Notes on the lattice of fuzzy rough sets with crisp reference sets
aa r X i v : . [ m a t h . G M ] J u l Notes on the lattice of fuzzy rough sets with crispreference sets ✩ Dávid Gégény a,1, ∗ , László Kovács b,1 , Sándor Radeleczki a,1 a Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary b Department of Information Technology, University of Miskolc, 3515Miskolc-Egyetemváros, Hungary
Abstract
Since the theory of rough sets was introduced by Zdzislaw Pawlak, severalapproaches have been proposed to combine rough set theory with fuzzy settheory. In this paper, we examine one of these approaches, namely fuzzyrough sets with crisp reference sets, from a lattice-theoretic point of view.We connect the lower and upper approximations of a fuzzy relation R to theapproximations of the core and support of R . We also show that the latticeof fuzzy rough sets corresponding to a fuzzy equivalence relation R and thecrisp subsets of its universe is isomorphic to the lattice of rough sets for the(crisp) equivalence relation E , where E is the core of R . We establish aconnection between the exact (fuzzy) sets of R and the exact (crisp) sets ofthe support of R . Keywords: fuzzy rough set, lower and upper approximation, fuzzy equivalence,uncertain knowledge, regular double Stone lattice, dually well-ordered set ∗ Corresponding author
Email addresses: [email protected] (Dávid Gégény), [email protected] (László Kovács), [email protected] (SándorRadeleczki)
URL: (Dávid Gégény), (LászlóKovács), (Sándor Radeleczki) This work was carried out as part of the grant EFOP-3.6.1-16-00011: "Younger andRenewing University - Innovative Knowledge City - intelligent specialization", in theframework of the Széchenyi 2020 program. The realization of this project is supported bythe European Union, co-financed by the European Social Fund.
010 MSC:
1. Introduction
The notion of fuzzy sets and rough sets both extend the concept of tra-ditional (crisp) sets by incorporating that our knowledge may be uncertainor incomplete. However, these approaches address the problem of imperfectinformation in a different way.Rough sets were introduced by Zdzislaw Pawlak [1], and they use thelower and upper approximations of a (crisp) set based on the indiscernibilityrelation of the elements. Given a reference set A in a universe U and anequivalence relation R ⊆ U × U , the lower approximation of the set A is A R = { x ∈ U | [ x ] R ⊆ A } and the upper approximation of A is A R = { x ∈ U | [ x ] R ∩ A = ∅} , where [ x ] R is the R -equivalence class of an element x . The pair ( A R , A R ) iscalled the rough set corresponding to the reference set A and ( U, R ) is calledan approximation space . The rough sets corresponding to this approximationspace ( U, R ) can be ordered with respect to the component-wise inclusion,and they form a complete lattice with several particular properties, denotedby RS ( U, R ) , see e.g. [2], [3] and [4].The theory of fuzzy sets was introduced by Lotfi Zadeh [5]. A fuzzy set A is defined by a membership function µ A : U −→ [0 , . The membershipdegree 0 means that the element is certainly not a member of the set A , andthe membership degree 1 means that the element is certainly in the set.One of the pioneer works to analyze the relationship between the twomain theories can be found in [6], where the author had shown that thereare significant differences between these concepts. The first approach tointegrate the two main theories relates to the work of Dubois and Prade[7]. The proposed lower and upper approximations for fuzzy sets are definedusing the t-norm Min and its dual co-norm Max. Using the symbolic notationintroduced by Yao in [8], the fuzzy rough set of a fuzzy set Γ is defined with µ apr R (Γ) ( x ) = inf { max [ µ Γ ( y ) , − µ R ( x, y )] | y ∈ U } , apr R (Γ) ( x ) = sup { min [ µ Γ ( y ) , µ R ( x, y )] | y ∈ U } , where U denotes the universe set and R is the symbol for a fuzzy similarityrelation. As the definition shows fuzzy rough sets are rough sets having fuzzysets as lower and upper approximations attached to a fuzzy reference set.As crisp sets are special cases of fuzzy sets (having two-valued membershipfunctions), the given definition can also be used to construct fuzzy rough setsfor crisp sets. A comparison of the two approaches can also be found in [9].Beside some other generalization approaches like Nanda and Majumdar[10], we can also find some different proposals for integration. The workin [11] had shown that fuzzy rough sets are, indeed, intuitionistic L-fuzzysets developed by Atanassov [12]. More general framework can be obtainedunder fuzzy environment based on fuzzy similarity relations defined by t-norms, see e.g. [13] or [14]. In [13], the upper and lower approximations of afuzzy subset with respect to an indistinguishability operator are studied, andtheir relations with fuzzy rough sets are pointed out. In [14], an axiomaticapproach is developed; using fuzzy similarity relations defined by a t-norm,the definition of the upper and lower approximation operator in case of fuzzyrough sets is generalized based on some axiomatic properties.The integration proposal of Yao [8] is based on the consideration thata fuzzy set can be represented by a family of crisp sets using its α -levelsets, whereas a rough set can be represented by three crisp sets. Yao hasanalyzed the relationship between the rough fuzzy set and fuzzy rough setmodels and proved that rough fuzzy sets are special cases of fuzzy roughsets as defined by Dubois and Prade. Another conclusion of [8] is that themembership functions of rough sets, rough fuzzy sets, and fuzzy rough setscan be computed uniformly using the same scheme: µ apr Γ (∆) ( x ) := inf { max [ µ ∆ ( y ) , − µ Γ ( x, y )] | y ∈ U } ,µ apr Γ (∆) ( x ) := sup { min [ µ ∆ ( y ) , µ Γ ( x, y )] | y ∈ U } , where Γ is a variable that takes either an equivalence relation or a fuzzysimilarity relation as its value, and ∆ is a variable that takes either a crisp setor a fuzzy set as its value. The properties of the general case that uses fuzzyreference sets in a fuzzy approximation space defined by a t-norm are alsoexamined in [15], where an application in query refinement is also presented.3he main application area of the fuzzy rough set theory relates to optimi-sation of knowledge engineering algorithms. Regarding the data preprocess-ing phase, the fuzzy rough set models are used mainly for attribute reduction[16], [17]. The main benefit of this approach is that fuzzy-rough feature ex-traction preserves the meaning, the semantics of the selected features afterelimination of the redundant attributes. The FRFS method works with dis-covering dependencies between the elements of the attribute set. The fuzzyrough set model can also be used for general data mining operations, likeclustering or classification in the case of uncertain input domains [18].The main focus of this paper is on fuzzy rough sets, using crisp sets asreference sets in a fuzzy approximation space. Fuzzy rough sets with crispreference sets are important modelling tools in machine learning applications,like in natural language processing, where the reference sets contain crisp val-ued feature vectors and we construct fuzzy concept categories correspondingto them (see e.g. [19] and [20]). Our aim is to examine the lattice-theoreticalproperties of fuzzy rough sets and to draw a comparison study to traditionalrough sets. We show that in case of crisp reference sets, the lattice of fuzzyrough sets corresponding to a fuzzy equivalence relation R is isomorphic tothe lattice of rough sets for the (crisp) equivalence relation E , where E is thecore of R , and this is a much investigated structure in the literature.Let ( U, R ) be a fuzzy approximation space, where U is the universe and R is a fuzzy equivalence relation defined by a mapping µ R : U −→ [0 , .A fuzzy equivalence relation is a reflexive, symmetric and transitive fuzzyrelation. As we are considering fuzzy relations, reflexive property means that µ R ( x, x ) = 1 for every x ∈ U and symmetry means that µ R ( x, y ) = µ R ( y, x ) for every x, y ∈ U . Initially, a fuzzy relation R was called transitive ifmin ( µ R ( x, y ) , µ R ( y, z )) ≤ µ R ( x, z ) , for all x, y, z ∈ U [21]. Later this notionwas generalized by using the notion of a t-norm (see e.g. [22]). A triangularnorm T ( t-norm for short) is an increasing commutative and associativemapping T : [0 , −→ [0 , satisfying T (1 , x ) = T ( x,
1) = x , for all x ∈ [0 , . The t-norm T is called positive (see e.g . [23]) , if T ( x, y ) > , whenever x, y > . We say that a fuzzy relation µ R : U −→ [0 , is T -transitive , if T ( µ R ( x, y ) , µ R ( y, z )) ≤ µ R ( x, z ) , for all x, y, z ∈ U. A reflexive, symmetric and T -transitive fuzzy relation R is called a T -equivalence, or a fuzzy T -similarity relation. It is well-known that T ( x, y ) = min ( x, y ) , x, y ∈ [0 , is a positive t-norm corresponding to the previous4otion of transitivity.Now, let A ⊆ U be a crisp set. A fuzzy rough set with reference set A is defined as a pair of two fuzzy sets corresponding to A [8]. The lowerapproximation of A is given by the membership function µ [ A ] R ( x ) = inf { − µ R ( x, y ) | y / ∈ A } , and the upper approximation of A is given by the membership function µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } . It is easy to check that µ [ ∅ ] R = µ [ ∅ ] R = , where denotes the constant mapping on U , and µ [ U ] R = µ [ U ] R = , where stands for the constant mapping on U . (Notice that sup ∅ = 0 , inf ∅ = 1 , and µ R ( x, x ) = 1 , for any x ∈ U .) The fuzzy rough set corresponding to the crisp set A is the pair ( µ [ A ] R , µ [ A ] R ) . We know that the set of all rough sets in approximation space ( U, R ) form a lattice with several interesting properties (see e.g. [2], [3], [4],[24]). The goal of this paper is to examine the algebraic structure of fuzzyrough sets for such favorable properties and to draw a comparison to the caseof traditional rough sets.
2. Preliminary observations
Let R be a fuzzy relation with a map µ R : U −→ [0 , . The set S R := { µ R ( x, y ) | x, y ∈ U } ⊆ [0 , is called the spectrum of R . We say that a fuzzyrelation R has a dually well-ordered spectrum , if any nonempty subset of S R has a maximal element. This is equivalent to the fact that for any x ∈ U andany crisp set B ⊆ U , B = ∅ there exists at an element m x ∈ B such thatsup { µ R ( x, y ) | y ∈ B } = max { µ R ( x, y ) | y ∈ B } = µ R ( x, m x ) .Observe that this is the case when the spectrum S R of R is a finite set. If R has a dually well-ordered spectrum, then for any crisp set A ⊆ U , A = ∅ µ [ A ] R ( x ) = 1 − max { µ R ( x, y ) | y / ∈ A } , µ [ A ] R ( x ) = max { µ R ( x, y ) | y ∈ A } .A similar approach in case of finite (crisp) base sets can be found in [25], fordecision attributes of decision tables in order to introduce distance measures5n fuzzy rough sets. As we pointed out previously the fuzzy rough set corre-sponding to a crisp set A is a pair of mappings ( µ [ A ] R , µ [ A ] R ) . Let us denotethe collection of these pairs by RS ( U, R ) , i.e. let RS ( U, R ) := (cid:8)(cid:0) µ [ A ] R , µ [ A ] R (cid:1) | A ⊆ U (cid:9) .The elements of RS ( U, R ) can be ordered by the component-wise order asfollows: (cid:0) µ [ A ] R , µ [ A ] R (cid:1) ≤ (cid:0) µ [ B ] R , µ [ B ] R (cid:1) ⇔⇔ µ [ A ] R ( x ) ≤ µ [ B ] R ( x ) and µ [ A ] R ( x ) ≤ µ [ B ] R ( x ) , for all x ∈ U ,obtaining a poset ( RS ( U, R ) , ≤ ) with least element ( , ) and greatest ele-ment ( , ) . In other words, this order is a particular case of the productlattice order. We will prove that for any fuzzy equivalence relation R with adually well-ordered spectrum, this poset is a complete lattice.For any number α ∈ [0 , , the crisp relation R α := { ( x, y ) ∈ U | µ R ( x, y ) ≥ α } is called an α -section ( α -level ) of the fuzzy relation R . If R is a fuzzyequivalence, then R α is a crisp equivalence for any α ∈ [0 , . Denote by E the crisp equivalence R , i.e. let E := { ( x, y ) ∈ U | µ R ( x, y ) = 1 } . The E -equivalence class of an element x ∈ U will be denoted by [ x ] E . Hence [ x ] E = { y ∈ U | ( x, y ) ∈ E } = { y ∈ U | µ R ( x, y ) = 1 } .The following lemma is well-known in the literature, see e.g. [26]: Lemma 1.
For any y ∈ [ x ] E and z ∈ U we have µ R ( z, x ) = µ R ( z, y ) . Now, let S be the support of the fuzzy equivalence relation R with mem-bership function µ R , i.e. let S = { ( x, y ) ∈ U | µ R ( x, y ) > } , and define S ( z ) = { y ∈ U | ( z, y ) ∈ S } , where U is the universe of R and z ∈ U is an arbitrary element. Obviously, the binary relation S is reflexiveand symmetric and S ( z ) = { y ∈ U | µ R ( z, y ) > } 6 = ∅ , for any z ∈ U .Next, assume that T is a positive t-norm and let ( x, y ) ∈ S and ( y, z ) ∈ S for some x, y, z ∈ U . This means that µ R ( x, y ) > and µ R ( y, z ) > .6f R is a T -equivalence we obtain: µ R ( x, z ) ≥ T ( µ R ( x, y ) , µ R ( y, z )) > .Therefore, µ R ( x, z ) > , from which it follows ( x, z ) ∈ S , meaning that S is an equivalence relation as well. As before, the S -equivalence class of anelement x will be denoted by [ x ] S , and clearly S ( x ) = [ x ] S .Using the above defined crisp relations E ⊆ U × U and S ⊆ U × U , wecan assign (crisp) rough sets to any reference set A ⊆ U , by defining its lowerand upper approximation with respect to E or S : A E = { x ∈ U | [ x ] E ⊆ A } , A E = { x ∈ U | [ x ] E ∩ A = ∅} ,A S = { x ∈ U | S ( x ) ⊆ A } , A S = { x ∈ U | S ( x ) ∩ A = ∅} . Lemma 2.
For any subset A ⊆ U we have(i) A E = { x ∈ U | µ [ A ] R ( x ) = 1 } ,(ii) A E = { x ∈ U | µ [ A ] R ( x ) > } ,(iii) A S = { x ∈ U | µ [ A ] R ( x ) > } ,(iv) A S = { x ∈ U | µ [ A ] R ( x ) = 1 } .In other words, assertions (i) and (ii) in Lemma 2 mean that A E is equalto the core of the fuzzy set corresponding to µ [ A ] R , whereas A E is equal tothe support of the fuzzy set corresponding to µ [ A ] R . Similarly, (iii) and (iv)mean that A S is equal to the support of the fuzzy set corresponding to µ [ A ] R ,whereas A S is equal to the core of the fuzzy set corresponding to µ [ A ] R .Proof. (i) If x ∈ A E , then there is a y ∈ A with ( x, y ) ∈ E , i.e. µ R ( x, y ) = 1 .Hence µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } = 1 . Conversely, suppose that µ [ A ] R ( x ) = 1 for some x ∈ U . Since R has a dually well-ordered spectrum,this means that max { µ R ( x, y ) | y ∈ A } = 1 , i.e. there exists a y x ∈ A , with µ R ( x, y x ) = 1 . Then ( x, y x ) ∈ E , whence [ x ] E ∩ A = ∅ . This yields x ∈ A E .(ii) If x ∈ A E , then [ x ] E ⊆ A . This means that there is no y / ∈ A with ( x, y ) ∈ E , i.e. such that µ R ( x, y ) = 1 . Since R has a dually well-orderedspectrum, the set { µ R ( x, y ) | y / ∈ A } has (at least one) maximal element µ R ( x, y m ) , where y m / ∈ A . Then µ R ( x, y m ) < , and we obtain µ [ A ] R ( x ) =1 − max { µ R ( x, y ) | y / ∈ A } = 1 − µ R ( x, y m ) > . Conversely, assume that µ [ A ] R ( x ) > , for some x ∈ U . Then for any y / ∈ A we get − µ R ( x, y ) ≥ inf { − µ R ( x, y ) | y / ∈ A } = µ [ A ] R ( x ) > . This implies µ R ( x, y ) < , for each y / ∈ A . Hence there is no y / ∈ A with µ R ( x, y ) = 1 , i.e.with ( x, y ) ∈ E . This yields [ x ] E ⊆ A . Hence x ∈ A E , and this proves (ii).7iii) Assume that x ∈ A S . This can only be if there exists a y m ∈ A with µ R ( x, y m ) > . Then µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } ≥ µ R ( x, y m ) > ,yielding that x is in the support of µ [ A ] R . Conversely, assume that x is in thesupport of µ [ A ] R , meaning that µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } > . Thiscan only happen if there exists y m ∈ A with µ R ( x, y m ) > , implying x ∈ A S .(iv) Let x ∈ A S , i.e. S ( x ) ⊆ A . Then, by the definition of S , for every y / ∈ A, µ R ( x, y ) = 0 . Thus, µ [ A ] R ( x ) = inf { − µ R ( x, y ) | y / ∈ A } = 1 − ,i.e. x is in the core of µ [ A ] R ( x ) . The reverse implication yielding S ( x ) ⊆ A can be easily checked.The assertion of the following proposition is implicitly contained in [15].In fact, it is based on the following observation:Having two fuzzy (or crisp) equivalence relations E and R with E ⊆ R , if wecalculate the upper approximation of a fuzzy (or crisp) set µ for E and thenthe upper approximation of the obtained fuzzy set for R , we get the upperapproximation for µ by R . Dually for lower approximations (see also [9]). Proposition 1.
Let R be a fuzzy T -equivalence on U with a dually well-ordered spectrum and E := { ( x, y ) ∈ U | µ R ( x, y ) = 1 } . Then for any set A ⊆ U we have µ [ A ] R = µ [ A E ] R and µ [ A ] R = µ [ A E ] R .
3. Main results
In what follows, denote as usually by ( RS ( U, E ) , ≤ ) the lattice of roughsets defined by the equivalence relation E . Theorem 1.
Let R be a fuzzy T -equivalence on U with a dually well-orderedspectrum. Then ( RS ( U, R ) , ≤ ) is a complete lattice isomorphic to ( RS ( U, E ) , ≤ ) .Proof. For each (crisp) rough set (cid:0) A E , A E (cid:1) ∈ RS ( U, E ) we will assign thefuzzy rough set corresponding to the crisp set A , i.e. the pair (cid:0) µ [ A ] R , µ [ A ] R (cid:1) .Observe, that the function f : RS ( U, E ) → RS ( U, R ) , f (cid:0)(cid:0) A E , A E (cid:1)(cid:1) = (cid:0) µ [ A ] R , µ [ A ] R (cid:1) , where (cid:0) A E , A E (cid:1) ∈ RS ( U, E ) ,is well-defined, because (cid:0) A E , A E (cid:1) = (cid:0) B E , B E (cid:1) for some A, B ⊆ U implies A E = B E , A E = B E , and hence, in view of Proposition 1, we obtain f (cid:0)(cid:0) A E , A E (cid:1)(cid:1) = (cid:0) µ [ A ] R , µ [ A ] R (cid:1) = (cid:0) µ [ A E ] R , µ [ A E ] R (cid:1) = (cid:0) µ [ B E ] R , µ [ B E ] R (cid:1) = (cid:0) µ [ B ] R , µ [ B ] R (cid:1) = f (cid:0)(cid:0) B E , B E (cid:1)(cid:1) . In addition, f is order-preserving because (cid:0) A E , A E (cid:1) ≤ (cid:0) B E , B E (cid:1) implies A E ⊆ B E , A E ⊆ B E , and this yields µ [ A E ] R ≤ µ [ B E ] R and µ [ A E ] R ≤ µ [ B E ] R .Thus we obtain: f (cid:0)(cid:0) A E , A E (cid:1)(cid:1) = (cid:0) µ [ A E ] R , µ [ A E ] R (cid:1) ≤ (cid:0) µ [ B E ] R , µ [ B E ] R (cid:1) = f (cid:0)(cid:0) B E , B E (cid:1)(cid:1) .Clearly, f is onto, since for any (cid:0) µ [ X ] R , µ [ X ] R (cid:1) ∈ RS ( U, R ) , X ⊆ U is a crispset, and hence f (cid:0)(cid:0) X E , X E (cid:1)(cid:1) = (cid:0) µ [ X ] R , µ [ X ] R (cid:1) . Now, to prove that f is anorder-isomorphism, it suffices to show that f (cid:0)(cid:0) A E , A E (cid:1)(cid:1) ≤ f (cid:0)(cid:0) B E , B E (cid:1)(cid:1) implies (cid:0) A E , A E (cid:1) ≤ (cid:0) B E , B E (cid:1) , for any (cid:0) A E , A E (cid:1) , (cid:0) B E , B E (cid:1) ∈ RS ( U, E ) .Indeed, f (cid:0)(cid:0) A E , A E (cid:1)(cid:1) ≤ f (cid:0)(cid:0) B E , B E (cid:1)(cid:1) yields that (cid:0) µ [ A ] R ( x ) , µ [ A ] R ( x ) (cid:1) ≤ (cid:0) µ [ B ] R ( x ) , µ [ B ] R ( x ) (cid:1) , for all x ∈ U . Hence we get µ [ A ] R ( x ) ≤ µ [ B ] R ( x ) and µ [ A ] R ( x ) ≤ µ [ B ] R ( x ) , for any x ∈ U . Now, in view of Lemma 2, we obtain: A E = { x ∈ U | µ [ A ] R ( x ) = 1 } ⊆ { x ∈ U | µ [ B ] R ( x ) = 1 } = B E , and A E = { x ∈ U | µ [ A ] R ( x ) > } ⊆ { x ∈ U | µ [ B ] R ( x ) > } = B E .Hence (cid:0) A E , A E (cid:1) ≤ (cid:0) B E , B E (cid:1) , and this proves that f is an order-isomorphism.Since ( RS ( U, E ) , ≤ ) is a complete lattice, we obtain that ( RS ( U, R ) , ≤ ) isalso a complete lattice isomorphic to ( RS ( U, E ) , ≤ ) .As an immediate consequence, in view of [2] [3] [24] we obtain: Corollary 1. If R is a fuzzy equivalence on the set U with a dually well-ordered spectrum, then ( RS ( U, R ) , ≤ ) is a completely distributive regular dou-ble Stone lattice.Proof. It is known that the rough set lattice ( RS ( U, E ) , ≤ ) is completelydistributive regular double Stone lattice. Hence Corollary 1 is obtained byapplying the isomorphism established in Theorem 1. Example 1.
Let the universe be U = { a, b, c, d, e } and the fuzzy equivalencerelation R be given by Table 1. The corresponding E = { ( x, y ) ∈ U | µ R ( x, y ) = 1 } relation can be seen on Figure 1 (loops are not noted forsimplicity). Table 2 shows the lower and upper approximations of fuzzyrelation R and of the (crisp) equivalence relation E . Figure 2 shows theHasse-diagram of the lattice ( RS ( U, R ) , ≤ ) . Here, the nodes are representedas tables, where the top row represents the membership function of the upperapproximation of R , and the bottom row represents the membership functionof the lower approximation of R . 9 a b c d ea b c d e Table 1: An example fuzzy equivalence relation R . ab c de Figure 1: The equivalence relation E corresponding to R . A A E A E µ [ A ] R ( x ) µ [ A ] R ( x ) a b c d e a b c d e ∅ ∅ ∅ ∅ ab 0 0 0 0 0 1 1 0.5 0 0b ∅ ab 0 0 0 0 0 1 1 0.5 0 0c c c 0 0 0.5 0 0 0.5 0.5 1 0 0d ∅ de 0 0 0 0 0 0 0 0 1 1e ∅ de 0 0 0 0 0 0 0 0 1 1ab ab ab 0.5 0.5 0 0 0 1 1 0.5 0 0ac c abc 0 0 0.5 0 0 1 1 1 0 0ad ∅ abde 0 0 0 0 0 1 1 0.5 1 1ae ∅ abde 0 0 0 0 0 1 1 0.5 1 1bc c abc 0 0 0.5 0 0 1 1 1 0 0bd ∅ abde 0 0 0 0 0 1 1 0.5 1 1be ∅ abde 0 0 0 0 0 1 1 0.5 1 1cd c cde 0 0 0.5 0 0 0.5 0.5 1 1 1ce c cde 0 0 0.5 0 0 0.5 0.5 1 1 1de de de 0 0 0 1 1 0 0 0 1 1abc abc abc 1 1 1 0 0 1 1 1 0 0abd ab abde 0.5 0.5 0 0 0 1 1 0.5 1 1abe ab abde 0.5 0.5 0 0 0 1 1 0.5 1 1acd c U 0 0 0.5 0 0 1 1 1 1 1ace c U 0 0 0.5 0 0 1 1 1 1 1ade de abde 0 0 0 1 1 1 1 0.5 1 1bcd c U 0 0 0.5 0 0 1 1 1 1 1bce c U 0 0 0.5 0 0 1 1 1 1 1bde de abde 0 0 0 1 1 1 1 0.5 1 1cde cde cde 0 0 0.5 1 1 0.5 0.5 1 1 1abcd abc U 1 1 1 0 0 1 1 1 1 1abce abc U 1 1 1 0 0 1 1 1 1 1abde abde abde 0.5 0.5 0 1 1 1 1 0.5 1 1acde cde U 0 0 0.5 1 1 1 1 1 1 1bcde cde U 0 0 0.5 1 1 1 1 1 1 1U U U 1 1 1 1 1 1 1 1 1 1 Table 2: Approximations on U given by the equivalence E and fuzzy relation R . Figure 2: Lattice of ( RS ( U, R ) , ≤ ) based on Example 1. A rough set is called exact if the lower approximation and the upperapproximation of the set are equal. This notion can be extended to fuzzyrough sets. A fuzzy rough set defined by the fuzzy equivalence R is exact iffor every x ∈ U , µ [ A ] R ( x ) = µ [ A ] R ( x ) holds, where U is the universe of R .The following proposition describes the relationship between exact fuzzyrough sets and the support of the fuzzy equivalence relation. Proposition 2.
Let A be a (crisp) subset of U . Then µ [ A ] R ( x ) = µ [ A ] R ( x ) for all x ∈ U ⇔ A S = A S . Proof.
Suppose that x ∈ A S = A S = A . This means that µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } = µ R ( x, x ) = 1 , since R is reflexive.Now let us examine the lower approximation. If y / ∈ A , then y / ∈ S ( x ) either,because S ( x ) ⊆ A . Since y / ∈ S ( x ) , according to the definition of S , it followsthat µ R ( x, y ) = 0 . This is true for every y / ∈ A , yielding µ [ A ] R ( x ) = inf { − µ R ( x, y ) | y / ∈ A } = 1 . So we obtain in this case that µ [ A ] R ( x ) = µ [ A ] R ( x ) = 1 .11ow, let x / ∈ A S = A S = A . Then µ R ( x, y ) = 0 for each y ∈ A , and we have µ [ A ] R ( x ) = sup { µ R ( x, y ) | y ∈ A } = 0 , and µ [ A ] R ( x ) = inf { − µ R ( x, y ) | y / ∈ A } = 1 − µ R ( x, x ) = 0 . Hence in this case we obtain µ [ A ] R ( x ) = µ [ A ] R ( x ) = 0 .Therefore, we proved that A S = A S yields µ [ A ] R ( x ) = µ [ A ] R ( x ) , for all x ∈ U .Conversely, assume that µ [ A ] R ( x ) = µ [ A ] R ( x ) , for all x ∈ U . Let x ∈ A bearbitrary. Thensup { µ R ( x, y ) | y ∈ A } = inf { − µ R ( x, y ) | y / ∈ A } = 1 , because µ R ( x, x ) = 1 and µ R ( x, y ) ≤ for all y ∈ A . We conclude µ R ( x, y ) =0 , for all y / ∈ A , otherwise the infimum on the right side would be strictlyless than .Assume ( x, y ) ∈ S , i.e. y ∈ S ( x ) . Then, µ R ( x, y ) > by the definition of S , so y / ∈ A is not possible. Thus, we get y ∈ A and this implies x ∈ A S .Hence, A = A S . Then A S = ( A S ) S ⊆ A implies A S = A = A S . Example 2.
Let U = { a, b, c, d } and let R be a fuzzy relation given by Table3. Table 4 shows the four sets for which this relation yields exact sets as lowerand upper approximations for R and for S (the support of R ). R a b c da b c d Table 3: An example fuzzy equivalence relation R . A A S = A S µ [ A ] R = µ [ A ] R ∅ ∅ { ( a, , ( b, , ( c, , ( d, }{ d } { d } { ( a, , ( b, , ( c, , ( d, }{ a, b, c } { a, b, c } { ( a, , ( b, , ( c, , ( d, }{ a, b, c, d } { a, b, c, d } { ( a, , ( b, , ( c, , ( d, } Table 4: Exact fuzzy sets of relation R from Table 3.
12t can be verified that the containment relationship between a base set A and its fuzzy rough approximations is similar to the containment relationshipbetween the base set and its crisp rough approximations, namely:core ( µ [ A ] R ) ⊆ support ( µ [ A ] R ) ⊆ A,A ⊆ core ( µ [ A ] R ) ⊆ support ( µ [ A ] R ) . Remark 1.
An important simple case should also be discussed: when wehave imperfect information and we are uncertain about setting up the rela-tion. In this simple case, the relationship between two elements can havethree possibilities: the elements are certainly related; the elements are cer-tainly not related; the elements might be related, but we are uncertain. Wemodel this with a fuzzy relation R , for which µ R ( x, y ) = , if x and y are certainly related , if x and y are certainly not related , if x and y might be related, but we are uncertain . It can be easily checked that for a crisp set A ⊆ U , the membershipfunctions of the lower and upper approximations can be given as follows:(i) µ [ A ] R ( x ) = , if x / ∈ A E , if x ∈ A E \ A S , if x ∈ A S , (ii) µ [ A ] R ( x ) = , if x / ∈ A S , if x ∈ A S \ A E , if x ∈ A E .
4. Conclusions and further work
In this paper, we examined the lattice of fuzzy rough sets correspondingto a fuzzy equivalence relation R . We also investigated the relationshipbetween the core/support of the approximations of a fuzzy rough set andthe (crisp) approximations corresponding to the core/support of R . We haveshown that the lattice of fuzzy rough sets is isomorphic to the lattice ofrough sets corresponding to E , the core of R . We also proved that the13embership function of an exact fuzzy set (where A ⊆ U is a crisp set and µ [ A ] R ( x ) = µ [ A ] R ( x ) for every x ∈ U ) is the same as the characteristic functionof a (regular) exact set corresponding to S , the support of R .We can extend the investigation of the E -based approximation to the α -cut R -approximation. The related R α crisp relation for different α -levels canbe defined in the following way: R α = { ( x, y ) | µ R ( x, y ) ≥ α } . It is known that R α is also an equivalence relation whenever R is a fuzzyequivalence. Then we can give the following result: A R α = { x ∈ U | µ [ A ] R ( x ) ≥ α } ,A R α = { x ∈ U | µ [ A ] R ( x ) > − α } . As a general case, we would like to extend our results using the frame-work presented in [15], and verifying the lattice-theoretical properties of thegenerated fuzzy rough sets.
Acknowledgement
The authors would like to thank the area editor and the reviewers fortheir valuable comments and suggestions.
References [1] Z. Pawlak, Rough sets, International Journal of Com-puter & Information Sciences 11 (5) (1982) 341–356. doi:https://doi.org/10.1007/BF01001956 .[2] J. Pomykała, J. Pomykała, The Stone algebra of rough sets, Bulletin ofPolish Academy of Sciences. Mathematics 36 (1988) 495––512.[3] S. D. Comer, On connections between information systems, rough setsand algebraic logic, Banach Center Publications 28 (1993) 117–124. doi:https://doi.org/10.4064/-28-1-117-124 .[4] I. Düntsch, A logic for rough sets, Theo-retical Computer Science 179 (1997) 427–436. doi:https://doi.org/10.1016/S0304-3975(96)00334-9 .145] L. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338–353. doi:https://doi.org/10.1016/S0019-9958(65)90241-X .[6] Z. Pawlak, Rough sets and fuzzy sets, FuzzySets and Systems 17 (1) (1985) 99–102. doi:https://doi.org/10.1016/S0165-0114(85)80029-4 .[7] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, In-ternational Journal of General Systems 17 (2-3) (1990) 191–209. doi:https://doi.org/10.1080/03081079008935107 .[8] Y. Y. Yao, Combination of rough sets and fuzzy sets based on α -level sets, Rough Sets and Data Mining (15) (1997) 301–321. doi:https://doi.org/10.1007/978-1-4613-1461-5\_15 .[9] J. Järvinen, L. Kovács, S. Radeleczki, Rough sets de-fined by multiple relations, Rough Sets. IJCRS 2019. Lec-ture Notes in Computer Science 11499 (2019) 40–51. doi:https://doi.org/10.1007/978-3-030-22815-6\_4 .[10] S. Nanda, S. Majumdar, Fuzzy rough sets,Fuzzy Sets and Systems 45 (2) (1992) 157–160. doi:https://doi.org/10.1016/0165-0114(92)90114-J .[11] D. Çoker, Fuzzy rough sets are intuitionistic l-fuzzysets, Fuzzy Sets and Systems 96 (3) (1998) 381–383. doi:https://doi.org/10.1016/S0165-0114(97)00249-2 .[12] K. T. Atanassov, Intuitionistic fuzzy sets,Fuzzy Sets and Systems 20 (1) (1986) 87–96. doi:https://doi.org/10.1016/S0165-0114(86)80034-3 .[13] D. Boixader, J. Jacas, J. Recasens, Upper and lower approximationsof fuzzy sets, International Journal of General Systems 29 (4) (2000)555–568. doi:https://doi.org/10.1080/03081070008960961 .[14] N. N. Morsi, M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Setsand Systems 100 (1998) 327–342. doi:10.1080/03081070008960961 .[15] C. Cornelis, M. De Cock, A. Radzikowska, Fuzzy RoughSets: From Theory into Practice, 2008, Ch. 24, pp. 533–552. doi:https://doi.org/10.1002/9780470724163.ch24 .1516] J. Dai, Q. Xu, Attribute selection based on information gainratio in fuzzy rough set theory with application to tumorclassification, Applied Soft Computing 13 (1) (2013) 211–221. doi:https://doi.org/10.1016/j.asoc.2012.07.029 .[17] L. I. Kuncheva, Fuzzy rough sets: Application to featureselection, Fuzzy Sets and Systems 51 (2) (1992) 147–153. doi:https://doi.org/10.1016/0165-0114(92)90187-9 .[18] H.-L. Yang, S.-G. Li, S. Wang, J. Wang, Bipolar fuzzyrough set model on two different universes and its ap-plication, Knowledge-Based Systems 35 (2012) 94–101. doi:https://doi.org/10.1016/j.knosys.2012.01.001 .[19] Y. Ji, L. Shang, X. Dai, R. Ma, Apply a rough set-based classifier to de-pendency parsing, International Conference on Rough Sets and Knowl-edge Technology. Lecture Notes in Computer Science 5009 (2008) 97–105. doi:https://doi.org/10.1007/978-3-540-79721-0\_18 .[20] D. Gupta, V. K, C. K. Singh, Using natural language processing tech-niques and fuzzy-semantic similarity for automatic external plagiarismdetection, in: 2014 International Conference on Advances in Computing,Communications and Informatics (ICACCI), 2014, pp. 2694–2699.[21] L. Zadeh, Similarity relations and fuzzy order-ings, Information Sciences 3 (2) (1971) 177–200. doi:https://doi.org/10.1016/S0020-0255(71)80005-1 .[22] L. Valverde, On the structure of f-indistinguishability op-erators, Fuzzy Sets and Systems 17 (3) (1985) 313–328. doi:https://doi.org/10.1016/0165-0114(85)90096-X .[23] I. Beg, S. Ashraf, Fuzzy equivalence relations, Kuwait Journal of Science& Engineering 35 (1A) (2008) 33–51.[24] M. Gehrke, E. Walker, On the structure of rough sets, Bulletin of PolishAcademy of Sciences. Mathematics 40 (1992) 235–245.[25] C. Wang, Y. Huang, M. Shao, X. Fan, Fuzzy roughset-based attribute reduction using distance measures,16nowledge-Based Systems 164 (15) (2019) 205–212. doi:https://doi.org/10.1016/j.knosys.2018.10.038 .[26] J. Recasens, Indistinguishability Operators, Modelling Fuzzy Equalitiesand Fuzzy Equivalence Relations, Vol. 260, Springer-Verlag Berlin Hei-delberg, 2011. doi:https://doi.org/10.1007/978-3-642-16222-0doi:https://doi.org/10.1007/978-3-642-16222-0