Lattice valued intuitionistic fuzzy sets

Transcription

1 CEJM 2(3) Lattice valued intuitionistic fuzzy sets Tadeusz Gerstenkorn 1, Andreja Tepavčević 2 1 Faculty of Mathematics, Lódź University, ul. S. Banacha 22, PL Lódź, Poland 2 Department of Mathematics and Informatics Fac. of Sci., University of Novi Sad Trg D. Obradovića 4, Novi Sad, Serbia and Montenegro Received 3 October 2003; accepted 11 June 2004 Abstract: In this paper a new definition of a lattice valued intuitionistic fuzzy set (LIFS) is introduced, in an attempt to overcome the disadvantages of earlier definitions. Some properties of this kind of fuzzy sets and their basic operations are given. The theorem of synthesis is proved: For every two families of subsets of a set satisfying certain conditions, there is an lattice valued intuitionistic fuzzy set for which these are families of level sets. c Central European Science Journals. All rights reserved. Keywords: Intuitionistic fuzzy sets, bifuzzy sets, lattice valued intuitionistic fuzzy sets MSC (2000): 03E72, 03B52, 06D72 1 Preliminaries Let X be a nonempty set, and f and g be two functions from X to [0, 1], such that for all x X, 0 f(x)+g(x) 1. By the original definition of Atanassov in [1], an intuitionistic fuzzy set is an object of the form: A = {(x, f(x),g(x)) x X}. This object is also called bifuzzy set tadger@math.uni.lodz.pl etepavce@eunet.yu The research supported by Serbian Ministry of Science and Technology, Grant No

2 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) (according to [5]). We consider it in a form of an ordered triple: A =(X, f, g), where X, f and g are as above. Definition of LIFS by involutive unary operation In the paper [2], Atanassov defined a lattice valued intuitionistic fuzzy set (intuitionistic L-fuzzy set), using a complete lattice L with an involutive order reversing unary operation N : L L. An intuitionistic L-fuzzy set is an object of the form A = {(x, f(x),g(x)) x X}, where f and g are functions f : X L, g : X L, such that for all x X, f(x) N(g(x)). We call this object intuitionistic L-fuzzy set of the first type (LIFS-1). The disadvantage of this definition is that very often such a required involutive, order reversing unary operation does not exist. LIFS-1 can not be defined for a large class of lattices ( there is e.g. a lattice of 6 elements without unary operation satisfying the desired properties). Moreover, some of the operators that are natural for ordinary intuitionistic fuzzy sets can not be defined over the LIFS (see [3], page 181). Definition of LIFS by a linearization function In the paper [8]a notion of an L-valued bifuzzy set is introduced in the following way. Let L be a lattice together with its linearization: L =(L,,,l), where (L,, ) isa complete lattice with the top element T, and the bottom element B. l is a mapping from L to the real interval [0, 1](l : L [0, 1]), for which from x y it follows that l(x) l(y) and l(t )=1,l(B) =0. A lattice valued bifuzzy (L-valued intuitionistic fuzzy) set is the ordered triple (X, f, g), where X is a nonempty set, f and g are functions from X to L and L is a lattice with a linearization function as above, such that l(f(x)+l(g(x)) 1. We call this object intuitionistic fuzzy set of the second type (LIFS-2). Some of the most important properties of such fuzzy sets are proved in [8], like the theorem of decomposition, synthesis and others. In spite of many advantages, this definition have one main disadvantage: it does not allow the natural definitions of basic set operations. This is the reason of introducing the new definition of lattice valued (L-valued) intuitionistic fuzzy sets in this paper. To be more precise, we distinguish a

3 390 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) special class of LIFS-2 and use the notion of L-valued intuitionistic fuzzy sets of the third type for elements of this class. The following example illustrates the unconvenience of the definition of LIFS-2 in defining the set operations. Example 1.1. LetLbealattice,giveninFigure1: T a c d B L Fig. 1 Let the linearization function l : L [0, 1]be given by l(t )=1,l(B) =0,l(a) = 0.5,l(c) =0.5,l(d) =0.2. Let X = {x, y}, andf A and g A be functions defined by: and xy f A = at xy g A =. cb Further, we define functions f B and g B by: xy f B = ct and xy g B =. ab A =(X, f A,g A )andb =(X, f B,g B ) are LIFS-2. However, an attempt to define a union of these bifuzzy sets (analogously as the case of ordinary intuitionistic fuzzy sets), would yield the following.

4 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) We would like A B to be bifuzzy set: (X, f A B,g A B ), where f A B (x) =f A (x) f B (x) =T, and g A B (x) =g A (x) g B (x) =d. However, l(f A (x) f B (x))+l(g A (x) g B (x)) = l(t )+l(d) =1+0.2 > 1, and the obtained functions would not satisfy the condition: l(f A B (x)) + l(g A B (x)) 1. The definition introduced in the next section will overcome this difficulty. 2 Definitions and basic properties A new definition of an L-valued intuitionistic fuzzy set is introduced in this section. Let L be a complete lattice with the top element T and the bottom element B and α a lattice homomorphism from L to [0,1], such that α(t )=1andα(B) = 0. Recall that the lattice homomorphism is a function α : L [0, 1]satisfying α(x y) =min(α(x),α(y)), α(x y) =max(α(x),α(y)). It is straightforward to prove that such a function is a linearization function as well, so this definition is a special case of LIFS-2. In the rest of the paper by L we will denote a lattice (as well as the underlying set), and by L the lattice together with its homomorphism as above. A lattice valued intuitionistic fuzzy (L-valued intuitionistic fuzzy) set of the type 3 (LIFS-3) is the ordered triple (X, f, g), where X is a nonempty set, f and g are functions from X to L and L a lattice with a homomorphism α as above, such that α(f(x)) + α(g(x)) 1. (1) A lattice valued intuinistic fuzzy set obtained in this way has some advantages when compared with other definitions. The notion is a generalization of the ordinary intuitionistic fuzzy set and its structure is richer. To every LIFS-3 there correspond two families of level subsets, which are lattices under inclusion. An ordinary intuitionistic fuzzy set is obtained by this homomorphism, in a natural way, which is not the case with LIFS-1. However, still for some lattices there is no a convenient lattice homomorphism, mapping the top element to 1 and the bottom element to 0. In comparison with the LIFS-2, in LIFS-3 basic operations can be introduced in a natural way. The following proposition is a straightforward corollary of the analogous proposition in [8]. Proposition 2.1. Let L be a lattice with a homomorphism α : L [0, 1], and (X, f, g) a LIFS-3. Then, (X, α f,α g) is an ordinary intuitionistic fuzzy set.

5 392 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) We define some basic relations and operations over the new introduced fuzzy sets, analogously as for ordinary intuitionistic fuzzy sets. 1. A B if and only if for all x X f A (x) f B (x) and g A (x) g B (x). 2. A = B if and only if A B and B A. 3. A =(X, f A,g A ), where f A (x) =g A (x) andg A (x) =f A (x). 4. A B =(X, f A B,g A B ), where for all x X, f A B (x) =f A (x) f B (x), and g A B (x) =g A (x) g B (x). 5. A B =(X, f A B,g A B ), where for all x X, f A B (x) =f A (x) f B (x), and g A B (x) =g A (x) g B (x). In the following, we prove that by formulas 3-5 the operations complement, union and intersection on the set of all LIFS-3 over the same set is well defined. Proposition 2.2. Let A =(X, f A,g A )andb =(X, f B,g B ) be LIFS-3, where L is a lattice and α a homomorphism from L to [0, 1]. Then, ordered triples A, A B and A B, defined as above are also LIFS-3. Proof. The proof for A is obvious. Further, we have to prove that for every x X, Indeed, α(f A B (x)) + α(g A B (x)) 1. α(f A B (x)) + α(g A B (x)) = α(f A (x) f B (x)) + α(g A (x) g B (x)) = min{α(f A (x)),α(f B (x))} + max{α(g A (x)),α(g B (x))}. Let max{α(g A (x)),α(g B (x))} = α(g A (x)) (the proof is similar in other case). Then min{α(f A (x)),α(f B (x))} α(f A (x)), and the statement is easily proved. The proof for union is similar. Now, it is not difficult to prove that the operations and satisfy commutative, associative and absorptive laws. Moreover, if L is a distributive lattice, then distributive laws are also satisfied.

6 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) We already mentioned that a LIFS-3 determines two L-valued fuzzy sets. However, the converse is not true: not every couple of L-valued fuzzy sets corresponds to a LIFS-3. The notions of cut (level) subsets and cut (level) functions for LIFS-3 are introduced similar to those of L-fuzzy sets. For each p L, there are two cut sets defined by: f p = {x X f(x) p} and g p = {x X g(x) p}. The corresponding cut (level) functions are denoted by f p and g p. Thus, for an L valued intuitionistic fuzzy set (X, f, g), there are two families of cut sets. The following two propositions are consequences of the fact that a LIFS-3 consists of two L-valued fuzzy sets. Proposition 2.3. Every LIFS-3 determines two families of cut sets. Each of these families is closed under intersections and contains set X, and thus it is a lattice under inclusion. Proposition 2.4. Let (X, f, g) be a LIFS-3. Then the following is satisfied. 1. f B = X, and g B = X, where B is the bottom element of L. 2. If p q, thenf q f p,andg q g p If M L, then f(x) = {p L f p (x) =1}; g(x) = {p L g p (x) =1}. (fp p M) =f {p p M} and (gp p M) =g {p p M}. 3 Theorem of synthesis for the L-valued intuitionistic fuzzy sets LIFS-3 In this section necessary and sufficient conditions under which two families of subsets of a set are families of cut sets of a LIFS-3 are given.

7 394 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) Theorem 3.1. Let X be a nonempty set and let F 1 and F 2 be two families of subsets of X, each of them closed under intersections, containing X, and satisfying the following condition: ( x X)( A F 1 F 2 )(x A). (2) Then, there is a lattice L with a homomorphism α : L [0, 1]and two mappings f and g from X to L, such that (X, f, g) is a LIFS-3, and F 1 and F 2 are its families of cut sets. Proof. The families F 1 and F 2 are closed under intersections, each of them containing X, thus they are lattices under inclusion. Consider lattices (L 1, 1 )and(l 2, 2 ), antiisomorphic to lattices F 1 and F 2, under the functions f 1 and f 2, respectively, such that L 1 L 2 =. WedenotebyT Fi and by B Fi, the top and the bottom element, respectively, of lattices F i, for i =1, 2. By T i, B i, we denote the top and the bottom element, respectively, of lattices L i, i =1, 2. Let (L, ) bealinearsumofl 1, L 2 and one element lattice ({T }, ), for T L 1 and T L 2. Namely, L = L 1 L 2 {T }, and x L 1 and y L 2 or x 1 y and x, y L 1 or x y if and only if x 2 y and x, y L 2 or y = T. Obviously T is the top element of the lattice L. Now, we define functions f : X L and g : X L, as follows. T, if x B F1 ; f(x) = f 1 ( {X F 1 x X}) otherwise. T, if x B F2 ; g(x) = f 2 ( {X F 2 x X}) otherwise. We have to prove that the families of cut sets of the LIFS-3 (X, f, g) aref 1 and F 2. First we will consider all p L and prove that every cut set f p coincides with a set from F 1.

8 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) Case 1. p = T.Sincef(x) =T for x B F1, and for all other x X, f(x) <T,wehave that f T = B F1. Case 2. p L 2.Sincef(x) L 1 or f(x) =T,wehavethatf p = f T, for all p L 2. Case 3. p = T 1. x f T1 if and only if f(x) T 1 ifandonlyif f(x) =T or f(x) =T 1 if and only if f 1 ( {X F 1 x X}) =T 1 or x B F1 if and only if {X F1 x X} = B F1 or x B F1 if and only if x B F1. Case 4. p L 1 \{T 1 }. We prove that f p = f2 1 (p). If x B F1, then obviously x f p and x f2 1 (p), hence x f p if and only if x f2 1 (p). Now, suppose that x B F1. x f p if and only if f(x) p if and only if f1 1 (f(x)) f1 1 (p) if and only if f1 1 (f 1 ( {X F 1 x X})) f1 1 (p) if and only if {X F1 x X} f1 1 (p) if and only if x f1 1 (p). Thereby we also proved that every element from family F 1 coincides with a cut set of f. This proves that the family of cut sets of f coincides with family F 1. The proof that the family of cut sets of g coincides with family F 2 is similar. Now, we have to define a homomorphism α from L to [0, 1], such that (1) is satisfied. We define a function satisfying (1), as follows: α(t ) = 1, and α(x) = 0, for all T x L. This function is obviously a homomorphism, satisfying the condition (1). Indeed, the condition (1) is not satisfied only in the case when there is an x X, such that f(x) =T,andg(x) =T. In this case, x f T,andx g T, and by Proposition 2.4(2), x f p,andx g p for all p L, which is in contrary to (2). It is evident that α need not be the only homomorphism satisfying (1), but here it is enough to prove that there exists such a function. Theorem 3.2. Necessary and sufficient conditions under which two families F 1 and F 2 of subsets of X are families of cut sets of a LIFS-3 are that both are closed under intersections, X belongs to both of them, and that condition (2) is satisfied. Proof. The first part follows by Theorem 3.1. Suppose that F 1 and F 2 are families of cut sets of a LIFS-3 (X, f, g). They are closed under intersections and contain X, by Proposition 2.3. Now, suppose that condition (2) is not satisfied, i.e., that there is x X, such that x belongs to all elements of F 1 and F 2.Now,f(x) =T and g(x) =T, by Proposition 2.4 (3). Since α(t ) = 1, α(f(x)) + α(g(x)) = 2, a contradiction. Example 3.3. Let X = {a, b, c, d}, F 1 = {{a, b, c, d}, {a, c}, {b, d}, {c}, } and F 2 = {{a, b, c, d}, {a, b, c}, {a, b, d}, {a, b}, {b, c}, {b}}. F 1 and F 2 satisfy conditions of Theo-

9 396 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) rem 3.1, i.e., both families are closed under intersections, contain X, and the condition (2) is also satisfied. By the construction in Theorem 3.1, we determine lattices L 1 and L 2, antiisomorphic to (F 1, ) and(f 2, ), respectively. The isomorphisms f 1 and f 2 are given by the following tables and corresponding lattices are presented in Figure 2a. Lattice L in Figure 2b is a linear sum of L 1, L 2 and {T }. {a, b, c, d} {a, c} {b, d} {c} f 1 = B 1 r t s T 1 {a, b, c, d} {a, b, c} {a, b, d} {a, b} {b, c}{b} f 2 = B 2 u v q p T 2 Now, by the mentioned construction, we have that: abcd f = rtst and abcd g =. qtpv One possibility for the homomorphism is the one from Theorem 3.1. We present here another homomorphism satisfying the conditions: x T p T 2 u q B 2 v T 1 s r t B 1 α(x)

10 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) T s r T 1 t B 1 L 2 T 2 p q v T u 2 B 2 p q T 1 s v u t r B 2 B 1 L 1 L Fig. 2a Fig. 2b The families of level sets of the obtained LIFS-3 (X, f, g) coincide with F 1 and F 2, which is straightfoward to check. References [1]K. Atanassov: Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, (1986), pp [2]K. Atanassov, S. Stoeva: Intuitionistic L-fuzzy sets, Cybernetics and Systems Research, Vol. 2, R. Trappl (ed.) Elsevier Science Publishers B.V., North-Holland, (1984), pp [3]K. Atanassov: Intuitionistic fuzzy sets, Theory and Applications, Physica-Verlag, Springer Company. Heilderberg, New York, [4]B.A. Davey, H.A. Priestly: Introduction to lattices and order, Cambridge University Press, [5]T. Gerstenkorn, J. Mańko: Bifuzzy probabilistic sets, Fuzzy Sets and Systems, Vol. 71, (1995), pp [6]T. Gerstenkorn, J. Mańko: Bifuzzy probability of intuitionistic fuzzy sets, Notes on Intuitionistic Fuzzy Sets, Vol. 4, (1998), pp [7]T. Gerstenkorn, J. Mańko: On probability and independence in intuitionistic fuzzy set theory, Notes on Intuitionistic Fuzzy Sets, Vol. 1, (1995), pp [8]T. Gerstenkorn, A. Tepavčević: Lattice valued bifuzzy sets, New Logic for the New Economy, VIII SIGEF Congress Proceedings, ed. by G. Zollo, pp [9]B. Šešelja, A. Tepavčević: Representation of lattices by fuzzy sets, Information Sciences, Vol. 79, (1993), pp [10]B. Šešelja, A. Tepavčević, G. Vojvodić: L-fuzzy sets and codes, Fuzzy sets and systems, Vol. 53, (1993), pp

11 398 T. Gerstenkorn, A. Tepavčević / Central European Journal of Mathematics 2(3) [11]B. Šešelja, A. Tepavčević: Completion of ordered structures by cuts of fuzzy sets, an overview, Fuzzy Sets and Systems, Vol. 136, (2003) pp [12]B. Šešelja, A. Tepavčević: Representing ordered structures by fuzzy sets, an overview, Fuzzy Sets and Systems, Vol. 136, (2003), pp