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On existence of P-valued fuzzy sets with a given collection of cuts
Fuzzy Sets and Systems 161 (2010) 763 – 768 www.elsevier.com/locate/fss
On existence of P-valued fuzzy sets with a given collection of cuts夡 Branimir Šešeljaa , Dragan Stoji´cb , Andreja Tepavˇcevi´ca,∗ a Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovi´ca 4, 21000 Novi Sad, Serbia b Faculty of Economics, University of Novi Sad, 24000 Subotica, Serbia
Received 15 April 2008; received in revised form 23 November 2008; accepted 23 September 2009 Available online 6 October 2009
Abstract A representation theorem of families of subsets by poset-valued fuzzy sets is presented. Namely, necessary and sufficient conditions are given under which for a given family of subsets F of a set X and a fixed poset P there is a fuzzy set : X −→ P, such that the collection of cuts of coincides with F. © 2009 Elsevier B.V. All rights reserved. MSC: 03E72 Keywords: Poset-valued fuzzy sets; Cut-sets; Representation by cuts
1. Introduction In basic investigations of fuzzy sets and relations, and also in dealing with fuzzy algebras and topologies, collections of cut sets are one of the most important tools. Cut sets (also called level sets, levels) show the dual nature of fuzzy sets. On one hand, fuzzy sets are mappings generalizing the characteristic function and on the other hand, they can be characterized by collections of crisp subsets of the domain—cut sets. Many properties of fuzzy structures are investigated using the transferability to cut sets. These are known as ‘cutworthy’ properties (see e.g., [6,5]). Due to this importance of cut sets, investigations have been focused to the problem of identification of a fuzzy set by the collection of its cuts. In the framework of classical, [0, 1]-interval-valued fuzzy sets, relevant results were obtained in the early period of fuzzy era by Negoita and Ralescu (see [9,10,12,13] and references in these). In addition, the same authors have investigated and solved analogue problems for lattice valued fuzzy sets (see [11]). Recently, many problems connected with the representation of collections of subsets as cuts of fuzzy sets (still for [0, 1]-valued fuzzy sets) have been formulated and solved by Jaballah and Saidi (see e.g. [14–16,4]).
夡 The research of the first and the third author was supported by Serbian Ministry of Science, Grant no. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant ‘Lattice methods and applications’. ∗ Corresponding author. Tel.: +38 1214852862; fax: +38 1216350458. E-mail addresses: [email protected] (B. Šešelja), [email protected] (D. Stoji´c), [email protected] (A. Tepavˇcevi´c). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.09.020
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For lattice-valued fuzzy sets, the following two representation problems have recently been investigated; in both L is a fixed complete lattice: Problem 1. Find conditions under which two fuzzy sets : X −→ L and : X −→ L have equal families of cuts. Problem 2. Characterize conditions under which F ⊆ P(X ) is a collection of cut sets of a fuzzy set : X −→ L. Problem 1 was posed and partially solved (for the co-domain being a finite chain) by Murali and Makamba in [8]; a general solution was given in [18]. Problem 2 has been investigated and solved by Liu and Luo (see [7]). Independently, the same problem was solved in a different way in [3]. The abovementioned and similar problems can be investigated in the more general settings, namely for poset-valued fuzzy sets. These are defined as mappings from a nonempty set X to an arbitrary poset P (see e.g, [19–21]); obviously, lattice valued fuzzy sets are a special case of poset-valued ones. Problem 1 formulated for poset-valued fuzzy sets was investigated and solved in [17]. In this paper we investigate a poset-valued version of Problem 2. Namely, we present necessary and sufficient conditions under which for a given family of subsets F of a set X and a fixed poset P there is a fuzzy set : X −→ P, such that the collection of cuts of coincides with F. 2. Preliminaries In this part, basic notions and some properties of posets and poset valued fuzzy sets are given. Since only the facts used in this article are listed, we refer to the relevant literature and articles for more comprehensive presentation: book [1] for posets, articles [19,20] for poset-valued fuzzy sets. 2.1. Posets A relational structure (P, ≤) is called a partially ordered set, poset, if the relation ‘≤’ is reflexive, antisymmetric and transitive on a nonempty set P. A poset (P, ≤) is sometimes denoted by the underlying set P only. If P is a poset and p ∈ P, then we denote by ↑ p the principal filter generated by p: ↑ p := {q ∈ P| p ≤ q}. Recall that the poset (P, ≤) is said to be of finite length if each chain in P is finite. If (P, ≤) and (Q, ≤) are posets, then the function f : P → Q is said to be isotone if x ≤ y implies f (x) ≤ f (y), for all x, y ∈ P. An injection f : P → Q is an order embedding from P into Q, if x ≤ y ←→ f (x) ≤ f (y), for all x, y ∈ P. A bijective order embedding is an order isomorphism. Let F be a collectionof subsets of a nonempty set X union of which is X. Let also F be closed under the componentwise intersections, i.e., let {y ∈ F|x ∈ y} ∈ F for every x ∈ X . Then F is called a point closure system or centralized system on X. The following set-theoretic proposition is used in the sequel. It is proved as Lemma 3 in [20]. Proposition 1. Let C be a collection of subsets of a nonempty set A. If a, b ∈ A and a ∈ { f ∈ C|a ∈ f } ⊆ { f ∈ C|b ∈ f }.
{ f ∈ C|b ∈ f }, then
2.2. Poset-valued fuzzy sets If X is a nonempty set and P a poset, then a poset valued, P-valued fuzzy set on X is a mapping : X −→ P. Special case of these are L-valued fuzzy sets, where L is a (complete) lattice [2]. Obviously, P- and L-valued fuzzy sets generalize the original definition by Zadeh who used as the co-domain the unit interval [0, 1]. By (X ) we denote the set of images of : (X ) = { p ∈ P| p = (x) for some x ∈ X }.
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If : X −→ P is a fuzzy set on X then for p ∈ P, the set p = {x ∈ X |(x) ≥ p} is a p-cut, or a cut set, cut of . The collection of all cuts of is denoted by P : P := { p | p ∈ P}. Next we present several propositions concerning properties of cut sets of P-valued fuzzy sets. These results are taken from [20]. Proposition 2. Let : X −→ P be a P-valued fuzzy set on X. Then: (1) if for P1 ⊆ P there is a supremum P1 in P, then { p | p ∈ P1 } = ∨P1 ; (2) { p | p ∈ P} = X ; (3) { p |x ∈ p } ∈ P for every x ∈ X . Proposition 3. Let : X −→ P be a P-valued fuzzy set on X. Then, for x, y ∈ X , (x) ≤ (y) if and only if (y) ⊆ (x) . Proposition 4. Let P be a family of subsets of a nonempty set X, union of which is also X and such that for every x ∈ X, { p ∈ P|x ∈ p} ∈ P. Let : X −→ P, where the co-domain is (P, ⊇), be defined by: (x) := { p ∈ P|x ∈ p}. Then, is a P-valued fuzzy set on X, and for every p ∈ P, p = p . The following result is taken from [17]. Proposition 5. If : X −→ P is a fuzzy set on X, then there exists an order isomorphism from the poset P to the poset P of cuts of , where P := ({↑ p ∩ (X )| p ∈ P}, ⊆). 3. Results As a main result of this paper, we present a solution of a cut set representation problem for a collection of subsets of a set. Namely, starting with a collection F of subsets of a set X and knowing a poset P, we give necessary and sufficient conditions for the existence of a P-valued fuzzy set whose collection of cuts coincides with F. We start with a property which plays an important role in the proof of the main theorem. Lemma 1. Let : X −→ P be a P-valued fuzzy set and P the collection of cuts of . Then, for every x ∈ X (x) = { p ∈ P |x ∈ p }. Proof. By the definition of a cut, we have that x ∈ (x) and hence (x) ⊇ { p ∈ P |x ∈ p }. By Proposition 2(3) centralized intersection of cuts is always a cut. So, there is q ∈ P such that { p ∈ P |x ∈ p } = q . Since x ∈ q , then (x) ≥ q. Therefore (x) = q and (x) = { p ∈ P |x ∈ p }. Next we give our main result. Observe that we denote by ⊇ the ordering relation on a collection of subsets of a set, which is dual to set inclusion. Theorem 1. Let F be a family of subsets of a nonempty set X and let Y = {Z x |x ∈ X and Z x = Let also (P, ≤) be a poset.
{ f ∈ F|x ∈ f }}.
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Then, there is a fuzzy set : X −→ P such that F is its collection of cuts if and only if the following are satisfied: 1. F is closed under centralized intersections and { f | f ∈ F} = X . 2. There is an isotone function E : Y −→ P from the poset (Y, ⊇) to (P, ≤), such that for every r ∈ P, {y|y ∈ E −1 (↑r ∩ E(Y ))} ∈ F, and the mapping P −→ F, defined by (r ) = {y|y ∈ E −1 (↑r ∩ E(Y ))} is ‘onto’. Proof. Let : X −→ P be a fuzzy set such that F is its collection of cuts. Then 1 is satisfied by Proposition 2. To prove 2, define E : Y −→ P by E(Z x ) := (x). This function is well defined since from Z x = Z y it follows that (x) = (y), by Proposition 3 and Lemma 1. This function is isotone (according to the orders in Y and P, the former being by definition dual to inclusion). Indeed, Z x is the cut set corresponding to (x) by Lemma 1. Hence, Z x = (x) and Z y = (y) . By Proposition 3, Z x ⊇ Z y ←→ (x) ≤ (y), implying that E is isotone. Next we prove that for an arbitrary r ∈ P, the set {y|y ∈ E −1 (↑r ∩ E(Y ))} is a cut of , i.e., that it belongs to F. As indicated above, for every x ∈ X we have Z x = (x) . Therefore E(Y ) = (X ) and ↑r ∩ E(Y ) = ↑r ∩ (X ). By the definition of a cut, r = {x ∈ X |(x) ≥ r }, i.e., r = {x ∈ X |↑r ∩ (X ) (x)}. We prove that this implies {y|y ∈ E −1 (↑r ∩ E(Y ))}. r = Indeed, x ∈ r ←→ (x)≥ r ←→ E(Z x ) ≥ r ←→ E(Z x ) ∈ ↑r ∩ E(Y ). Since x ∈ Z x and Z x ∈ E −1 (↑r ∩ E(Y )), finally we have that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}, proving ‘⊆’. To prove the dual inclusion, suppose x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then, there is Z y ∈ E −1 (↑r ∩ E(Y )) such that x ∈ Z y . Then E(Z y ) ∈ ↑r ∩ E(Y ), i.e., E(Z y ) ≥ r , and (y) ≥ r . Now, by Proposition 1, from x ∈ Z y , it follows that Z x ⊆ Z y , and since E is proved to be isotone we obtain (x) ≥ (y) ≥ r . Hence (x) ≥ r and x ∈ r . Finally, we prove that the mapping : P −→ F is ‘onto’. This follows immediately from the above proof of 2, since for every r ∈ P, r = {y|y ∈ E −1 (↑r ∩ E(Y ))}. The converse: Suppose that F is a collection of subsets of a set X and (P, ≤) a poset, such that conditions 1 and 2 are fulfilled. Then, : X −→ P defined by (x) := E(Z x ) is a P-valued fuzzy set. We have to prove that F is the collection of cuts of . We show that for every r ∈ P, r = (r ). (x) ≥ r ←→ E(Z x ) ≥ r . Therefore, E(Z x ) ∈ ↑r ∩ E(Y ) and Z x ∈ E −1 (↑r ∩ E(Y )). Let r ∈ P. Then, x ∈ r ←→ Since x ∈ Z x , we have that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. We have proved r ⊆ (r ). To prove the dual inclusion, let x ∈ (r ) i.e., x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then, there is Z y ∈ E −1 (↑r ∩ E(Y )) such that x ∈ Z y . Observe that also E(Z y ) ∈ ↑r ∩ E(Y ), i.e., E(Z y ) ≥ r . From x ∈ Z y , it follows that Z x ⊆ Z y (again by Proposition 1). Since E is isotone, E(Z x ) ≥ E(Z y ) ≥ r . Hence we have E(Z x ) ≥ r , i.e., (x) ≥ r and finally x ∈ r . Thus we have that for every r ∈ P, r = (r ). Since is ‘onto’ the collection of cuts of is F, concluding the proof. Remark. The function E appearing in Theorem 1 is not only isotone, but also an order embedding from (Y, ⊇) to (P, ≤). This can be deduced as a consequence of Proposition 5 and the fact that the function is a surjection. Example. Let F = {{x}, {z}, {x, y}, {x, z}, {z, t}, {x, y, z}}. The poset (F, ⊇) is presented by the diagram in Fig. 1(a). Observe that F is a point closure system. In addition, Z x = {x}, Z y = {x, y}, Z z = {z} and Z t = {z, t}, according to the definition given in Theorem 1. These elements constitute the set Y, which is as the poset (Y, ⊇) denoted by filled circles in the same diagram (Fig. 1(a)). The poset (P, ≤) is given in Fig. 1(b). The mapping E : Y −→ P is defined as follows: E(Z x ) = a, E(Z y ) = b, E(Z z ) = c, E(Z t ) = d. Obviously, E is isotone and the mapping from P to F defined by (a) = (e) = {x}, (b) = {x, y}, (c) = (g) = {z}, (d) =
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Fig. 1.
{z, t}, ( f ) = (i) = {x, z}, (h) = {x, y, z}, is ‘onto’. According to Theorem 1 there exists a fuzzy set : X −→ P defined by (x) := E(Z x ), i.e., (x) = a, (y) = b, (z) = c, (t) = d. Now it is easily verifiable that the collection of cut sets of equals F. The above example motivates the following consequence of Theorem 1. Namely, we present some sufficient conditions for the existence of a poset-valued fuzzy set whose family of cuts is isomorphic to a given collection of subsets of a set. These conditions are easy to handle and could be considered as analogue to those formulated for the lattice-valued case in [3]. Corollary 1. Let (P, ≤) be a poset, X a nonempty set and F a collection of some of its subsets whose union is X and which is a point closure system. Assume also that there exists an order embedding from the poset (F, ⊇) into (P, ≤), such that for every r ∈ P, ↑r ∩ (F) = ↑( f ) for some f ∈ F.
(1)
Then there is a P-valued fuzzy set : X −→ P, such that F coincides with its collection of cut sets. Proof. F fulfils condition 1 from Theorem 1 by assumption. We prove that also condition 2 holds. Indeed, we define E : Y −→ P to be the restriction |Y of to the set Y = {Z x |Z x = { f ∈ F|x ∈ f }}. Since is an embedding, its restriction |Y is obviously isotone. Now let r ∈ P be fixed. By the assumption, there is an element f ∈ F such that ↑r ∩ (F) =↑ ( f ). We prove that {y|y ∈ E−1 (↑r ∩ E(Y ))} equals f, which obviously fulfils our task. Observe first that f can x ∈ f is by definition subset of be expressed as f = x∈ f Z x . Indeed, every Z x for which f. Therefore, we have x∈ f Z x ⊆ f . On the other hand, if x ∈ f , then x ∈ Z x ⊆ x∈ f Z x . We now prove {y|y ∈ E −1 (↑r ∩ E(Y ))} = f. Let x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then there is Z y such that x ∈ Z y and Z y ∈ E −1 (↑r ∩ E(Y )). Therefore E(Z y ) ∈ ↑r ∩ E(Y ), i.e., (Z y ) ∈ ↑r ∩ E(Y ) ⊆ ↑r ∩ (F) = ↑( f ). Now, we have ( f ) ≤ (Z y ) and since is an order embedding, we obtain x ∈ Z y ⊆ f, i.e., x ∈ f.
On the other hand, if x ∈ f , we know that f = x∈ f Z x , so there is Z y ⊆ f such that x ∈ Z y . Since is an order embedding, we have (Z y ) ≥ ( f ). By the condition (1) we obtain (Z y ) ∈ ↑r ∩ (F). Now, since Z y ∈ Y , we have (Z y ) = E(Z y ) ∈ E(Y ) and E(Z y ) ∈ ↑r ∩ E(Y ). Finally, since Z y ∈ E −1 (↑r ∩ E(Y )) and x ∈ Z y , we conclude that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Since both conditions from Theorem 1 are satisfied, there is a P-valued fuzzy set such that F coincides with its collection of cuts.
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4. Conclusion A representation theorem for a collection of subsets of a set to be the family of cuts of a poset valued fuzzy set is presented. The theorem is proved under general conditions, for any given poset. It shows that, in order to fulfil the above representation, a collection of subsets and a poset should be connected by particular properties. We also give some sufficient conditions which are more transparent. Hence, the problem of existence of a fuzzy set in the mentioned framework is solved. Our next task would be to analyze the equally important problem of uniqueness of the above representation. We have to mention that analogue problems for unit-interval valued fuzzy sets are particular cases of poset-valued ones. Still, the way these are formulated and solved in terms of real functions [9,10] are not equivalent to our approach. Acknowledgment The referees’ helpful comments are appreciated. References [1] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1992. [2] A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. [3] M. Gorjanac-Ranitovi´c, A. Tepavˇcevi´c, General form of lattice-valued fuzzy sets under the cutworthy approach, Fuzzy Sets and Systems 158 (2007) 1213–1216. [4] A. Jaballah, F.B. Saidi, Uniqueness results in the representation of families of sets by fuzzy sets, Fuzzy Sets and Systems 157 (2006) 964–975. [5] V. Janiš, A. Tepavˇcevi´c, B. Šešelja, Non-standard cut classification of fuzzy sets, Inform. Sci. 177 (2007) 161–169. [6] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-Hall P T R, Englewood Cliffs, NJ, 1995. [7] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997. [8] V. Murali, B.B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and Systems 123 (2001) 259–264. [9] C.V. Negoita, D.A. Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. [10] C.V. Negoita, D.A. Ralescu, Simulation, Knowledge-Based Computing, and Fuzzy Statistics, Van Nostrand Reinhold, New York, 1987, pp. 89–91. [11] C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. [12] D.A. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M.M. Gupta, R.K. Ragade, R. Yager (Eds.), Advances in Fuzzy Sets Theory and Applications, North-Holland, Amsterdam, 1979, pp. 77–91. [13] D.A. Ralescu, A generalization of the representation theorem, Fuzzy Sets and Systems 51 (1992) 309–311. [14] F.B. Saidi, A. Jaballah, Existence and uniqueness of fuzzy ideals, Fuzzy Sets and Systems 149 (2005) 527–541. [15] F.B. Saidi, A. Jaballah, Alternative characterizations for the representation of families of sets by fuzzy sets, Inform. Sci. 178 (12) (2008) 2639–2647. [16] F.B. Saidi, A. Jaballah, Uniqueness in the generalized representation by fuzzy sets, Fuzzy Sets and Systems 159 (16) (2008) 2176–2184. [17] B. Šešelja, A. Tepavˇcevi´c, Equivalent fuzzy sets, Kybernetika 41 (2) (2005) 115–128. [18] B. Šešelja, A. Tepavˇcevi´c, A note on natural equivalence relation on fuzzy power set, Fuzzy Sets and Systems 148 (2) (2004) 201–210. [19] B. Šešelja, A. Tepavˇcevi´c, On a construction of codes by P-fuzzy sets, Rev. Res. Fac. Sci. Univ. Novi Sad 20 (2) (1990) 71–80. [20] B. Šešelja, A. Tepavˇcevi´c, Completion of ordered structures by cuts of fuzzy sets: an overview, Fuzzy Sets and Systems 136 (2003) 1–19. [21] B. Šeselja, A. Tepavˇcevi´c, Representing ordered structures by fuzzy sets, an overview, Fuzzy Sets and Systems 136 (2003) 21–39.
On existence of P-valued fuzzy sets with a given collection of cuts夡 Branimir Šešeljaa , Dragan Stoji´cb , Andreja Tepavˇcevi´ca,∗ a Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovi´ca 4, 21000 Novi Sad, Serbia b Faculty of Economics, University of Novi Sad, 24000 Subotica, Serbia
Received 15 April 2008; received in revised form 23 November 2008; accepted 23 September 2009 Available online 6 October 2009
Abstract A representation theorem of families of subsets by poset-valued fuzzy sets is presented. Namely, necessary and sufficient conditions are given under which for a given family of subsets F of a set X and a fixed poset P there is a fuzzy set : X −→ P, such that the collection of cuts of coincides with F. © 2009 Elsevier B.V. All rights reserved. MSC: 03E72 Keywords: Poset-valued fuzzy sets; Cut-sets; Representation by cuts
1. Introduction In basic investigations of fuzzy sets and relations, and also in dealing with fuzzy algebras and topologies, collections of cut sets are one of the most important tools. Cut sets (also called level sets, levels) show the dual nature of fuzzy sets. On one hand, fuzzy sets are mappings generalizing the characteristic function and on the other hand, they can be characterized by collections of crisp subsets of the domain—cut sets. Many properties of fuzzy structures are investigated using the transferability to cut sets. These are known as ‘cutworthy’ properties (see e.g., [6,5]). Due to this importance of cut sets, investigations have been focused to the problem of identification of a fuzzy set by the collection of its cuts. In the framework of classical, [0, 1]-interval-valued fuzzy sets, relevant results were obtained in the early period of fuzzy era by Negoita and Ralescu (see [9,10,12,13] and references in these). In addition, the same authors have investigated and solved analogue problems for lattice valued fuzzy sets (see [11]). Recently, many problems connected with the representation of collections of subsets as cuts of fuzzy sets (still for [0, 1]-valued fuzzy sets) have been formulated and solved by Jaballah and Saidi (see e.g. [14–16,4]).
夡 The research of the first and the third author was supported by Serbian Ministry of Science, Grant no. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant ‘Lattice methods and applications’. ∗ Corresponding author. Tel.: +38 1214852862; fax: +38 1216350458. E-mail addresses: [email protected] (B. Šešelja), [email protected] (D. Stoji´c), [email protected] (A. Tepavˇcevi´c). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.09.020
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For lattice-valued fuzzy sets, the following two representation problems have recently been investigated; in both L is a fixed complete lattice: Problem 1. Find conditions under which two fuzzy sets : X −→ L and : X −→ L have equal families of cuts. Problem 2. Characterize conditions under which F ⊆ P(X ) is a collection of cut sets of a fuzzy set : X −→ L. Problem 1 was posed and partially solved (for the co-domain being a finite chain) by Murali and Makamba in [8]; a general solution was given in [18]. Problem 2 has been investigated and solved by Liu and Luo (see [7]). Independently, the same problem was solved in a different way in [3]. The abovementioned and similar problems can be investigated in the more general settings, namely for poset-valued fuzzy sets. These are defined as mappings from a nonempty set X to an arbitrary poset P (see e.g, [19–21]); obviously, lattice valued fuzzy sets are a special case of poset-valued ones. Problem 1 formulated for poset-valued fuzzy sets was investigated and solved in [17]. In this paper we investigate a poset-valued version of Problem 2. Namely, we present necessary and sufficient conditions under which for a given family of subsets F of a set X and a fixed poset P there is a fuzzy set : X −→ P, such that the collection of cuts of coincides with F. 2. Preliminaries In this part, basic notions and some properties of posets and poset valued fuzzy sets are given. Since only the facts used in this article are listed, we refer to the relevant literature and articles for more comprehensive presentation: book [1] for posets, articles [19,20] for poset-valued fuzzy sets. 2.1. Posets A relational structure (P, ≤) is called a partially ordered set, poset, if the relation ‘≤’ is reflexive, antisymmetric and transitive on a nonempty set P. A poset (P, ≤) is sometimes denoted by the underlying set P only. If P is a poset and p ∈ P, then we denote by ↑ p the principal filter generated by p: ↑ p := {q ∈ P| p ≤ q}. Recall that the poset (P, ≤) is said to be of finite length if each chain in P is finite. If (P, ≤) and (Q, ≤) are posets, then the function f : P → Q is said to be isotone if x ≤ y implies f (x) ≤ f (y), for all x, y ∈ P. An injection f : P → Q is an order embedding from P into Q, if x ≤ y ←→ f (x) ≤ f (y), for all x, y ∈ P. A bijective order embedding is an order isomorphism. Let F be a collectionof subsets of a nonempty set X union of which is X. Let also F be closed under the componentwise intersections, i.e., let {y ∈ F|x ∈ y} ∈ F for every x ∈ X . Then F is called a point closure system or centralized system on X. The following set-theoretic proposition is used in the sequel. It is proved as Lemma 3 in [20]. Proposition 1. Let C be a collection of subsets of a nonempty set A. If a, b ∈ A and a ∈ { f ∈ C|a ∈ f } ⊆ { f ∈ C|b ∈ f }.
{ f ∈ C|b ∈ f }, then
2.2. Poset-valued fuzzy sets If X is a nonempty set and P a poset, then a poset valued, P-valued fuzzy set on X is a mapping : X −→ P. Special case of these are L-valued fuzzy sets, where L is a (complete) lattice [2]. Obviously, P- and L-valued fuzzy sets generalize the original definition by Zadeh who used as the co-domain the unit interval [0, 1]. By (X ) we denote the set of images of : (X ) = { p ∈ P| p = (x) for some x ∈ X }.
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If : X −→ P is a fuzzy set on X then for p ∈ P, the set p = {x ∈ X |(x) ≥ p} is a p-cut, or a cut set, cut of . The collection of all cuts of is denoted by P : P := { p | p ∈ P}. Next we present several propositions concerning properties of cut sets of P-valued fuzzy sets. These results are taken from [20]. Proposition 2. Let : X −→ P be a P-valued fuzzy set on X. Then: (1) if for P1 ⊆ P there is a supremum P1 in P, then { p | p ∈ P1 } = ∨P1 ; (2) { p | p ∈ P} = X ; (3) { p |x ∈ p } ∈ P for every x ∈ X . Proposition 3. Let : X −→ P be a P-valued fuzzy set on X. Then, for x, y ∈ X , (x) ≤ (y) if and only if (y) ⊆ (x) . Proposition 4. Let P be a family of subsets of a nonempty set X, union of which is also X and such that for every x ∈ X, { p ∈ P|x ∈ p} ∈ P. Let : X −→ P, where the co-domain is (P, ⊇), be defined by: (x) := { p ∈ P|x ∈ p}. Then, is a P-valued fuzzy set on X, and for every p ∈ P, p = p . The following result is taken from [17]. Proposition 5. If : X −→ P is a fuzzy set on X, then there exists an order isomorphism from the poset P to the poset P of cuts of , where P := ({↑ p ∩ (X )| p ∈ P}, ⊆). 3. Results As a main result of this paper, we present a solution of a cut set representation problem for a collection of subsets of a set. Namely, starting with a collection F of subsets of a set X and knowing a poset P, we give necessary and sufficient conditions for the existence of a P-valued fuzzy set whose collection of cuts coincides with F. We start with a property which plays an important role in the proof of the main theorem. Lemma 1. Let : X −→ P be a P-valued fuzzy set and P the collection of cuts of . Then, for every x ∈ X (x) = { p ∈ P |x ∈ p }. Proof. By the definition of a cut, we have that x ∈ (x) and hence (x) ⊇ { p ∈ P |x ∈ p }. By Proposition 2(3) centralized intersection of cuts is always a cut. So, there is q ∈ P such that { p ∈ P |x ∈ p } = q . Since x ∈ q , then (x) ≥ q. Therefore (x) = q and (x) = { p ∈ P |x ∈ p }. Next we give our main result. Observe that we denote by ⊇ the ordering relation on a collection of subsets of a set, which is dual to set inclusion. Theorem 1. Let F be a family of subsets of a nonempty set X and let Y = {Z x |x ∈ X and Z x = Let also (P, ≤) be a poset.
{ f ∈ F|x ∈ f }}.
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Then, there is a fuzzy set : X −→ P such that F is its collection of cuts if and only if the following are satisfied: 1. F is closed under centralized intersections and { f | f ∈ F} = X . 2. There is an isotone function E : Y −→ P from the poset (Y, ⊇) to (P, ≤), such that for every r ∈ P, {y|y ∈ E −1 (↑r ∩ E(Y ))} ∈ F, and the mapping P −→ F, defined by (r ) = {y|y ∈ E −1 (↑r ∩ E(Y ))} is ‘onto’. Proof. Let : X −→ P be a fuzzy set such that F is its collection of cuts. Then 1 is satisfied by Proposition 2. To prove 2, define E : Y −→ P by E(Z x ) := (x). This function is well defined since from Z x = Z y it follows that (x) = (y), by Proposition 3 and Lemma 1. This function is isotone (according to the orders in Y and P, the former being by definition dual to inclusion). Indeed, Z x is the cut set corresponding to (x) by Lemma 1. Hence, Z x = (x) and Z y = (y) . By Proposition 3, Z x ⊇ Z y ←→ (x) ≤ (y), implying that E is isotone. Next we prove that for an arbitrary r ∈ P, the set {y|y ∈ E −1 (↑r ∩ E(Y ))} is a cut of , i.e., that it belongs to F. As indicated above, for every x ∈ X we have Z x = (x) . Therefore E(Y ) = (X ) and ↑r ∩ E(Y ) = ↑r ∩ (X ). By the definition of a cut, r = {x ∈ X |(x) ≥ r }, i.e., r = {x ∈ X |↑r ∩ (X ) (x)}. We prove that this implies {y|y ∈ E −1 (↑r ∩ E(Y ))}. r = Indeed, x ∈ r ←→ (x)≥ r ←→ E(Z x ) ≥ r ←→ E(Z x ) ∈ ↑r ∩ E(Y ). Since x ∈ Z x and Z x ∈ E −1 (↑r ∩ E(Y )), finally we have that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}, proving ‘⊆’. To prove the dual inclusion, suppose x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then, there is Z y ∈ E −1 (↑r ∩ E(Y )) such that x ∈ Z y . Then E(Z y ) ∈ ↑r ∩ E(Y ), i.e., E(Z y ) ≥ r , and (y) ≥ r . Now, by Proposition 1, from x ∈ Z y , it follows that Z x ⊆ Z y , and since E is proved to be isotone we obtain (x) ≥ (y) ≥ r . Hence (x) ≥ r and x ∈ r . Finally, we prove that the mapping : P −→ F is ‘onto’. This follows immediately from the above proof of 2, since for every r ∈ P, r = {y|y ∈ E −1 (↑r ∩ E(Y ))}. The converse: Suppose that F is a collection of subsets of a set X and (P, ≤) a poset, such that conditions 1 and 2 are fulfilled. Then, : X −→ P defined by (x) := E(Z x ) is a P-valued fuzzy set. We have to prove that F is the collection of cuts of . We show that for every r ∈ P, r = (r ). (x) ≥ r ←→ E(Z x ) ≥ r . Therefore, E(Z x ) ∈ ↑r ∩ E(Y ) and Z x ∈ E −1 (↑r ∩ E(Y )). Let r ∈ P. Then, x ∈ r ←→ Since x ∈ Z x , we have that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. We have proved r ⊆ (r ). To prove the dual inclusion, let x ∈ (r ) i.e., x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then, there is Z y ∈ E −1 (↑r ∩ E(Y )) such that x ∈ Z y . Observe that also E(Z y ) ∈ ↑r ∩ E(Y ), i.e., E(Z y ) ≥ r . From x ∈ Z y , it follows that Z x ⊆ Z y (again by Proposition 1). Since E is isotone, E(Z x ) ≥ E(Z y ) ≥ r . Hence we have E(Z x ) ≥ r , i.e., (x) ≥ r and finally x ∈ r . Thus we have that for every r ∈ P, r = (r ). Since is ‘onto’ the collection of cuts of is F, concluding the proof. Remark. The function E appearing in Theorem 1 is not only isotone, but also an order embedding from (Y, ⊇) to (P, ≤). This can be deduced as a consequence of Proposition 5 and the fact that the function is a surjection. Example. Let F = {{x}, {z}, {x, y}, {x, z}, {z, t}, {x, y, z}}. The poset (F, ⊇) is presented by the diagram in Fig. 1(a). Observe that F is a point closure system. In addition, Z x = {x}, Z y = {x, y}, Z z = {z} and Z t = {z, t}, according to the definition given in Theorem 1. These elements constitute the set Y, which is as the poset (Y, ⊇) denoted by filled circles in the same diagram (Fig. 1(a)). The poset (P, ≤) is given in Fig. 1(b). The mapping E : Y −→ P is defined as follows: E(Z x ) = a, E(Z y ) = b, E(Z z ) = c, E(Z t ) = d. Obviously, E is isotone and the mapping from P to F defined by (a) = (e) = {x}, (b) = {x, y}, (c) = (g) = {z}, (d) =
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Fig. 1.
{z, t}, ( f ) = (i) = {x, z}, (h) = {x, y, z}, is ‘onto’. According to Theorem 1 there exists a fuzzy set : X −→ P defined by (x) := E(Z x ), i.e., (x) = a, (y) = b, (z) = c, (t) = d. Now it is easily verifiable that the collection of cut sets of equals F. The above example motivates the following consequence of Theorem 1. Namely, we present some sufficient conditions for the existence of a poset-valued fuzzy set whose family of cuts is isomorphic to a given collection of subsets of a set. These conditions are easy to handle and could be considered as analogue to those formulated for the lattice-valued case in [3]. Corollary 1. Let (P, ≤) be a poset, X a nonempty set and F a collection of some of its subsets whose union is X and which is a point closure system. Assume also that there exists an order embedding from the poset (F, ⊇) into (P, ≤), such that for every r ∈ P, ↑r ∩ (F) = ↑( f ) for some f ∈ F.
(1)
Then there is a P-valued fuzzy set : X −→ P, such that F coincides with its collection of cut sets. Proof. F fulfils condition 1 from Theorem 1 by assumption. We prove that also condition 2 holds. Indeed, we define E : Y −→ P to be the restriction |Y of to the set Y = {Z x |Z x = { f ∈ F|x ∈ f }}. Since is an embedding, its restriction |Y is obviously isotone. Now let r ∈ P be fixed. By the assumption, there is an element f ∈ F such that ↑r ∩ (F) =↑ ( f ). We prove that {y|y ∈ E−1 (↑r ∩ E(Y ))} equals f, which obviously fulfils our task. Observe first that f can x ∈ f is by definition subset of be expressed as f = x∈ f Z x . Indeed, every Z x for which f. Therefore, we have x∈ f Z x ⊆ f . On the other hand, if x ∈ f , then x ∈ Z x ⊆ x∈ f Z x . We now prove {y|y ∈ E −1 (↑r ∩ E(Y ))} = f. Let x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Then there is Z y such that x ∈ Z y and Z y ∈ E −1 (↑r ∩ E(Y )). Therefore E(Z y ) ∈ ↑r ∩ E(Y ), i.e., (Z y ) ∈ ↑r ∩ E(Y ) ⊆ ↑r ∩ (F) = ↑( f ). Now, we have ( f ) ≤ (Z y ) and since is an order embedding, we obtain x ∈ Z y ⊆ f, i.e., x ∈ f.
On the other hand, if x ∈ f , we know that f = x∈ f Z x , so there is Z y ⊆ f such that x ∈ Z y . Since is an order embedding, we have (Z y ) ≥ ( f ). By the condition (1) we obtain (Z y ) ∈ ↑r ∩ (F). Now, since Z y ∈ Y , we have (Z y ) = E(Z y ) ∈ E(Y ) and E(Z y ) ∈ ↑r ∩ E(Y ). Finally, since Z y ∈ E −1 (↑r ∩ E(Y )) and x ∈ Z y , we conclude that x ∈ {y|y ∈ E −1 (↑r ∩ E(Y ))}. Since both conditions from Theorem 1 are satisfied, there is a P-valued fuzzy set such that F coincides with its collection of cuts.
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4. Conclusion A representation theorem for a collection of subsets of a set to be the family of cuts of a poset valued fuzzy set is presented. The theorem is proved under general conditions, for any given poset. It shows that, in order to fulfil the above representation, a collection of subsets and a poset should be connected by particular properties. We also give some sufficient conditions which are more transparent. Hence, the problem of existence of a fuzzy set in the mentioned framework is solved. Our next task would be to analyze the equally important problem of uniqueness of the above representation. We have to mention that analogue problems for unit-interval valued fuzzy sets are particular cases of poset-valued ones. Still, the way these are formulated and solved in terms of real functions [9,10] are not equivalent to our approach. Acknowledgment The referees’ helpful comments are appreciated. References [1] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1992. [2] A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. [3] M. Gorjanac-Ranitovi´c, A. Tepavˇcevi´c, General form of lattice-valued fuzzy sets under the cutworthy approach, Fuzzy Sets and Systems 158 (2007) 1213–1216. [4] A. Jaballah, F.B. Saidi, Uniqueness results in the representation of families of sets by fuzzy sets, Fuzzy Sets and Systems 157 (2006) 964–975. [5] V. Janiš, A. Tepavˇcevi´c, B. Šešelja, Non-standard cut classification of fuzzy sets, Inform. Sci. 177 (2007) 161–169. [6] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-Hall P T R, Englewood Cliffs, NJ, 1995. [7] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997. [8] V. Murali, B.B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and Systems 123 (2001) 259–264. [9] C.V. Negoita, D.A. Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. [10] C.V. Negoita, D.A. Ralescu, Simulation, Knowledge-Based Computing, and Fuzzy Statistics, Van Nostrand Reinhold, New York, 1987, pp. 89–91. [11] C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. [12] D.A. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M.M. Gupta, R.K. Ragade, R. Yager (Eds.), Advances in Fuzzy Sets Theory and Applications, North-Holland, Amsterdam, 1979, pp. 77–91. [13] D.A. Ralescu, A generalization of the representation theorem, Fuzzy Sets and Systems 51 (1992) 309–311. [14] F.B. Saidi, A. Jaballah, Existence and uniqueness of fuzzy ideals, Fuzzy Sets and Systems 149 (2005) 527–541. [15] F.B. Saidi, A. Jaballah, Alternative characterizations for the representation of families of sets by fuzzy sets, Inform. Sci. 178 (12) (2008) 2639–2647. [16] F.B. Saidi, A. Jaballah, Uniqueness in the generalized representation by fuzzy sets, Fuzzy Sets and Systems 159 (16) (2008) 2176–2184. [17] B. Šešelja, A. Tepavˇcevi´c, Equivalent fuzzy sets, Kybernetika 41 (2) (2005) 115–128. [18] B. Šešelja, A. Tepavˇcevi´c, A note on natural equivalence relation on fuzzy power set, Fuzzy Sets and Systems 148 (2) (2004) 201–210. [19] B. Šešelja, A. Tepavˇcevi´c, On a construction of codes by P-fuzzy sets, Rev. Res. Fac. Sci. Univ. Novi Sad 20 (2) (1990) 71–80. [20] B. Šešelja, A. Tepavˇcevi´c, Completion of ordered structures by cuts of fuzzy sets: an overview, Fuzzy Sets and Systems 136 (2003) 1–19. [21] B. Šeselja, A. Tepavˇcevi´c, Representing ordered structures by fuzzy sets, an overview, Fuzzy Sets and Systems 136 (2003) 21–39.