Gégény, D.

Piller, I.

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 20 (2019), No. 1, pp. 245–254 DOI: 10.18514/MMN.2019.2891APPROXIMATION LATTICES DEFINED BY TOLERANCESINDUCED BY IRREDUNDANT COVERINGSD. GE´GE´NY AND I. PILLERReceived 12 March, 2019Abstract. The topic of rough set theory considers a relation to determine the lower and upperapproximations of a set X . Originally, this relation was assumed to be an equivalence relation.This research focuses on using tolerance relations instead of equivalences, i.e. we do not assumethe transitivity of the relations. More specifically, in this paper we investigate tolerances inducedby irredundant coverings. We characterize the interrelation between the lattices of lower andupper approximations of such tolerances R and . The theory of Formal Concept Analysismakes it possible to examine the inclusions of the resulting concepts. We also use quasiorders(denoted by E ./ and D ./) and an equivalence relation (denoted by ker) for summarizingthe connection between tolerances and lattices in a theorem.2010 Mathematics Subject Classification: 06B15; 68T37; 06B23Keywords: rough set, tolerance relation, irredundant coverings1. INTRODUCTIONThe notion of rough sets was introduced by Z. Pawlak [8]. His idea was that ourknowledge about the elements of a universe U is given in terms of an informationrelation R  U U reflecting their indiscernibility. Originally, Pawlak assumed thatthis binary relation is an equivalence, but later several other types of relations werealso examined (see e.g. [4, 12], or [6]). For any binary relation R  U U and anyelement u 2 U , denote by R.u/ the R-neighbourhood of u, i.e. R.u/ WD fx 2 U j.u;x/ 2Rg. Now, for any subset X  U the lower approximation of X is defined asXR WD fx 2 U jR.x/ U g;and the upper approximation of X is given byXR WD fx 2 U jR.x/\X ¤¿g:If R is a reflexive relation then XR  X  XR. The rough set of X is the pair.XR;XR/, and the set of all rough sets isRS.U;R/D f.XR;XR/ jX  U g.c 2019 Miskolc University Press246 D. GE´GE´NY AND I. PILLERThe set RS.U;R/ may be canonically ordered by the component-wise inclusion:.XR;XR/ .YR;Y R/”XR  YR and XR  Y R,obtaining a partially ordered set RS.U;R/ WD .RS.U;R/;/. If R is an equivalence,then RS.U;R/ is a particular complete distributive lattice.Ordering the sets }.U /RDfXR jX U g and }.U /RDfXR jX U g by the re-lationwe obtain dually isomorphic complete lattices .}.U /R;/ and .}.U /R;/,called the lattice of upper approximations, respectively the lattice of lower approx-imations (see [6]). Let R be a tolerance, that is, a reflexive and symmetric relation.In [6] it was shown that .}.U /R;/ is isomorphic to the concept lattice of the con-text .U;U;Rc/, where Rc D .U U / nR is the complement of the relation R. Byusing this observation, in [2], we applied FCA methods to describe the sublattices ofthe lattices of upper (lower) approximations. These lattices play an important role inseveral applications of rough set theory (see [3, 9–11, 13]).This paper can be considered as a continuation of [2] where we deduced suffi-cient conditions which guarantee that for some tolerances R U U , the lattice}.U / (}.U /) is a complete sublattice of }.U /R (of }.U /R). The focus of thispaper is on the approximation lattices defined by tolerances induced by irredundantcoverings of U . These relations can be considered as a natural generalization ofequivalences. If R    U U are tolerance relations and R is induced by an irre-dundant covering ofU , we characterize the case when the concept latticeL.U;U;c/is a complete sublattice of the concept latticeL.U;U;Rc/. Then this characterizationis applied to compare the lattices .}.U /R;/ and .}.U /R;/.2. PRELIMINARIESFirst, we note that the above defined approximations for any X  U and anyH P .U / have the following properties:(a) SX2HX!RD SX2HXR and TX2HX!RD TX2HXR;(b) .Xc/R D .XR/c , .Xc/R D XRc.In view of (a), }.U /R is a closure system, being closed under arbitrary intersectionsand }.U /R is an interior system, because it is closed under any union. Therefore,}.U /R and }.U /R are complete lattices with respect to . If R is a tolerance rela-tion, then for any X;Y  U we have: XR  Y ,X  YR.Property (b) implies that the lattices .}.U /R;/ and .}.U /R;/ are dually iso-morphic via the map H W}.U /R! }.U /R, H WX ! Xc , since H.XR/D .XR/c D.Xc/R. If R is an equivalence, then }.U /R D }.U /R and they form the sameBoolean lattice.A formal context is a tripleK D .G;M;I /, where G is a set of objects, M is a setof attributes and I  G M is a relation, called incidence relation. The notationsAPPROXIMATION LATTICES DEFINED BY TOLERANCES INDUCED 247.g;m/ 2 I and gIm both express that an object g is in relation I with an attributem.The basics of Formal Concept Analysis (FCA) can be found e.g. in [1]. By definingfor all subsets AG and B MAI D fm 2M j .g;m/ 2 I; for all g 2 Ag;BI D fg 2G j .g;m/ 2 I; for all m 2 Bgwe establish a Galois connection between the power-set lattices }.G/ and }.M/ andthe maps A! AII , A  G and B ! BII , B M are closure operators on }.G/;respectively }.M/.A formal concept of the contextK is a pair .A;B/ 2 }.G/}.M/ with AI D Band BI D A, where the set A is called the extent and B is called the intent of theconcept .A;B/. It is easy to check that a pair .A;B/ 2 }.G/}.M/ is a concept ifand only if .A;B/D .AII ;AI /D .BI ;BII /. The set of all concepts of the contextK is denoted by L.K/. This set L.K/ is ordered by.A1;B1/ .A2;B2/, A1  A2, B1  B2:With respect to this order, L.K/ forms a complete lattice, called the concept latticeof the context K D .G;M;I /, denoted by L.G;M;I /.A relation J  I is called a closed subrelation of the context .G;M;I / if everyconcept of the context .G;M;J / is also a concept of .G;M;I /. In [1] it is provedthat this definition is equivalent to the condition that the concept lattice L.G;M;J /is a complete sublattice of L.G;M;I /.For a tolerance relation R  U U the relationship between the lattices of ap-proximations and the concept lattice L.U;U;Rc/ was described in [6]. Indeed, letI DRc . Then for any X  U we haveXI D fu 2 U j xRcu; for all x 2XgD fu 2 U j .x;u/ …R; for all x 2Xg D U nXR DXRc:Thus XR D  XI c , and XR D .Xc/Rc D .Xc/I , according to (b).In [6] it is also proved that }.U /R;Š .}.U /R;/ŠL.U;U;Rc/.3. COMPLETE SUBLATTICES OF APPROXIMATION LATTICESNow let ;R be two tolerance relations such that R    U U . Consider theformal contexts KR D .U;U;Rc/ and K D .U;U;c/. Since J WD c  Rc WD I ,K is a subcontext of KR. We intend to characterize the case when the lattice}.U / (}.U /) is isomorphic (dually isomorphic) to a complete sublattice of}.U /R(}.U /R, respectively). In [2] we proved that .}.U /;/ is a complete sublatticeof }.U /R;, respectively  }.U /; is a complete sublattice of .}.U /R;/,whenever L.U;U;c/ is a complete sublattice of L.U;U;Rc/. Unfortunately, the248 D. GE´GE´NY AND I. PILLERconverse implication does not necessarily hold. For instance, in [2] we construc-ted an example where .}.U /;/ is a complete sublattice of  }.U /R;, howeverL.U;U;c/ is not even a subset of the lattice L.U;U;Rc/. All we can say is that ingeneral the following conditions are equivalent:(1) }.U /R; is isomorphic to a complete sublattice of .}.U /;/;(2) .}.U /R;/ is isomorphic to a complete sublattice of }.U /;;(3) L.U;U;c/ is isomorphic to a complete sublattice of L.U;U;Rc/.Let  be a tolerance on U . Let us defineE ./ WD f.x;y/ 2 U U j .x/ .y/g;D ./ WD f.x;y/ 2 U U j .x/ .y/g;ker WD f.x;y/ 2 U U j .x/D .y/g:Clearly, E ./ and D ./ are reflexive and transitive relations, i.e. they are quasi-orders and D ./ is the inverse relation of E ./. ker is an equivalence relation,called the kernel of the tolerance . Clearly, ker DE ./\ D ./. Let the symbolı stand for the relational product, in what follows. It is easy to check that in thecase R   the relations RıE ./  and D ./ıR   always hold. Using thesenotions, in [2] we proved the following characterization:Theorem 1. Let ;R be two tolerance relations satisfying R    U U . Thenthe following conditions are equivalent:(C): L.U;U;c/ is a complete sublattice of L.U;U;Rc/;(D): For any .a;b/ 2  nR there exist some elements c;d 2 U such that.b;c/; .a;d/ 2R and .c/ .a/, .d/ .b/;(E): RıE ./D ;(E’): D ./ıRD .Now, the next corollary is immediate:Corollary 1. LetR; be two tolerance relations on U such thatR . IfR and satisfy one of the equivalent conditions of Theorem 1, then .}.U /;/ is a completesublattice of }.U /R; and  }.U /; is a complete sublattice of .}.U /R;/.In [2] we proved that in the particular case when  is an equivalence relation on Usuch that R  , then condition (D) is satisfied. Hence in such a case .}.U /;/ isobviously a sublattice of }.U /R; and .}.U /R;/.APPROXIMATION LATTICES DEFINED BY TOLERANCES INDUCED 249Here we give an algorithm (Algorithm 1) that is checking on a finite set U and tworelationsR U U whetherL.U;U;c/ is a complete sublattice ofL.U;U;Rc/by using condition (D).CD EXIST.R;;a;b/for .b;c/ 2R dofor .a;d/ 2R doif .d/ .b/^.c/ .a/ thenreturn TRUEendendendreturn FALSESATISFIES D.R;/for .a;b/ 2  nR doif :CD EXIST .R;;a;b/ thenreturn FALSEendendreturn TRUEAlgorithm 1: Algorithm for checking condition (D)4. TOLERANCES INDUCED BY IRREDUNDANT COVERINGSA collection C of nonempty subsets of U is called a covering of U ifSC D U .The covering C is called irredundant if removing any member X of C , the collectionC n fXg remains no longer a covering of U . For instance, the classes of an equival-ence relation E  U U provide a simple example of an irredundant covering of U .Each covering C of U defines a tolerance relation C DSfX X j X 2 Cg, calledthe tolerance induced by C . If C is an irredundant covering of U , then we say thatC is a tolerance induced by an irredundant covering. In [6] the authors proved thatthe lattices }.U /,}.U / and RS.U;/ are completely distributive if and only if is induced by an irredundant covering of U . It was shown that this condition is alsoequivalent to the condition that the lattice L.U;U;c/ is completely distributive. Acomplete lattice L is called completely distributive (see e.g. [1]) if for any doublyindexed family of elements fxi;j gi2I;j2J , .I;J ¤¿/ we have^i2I0@_j2Jxi;j1AD _f WI!J ^i2Ixi;f .i/!:250 D. GE´GE´NY AND I. PILLERWe note that any complete sublattice of a completely distributive lattice is also com-pletely distributive. As immediate consequence of the mentioned results we obtainthe following Lemma.Lemma 1. Let R; be two tolerance relations on U with R   and such that }.U /R; is isomorphic to a complete sublattice of .}.U /;/. If R is a toler-ance induced by an irredundant covering, then  is also a tolerance induced by anirredundant covering.Proof. Since in view of [6] }.U /R; is a completely distributive lattice,.}.U /;/, being a complete sublattice of  }.U /R;, is also completely dis-tributive. Therefore,  is a tolerance induced by an irredundant covering. Since any equivalence relation is a particular tolerance induced by an irredundantcovering, the following corollary is immediate.Corollary 2. Let R be an equivalence relation and  a tolerance relation on Uwith R   and such  }.U /R; is a complete sublattice of .}.U /;/. Then  isinduced by an irredundant covering.Let  be a tolerance on U . A nonempty subset X of U is called a preblock of if X X  . Note that in this case B  .x/ for all x 2 B . A preblock of  that ismaximal with respect to the inclusion is called a block of .Remark 1. It is well known that any tolerance relation  is determined by itsblocks, that is for any a;b 2 U , .a;b/ 2 , a;b 2 B , for some block B of .In [6] and in [7] it is shown that if  is induced by an irredundant covering C , then Bcan be chosen as a member of C having the property B D .k/, for some k 2 B . Itis also proved that in this case DD ./ı E ./ (see [5]).Theorem 2. Let ;R  U U be two tolerance relations with R   and assumethat R is induced by an irredundant covering. Then condition (E) is equivalent to thecondition(F): D .R\D .//ı ker ı .R\E .//.Proof. First we will show that condition (E) implies condition (F). If (E) holds,then L.U;U;c/ is a complete sublattice of L.U;U;Rc/, according to Theorem 1.By Corollary 1 this yields that .}.U /;/ is a complete sublattice of  }.U /R;.Then by Lemma 1,  is also a tolerance induced by an irredundant covering of U .Then DD ./ı E ./, according to [5]. It is easy to check thatD ./ı kerD ./.Hence.R\D .//ı ker ı .R\E .//D ./ı kerıE ./D ./ıE ./D .In order to prove the converse inclusion, take any .a;b/ 2 . Then, in view ofRemark 1, there exists a k 2 U and a block B of  such that a;b 2 B D .k/.Then B  .a/;.b/. As .a;k/; .b;k/ 2 , now condition (E), i.e. Rı E ./ D APPROXIMATION LATTICES DEFINED BY TOLERANCES INDUCED 251implies that there exist some elements c;d 2 U such that .a;c/ 2 R, .c/  .k/and .b;d/ 2 R, .d/  .k/. Since c;d 2 .k/ D B , we have B  .c/;.d/,whence we get .c/D .d/DB , proving .c;d/ 2 ker. Then .c/DB  .a/ alsoyields .a;c/ 2D ./. Hence .a;c/ 2 R\D ./. Similarly, .d/D B  .b/ yields.d;b/ 2E ./. Thus .d;b/ 2 R\E ./. Now, .a;c/ 2 R\D ./, .c;d/ 2 ker and.d;b/ 2R\E ./ together imply .a;b/ 2 .R\D .//ı ker ı .R\E .//, proving  .R\D .//ı ker ı .R\E .//.Conversely, assume that (F) holds, i.e.  D .R\D .//ı ker ı .R\E .//. Wewill prove (E’), which is equivalent to (E) by Theorem 1. Since D ./ ıR   isalways true, we have to show only the converse inclusion. Indeed, take any .a;b/ 2D./ıR. Then there exist some elements d;c 2U such that .a;c/2R\D ./, .c;d/2ker, and .d;b/ 2 R\ E ./. Hence .c/ D .d/. As .a;c/ 2D ./ means that.c/  .a/, we get also .d/  .a/, i.e. .a;d/ 2D ./. Since .d;b/ 2 R, weobtain .a;b/ 2D ./ıR. Hence  D ./ıR, and this means that condition (E’) issatisfied. ab cd ab cdFIGURE 1. Relations R and  satisfying condition (F)TABLE 1. The contexts .U;U;Rc/ and .U;U;c/Rc a b c dab  c  d  c a b c dab c d  Corollary 3. Let R; be two tolerance relations on U such that R  . If Rand  satisfy condition .F / and R is induced by an irredundant covering of U then.}.U /;/ is a complete sublattice of  }.U /R; and  }.U /; is a completesublattice of .}.U /R;/.Corollary 4. Let R; be two tolerance relations on U such that R  , condition.F / holds and R  ker. Then  is an equivalence.252 D. GE´GE´NY AND I. PILLERProof. If R  ker then R\D ./  ker and R\E ./  ker. This impliesthat .R\D .//ı ker ı .R\E .//  ker ı ker ı ker D ker, since ker isan equivalence. Thus, we get   ker using condition .F /. Since  is a tolerancerelation, ker   also holds, meaning D ker, therefore  is an equivalence. .¿;U /.fbg;fc;dg/.fb;cg;fdg/.fcg;fb;dg/.fb;dg;fcg/.fdg;fb;cg/.fc;dg;fbg/.U;¿/.¿;U /.U;¿/.fdg;fb;cg/.fb;cg;fdg/FIGURE 2. The Hasse-diagrams of the concept latticesL.U;U;Rc/and L.U;U;c/Corollary 5. Let R; be two tolerance relations on U such that R  . If  is anequivalence, condition (F) automatically holds.Proof. Since  is an equivalence,  D ker DD ./ DE ./. Therefore R\ D./DR\E ./DR\DR. Then it follows that .R\D .//ı kerı .R\E .//D R ı  ıR. However, R ı  ıR   ı  ı  D . On the other hand, R ı  ıR ııD , where is the identity relation. Combining them yieldsRııRD ,which proves condition .F /. APPROXIMATION LATTICES DEFINED BY TOLERANCES INDUCED 2535. CONCLUSIONIn this paper, we were aiming to extend the characterization found in [2] by fur-ther investigating the tolerance relations R  . We deduced condition (F), which isequivalent to the conditions in Theorem 1 whenever R is a tolerance induced by anirredundant covering. Additionally, an algorithm for checking condition (D) was alsoprovided. An example of two relations satisfying condition (F) can be seen in Figure1. Since tolerance relations are always reflexive, loops are not noted on the figurefor simplicity. Figure 2 shows that the concept lattice L.U;U;c/ is a complete sub-lattice of L.U;U;Rc/, i.e. condition (C) holds. We also proved some consequencesfor special cases, e.g. R being a subrelation of ker. As a future work, we pro-pose investigating the results in [2] regarding the so-called compatibility condition incombination with tolerances induced by an irredundant covering.REFERENCES[1] B. Ganter and R. Wille, Formal concept analysis: mathematical foundations. Springer Science& Business Media, 2012.[2] D. Ge´ge´ny, I. Piller, S. Radeleczki, and L. Veres, “Approximations induced by tolerance relations,”2019.[3] J. Ja¨rvinen, L. Kova´cs, and S. Radeleczki, “Defining rough sets using tolerances compatible withan equivalence,” 2019.[4] J. Ja¨rvinen, P. Pagliani, and S. Radeleczki, “Information completeness in Nelson algebras ofrough sets induced by quasiorders,” Studia logica, vol. 101, no. 5, pp. 1073–1092, 2013, doi:10.1007/s11225-012-9421-z.[5] J. Ja¨rvinen and S. 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Pomykala, “On similarity based approximation of information,” Demonstratio Mathematica,vol. 27, no. 3-4, pp. 663–671, 1994, doi: 10.1515/dema-1994-3-412.[11] S. Rissino and G. Lambert-Torres, “Rough set theory-fundamental concepts, principals, data ex-traction, and applications,” in Data mining and knowledge discovery in real life applications.InTech, 2009. doi: 10.5772/6440.[12] Y. Yao, “Generalized rough set models,” Rough sets in knowledge discovery, vol. 1, pp. 286–318,1998.[13] W. Zakowski, “Approximations in the space (u, ),” Demonstratio mathematica, vol. 16, no. 3,pp. 761–770, 1983, doi: 10.1515/dema-1983-0319.254 D. GE´GE´NY AND I. PILLERAuthors’ addressesD. Ge´ge´nyUniversity of Miskolc, Institute of Mathematics, H-3515 Miskolc-Egyetemva´ros, HungaryE-mail address: matgd@uni-miskolc.huI. PillerUniversity of Miskolc, Institute of Mathematics, H-3515 Miskolc-Egyetemva´ros, HungaryE-mail address: imre.piller@uni-miskolc.hu

Approximation lattices defined by tolerances induced by irredundant coverings

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