
lowast= ⋁
infin
119896=1119886119901119903
(119896) is a closure operator on 119871
4 The Relationships between Pawlak Algebraand Auxiliary Ordering
Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately
The following results present the relationships betweenPawlak algebra and strong auxiliary ordering
Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903
≺
119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows
(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺
120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
Then ldquo≺rdquo is an auxiliary ordering on 119871
Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573
(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582
(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus
119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)
implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary
ordering on 119871
5 Molecular Pawlak Algebraand Rough Topology
In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set
Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular
Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863
a molec-ular net and 120572 isin 120587 If there exists 119889
0isin 119863 such that 119886119901119903(120572) le
119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit
of 119878 denoted by 119878119886119901119903
997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878
In the sequel we provide two examples of molecularPawlak algebra in topology space
Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =
119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define
for fuzzy point 119909120582and it is know that 119878
119886119901119903
997888997888rarr 119909120582if and only if
there exists 1198890isin 119863 such that 119878(119889) = 119909
120582for 119889 ge 119889
0
Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap
119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)
The following is now straightforward
Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers
Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra
The Scientific World Journal 7
Proof Set ℎ(0) = 0 and define
ℎ (120574) = ⋁120572le120574120572isin120587
ℎ (120572) ℎ (120574) = (ℎ (120574119888
))119888
(35)
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra
6 The Application of ApproximationOperators to Rough Sets
Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880
and defined general dual approximation operators 119886119901119903119877
and119886119901119903
119877as for all119883 sube 119880
119886119901119903119877
(119883) = 119909 | 119903 (119909) sube 119883
119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =
(37)
In this section we further investigate the properties ofapproximation operators induced by a binary relation
The following results recall some basic properties ofapproximation operators induced by a binary relation
Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880
Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following
) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =
1199101 119910
2 119910
119898 sube 119880 Then we have
119901 (119883) = ⋃119910119896isin119883
119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910
= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901
= 119909 | [119909]119877119901
cap 119883 =
= 119886119901119903119877119901
(119883)
(42)
Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901
(119883) It
follows that (119886119901119903119877119901
119886119901119903119877119901
) = (119901 119901)
Now suppose that the cover 119901119909119909isin119880
of 119880 is finitethat is there exist 119909
119896isin 119880 119896 = 1 2 119898 such that
8 The Scientific World Journal
119901119909119896119896isin12119898
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
is a cover of119880 Analogous to the above proofto prove that (119886119901119903
119877119901
119886119901119903119877119901
) = (119901 119901) it suffices to prove that
for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from
119901119909119896119896isin12119898
being a cover of119880 that119883 sube 119880 sube ⋃119898
119896=1119901119909
119896
Hence
119901 (119883) sube 119901(
119898
⋃119896=1
119901 119909119896)
=
119898
⋃119896=1
119901 119909119896
= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909
119896
= 119909 | there exists 119909119896isin 119883 such that 119909
119896isin [119909]
119877119901
= 119909 | [119909]119877119901
cap 119883 = = 119886119901119903119877119901
(119883)
(43)
as required
In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds
Proposition 29 LetR be the set of all similar relations on 119880Then we have
sube 119883 that is 119909 isin 1198861199011199031198771
(119883)Therefore 119886119901119903
1198772
(119883) sube 1198861199011199031198771
(119883) and 1198861199011199031198771
(119883) sube 1198861199011199031198772
(119883)
by the duality It follows that (1198861199011199031198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)Conversely suppose that (119886119901119903
1198771
1198861199011199031198771
) ≺ (1198861199011199031198772
1198861199011199031198772
)
and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903
1198771
119910 Since119886119901119903
1198771
(119883) sube 1198861199011199031198772
(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903
1198771
119910 sube 1198861199011199031198772
119910 and hence 119909 isin 1198861199011199031198772
119910implying that (119909 119910) isin 119877
2 Therefore 119877
1sube 119877
2
Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) and that 119877 is the transitive
closure of 119877
Proof It follows from the proof of Theorem 10 thatcl(119886119901119903
119877
119886119901119903119877) = (⋀
infin
119896=1119886119901119903
119877
(119896) ⋁infin
119896=1119886119901119903
119877
(119896)
) and that
⋁infin
119896=1119886119901119903
119877
(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows
119877 = (119909 119910) | 119909 isin (
infin
⋃119896=1
119886119901119903119877
(119896)
)119910 (44)
Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903
119877
119886119901119903119877) = (119886119901119903
119877
119886119901119903119877) by Theorem 10 In the sequel
we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880
Suppose that 119877lowast is an equivalent relation such that 119877 sube
119877lowast By Proposition 29 we have (119886119901119903119877
119886119901119903119877) ≺ (119886119901119903
119877lowast 119886119901119903
119877lowast)
and 119886119901119903119877lowast is a closure operator and hence (119886119901119903
119877lowast) Thus by Proposition 29 119877 sube 119877lowast
7 Conclusions
In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)
References
[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982
[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013
[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012
[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014
The Scientific World Journal 9
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998
[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013
[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011
[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011
[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011
[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013
[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011
[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012
[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014
[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013
[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008
[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012
[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996
[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009
[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007
[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980
[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986
[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005
[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998