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Completion of ordered structures by cuts of fuzzy sets: an overview
Fuzzy Sets and Systems 136 (2003) 1 – 19 www.elsevier.com/locate/fss
Completion of ordered structures by cuts of fuzzy sets: an overview % selja∗ , Andreja Tepav%cevi*c Branimir Se% Technical Faculty, Department of Mathematics, Institute of Mathematics, University of Novi Sad, Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia Received 23 July 2001; received in revised form 17 May 2002; accepted 9 July 2002
Abstract The aim of the paper is to present a role of fuzzy sets in the theory of ordered structures. Main algebraic properties of cuts of fuzzy sets are given, and a completion of partially ordered sets to complete lattices is described. It turns out that this completion is equivalent with the famous Dedekind–MacNeille completion, but the algorithm presented here is much simpler. c 2002 Elsevier Science B.V. All rights reserved. MSC: 04A72; 06A23 Keywords: Algebra; Partially ordered set; Fuzzy set; Cut; Canonical representation; Completion
1. Introduction One of the basic properties of fuzzy structures is that they are mappings, generalizations of characteristic functions in set theory. Another important property is algebraic: every fuzzy structure is uniquely determined by the collection of subsets of the support, known as cuts. These cuts can be ordered naturally: by inclusion, or by the reverse ordering. The obtained collection of sets is therefore an ordered structure. Algorithms for transitions from the function (fuzzy structure) to the ordered family of cuts and vice versa are based on closure operators, particular set-theoretic completions, and structure properties of ordered sets. Techniques and methods obtained for such purposes are useful tools for investigating fuzzy structures. On the other hand, these can be applied back in the classical theory of ordered structures: it is possible to represent ordered structure by cuts of a suitable fuzzy set , and consequently by the mapping itself. ∗
The research supported by Serbian Ministry of Science and Technology, Grant No. 1227. Corresponding author. % selja). E-mail address: [email protected] (B. Se%
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 6 5 - 2
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By the above, the aim of this article is to present systematically algebraic background and basic properties of cuts and ordering on the collection of these. To provide compact presentation, we introduce the canonical representation of fuzzy structures as mappings, which make them independent of the co-domain. By this approach, every fuzzy set : A → P, where P is an ordered structure (interval [0; 1], in Zadeh’s sense [34], lattice as introduced by Goguen [4], poset, etc., see [14,19]), is replaced by the suitable mapping from A into its power set. Still, main ordering properties of the original function are preserved. Next we introduce particular completion in the collection of subsets of a given set, in order to obtain a collection of cuts of the “nearest” fuzzy set. In this way we get the famous Dedekind–McNeille completion, best known as a procedure to construct reals out of rationals (see e.g., [3]). This overview is a result of investigations conducted by the authors over several years. Some of the presented results are known some are new. Our intention was to collect and unify these results and present them as an algebraic background for investigating fuzzy structures. Naturally, in our research we have been using relevant known results from the algebraic theory of fuzzy structures. Some of these are cited in references; we mention particularly the early work of Negoita and Ralescu [11,12,13], the paper [1] and as a new and signiHcant contribution, the book by Liu and Luo [9]. The rest of the papers listed in references are connected with generalizations of the characteristic function ([6,7,8,18,29]), with some applications in mathematics and other sciences ([5,10,15,21,24,25,26,27,33]) and also with particular developing of fuzzy algebraic structures ([16,19,20,23,30,31,32]). In addition to our papers, we added only those which are closely related to our investigations. Therefore the list should not be considered as complete in any sense. 2. Preliminaries Let (P; 6) be a partially ordered set (poset), denoted also by the underlying set P only. The bottom element of a poset, if it exists, is denoted by 0; similarly, the top element is denoted by 1. If S ⊆ P, the set of all upper bounds of S is denoted by S u , and the set of all lower bounds is denoted by S ‘ : S u := {x ∈ P | (∀s ∈ S)s 6 x} and S ‘ := {x ∈ P | (∀s ∈ S)s ¿ x}: The greatest lower bound (inHmum) of a set S ⊆ P, if it exists, is denoted by S and dually, the supremum (least upper bound) by S. The inHmum of x and y, if it exists, is denoted by x ∧ y, and is said to be their meet. Similarly, supremum of x and y, if it exists, is their join, x ∨ y. If x ∈ P, then the ideal generated by x is denoted by ↓ x: ↓ x := {y ∈ P | y 6 x}: Dually, the Hlter generated by x is denoted by ↑ x: ↑ x := {y ∈ P | y ¿ x}:
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If supremum and inHmum exist for each pair x; y of elements in a poset, then this poset is a lattice and it is denoted by L. Recall that lattice L is complete if for every S ⊆ L, there exist supremum and inHmum. If P is a poset, L a complete lattice and ’ : P → L an order embedding (i.e., an injection preserving the order), than L is a completion of P. If (P; 6) is a poset, then DM (P) := {Q ⊆ P | Qu‘ = Q} is a complete lattice under inclusion, and it is Dedekind–MacNeille completion of P. In this case ’(x) =↓ x:
DM preserves all suprema and inHma existing in P, i.e., if Q ⊆ P and Q exists in P, then (P) ’( Q) = ’(Q), and analogously for inHmum (note that ’(Q) = {’(q) | q ∈ Q}). 3. Cuts The two-element poset ({0; 1}; 6) as the codomain of characteristic functions, is the starting point for all deHnitions of fuzzy sets. Zadeh’s deHnition follows the fact that 0 and 1 are numbers; thus, fuzzy sets as mappings from a nonempty set to the unit interval [0; 1] enable the use of real functions in this generalization of set theory. However, this interval is still a poset under the usual ordering, moreover it is a bounded distributive lattice. It turns out that some very important notions concerning fuzziness (such as cut functions and their structure) depend only on the fact that the codomain of the membership function is a poset. Moreover, some algebraic methods, particularly from the lattice theory, can be successfully used in applications of fuzzy structures, provided that this codomain is some special poset or a lattice. This motivates the following main deHnition. If A is a nonempty set and (P; 6) a partially ordered one, then a function AM : A → P is a partially ordered fuzzy set (P-fuzzy set) on A, or a fuzzy subset of A. M for p ∈ P, is a mapping A cut function (p-cut, cut) of A, AM p : A → {0; 1}; such that for x ∈ A, AM p (x) = 1
M if and only if A(x) ¿ p:
Obviously, AMp is the characteristic function of the following subset of A, called also a cut (p-cut, M cut subset) of A: M ¿ p}: Ap = {x ∈ A | A(x) If the partially ordered set is a complete lattice L with the bottom element 0 and the top element 1, then the corresponding fuzzy set AM : A → L is said to be lattice-valued (L-valued). Main properties of fuzzy sets are consequences of two basic facts: (a) they are functions, and (b) they can be uniquely represented by collections of cut subsets. More details are given in the sequel.
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All the statements formulated for partially ordered fuzzy sets are also fulHlled for the lattice-valued ones. Particular statements for L-fuzzy sets are given in cases when they diNer from the properties of P-valued ones. The Hrst proposition shows that for a given P-fuzzy set, every element of the supporting set provides the existence of a particular supremum in P. Proposition 1. If AM : A → P is a P-fuzzy set on A, then for every x ∈ A M A(x) = (p ∈ P | AM p (x) = 1); M i.e., the supremum of the set {p ∈ P | AMp (x) = 1} exists and is equal to A(x). M M = r ∈ P. Then, AMr (x) = 1. If for p ∈ P, AMp (x) = 1, then A(x)¿p, i.e., r¿p. Proof. For x ∈ A, let A(x) M Since r ∈ {p ∈ P | Ap (x) = 1}; r is the supremum of that set. It follows that M A(x) = r = {p | AM p (x) = 1}: In the case of lattice-valued fuzzy sets, the above property is also satisHed, however it can be formulated in terms of lattice operations. In that way, the synthesis of a fuzzy set is obtained. The following proposition is a direct consequence of Proposition 1. Proposition 2. If AM is an L-fuzzy set on A, then for every x ∈ A M p ◦ AM p (x); A(x) = p ∈L
where
p ◦ AM p (x) =
p
if AM p (x) = 1;
0
otherwise:
The inHnite join appearing in the formulation of the preceding proposition always exists, as proved in Proposition 1. However, the lattice L is usually taken to be complete, since most of the operations on fuzzy algebraic structures require arbitrary meets (see, e.g., [22]). One of the best known properties of cut subsets is their connection with the order in P, precisely the fact that smaller cuts (under the set inclusion) correspond to greater elements in P. The following proposition can be easily proved. Proposition 3. Let AM : A → P be a P-fuzzy set on A. Now, if p; q ∈ P and p 6 q; then Aq ⊆ Ap : It is easy to check that the opposite implication is not true in general. Conditions under which it is satisHed, and some other, more detailed descriptions of cuts are given in the sequel. Actually, elements of a poset behave exactly like cuts of a fuzzy set, provided that these elements are values of the fuzzy set.
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Proposition 4. Let AM : A → P be a P-valued fuzzy set on A. Then M = A(y) M (a) for x; y ∈ A; A(x) if and only if AA(x) M = AA(y) M ; M (b) for p ∈ P and x ∈ A; A(x)¿p if and only if AA(x) M ⊆ Ap . M = A(y). M Proof. (a) Let A(x) Then obviously AA(x) M = AA(y) M . M = A(y), M M ¿A(y), M M 6A(y). M M M On the other hand, if A(x) then A(x) or A(x) In both cases, {z | A(z)¿ A(x)} M M ={z | A(z)¿A(y)}, since x or y does not belong to both sets. (b) ⇒ Obvious, by Proposition 3. M ¿p, then x ∈ M M ⇐ If A(x) = Ap , and hence AA(x) (obviously, A(x)¿ A(x)). M ⊆ Ap , since x ∈ AA(x) M Corollary 1. Let AM : A → P be a P-valued fuzzy set on A. Then, for x; y ∈ A, M M ⊆ AA(y) A(y) 6 A(x) if and only if AA(x) M M : M Proof. Straightforward by Proposition 4(b), for p = A(y). General properties of P-fuzzy sets are also connected with cut subsets. The poset of cuts is closed under particular intersections: for every element, the intersection of all cuts containing it, is again a cut set. Moreover, the union of all cuts is the whole set. This is proved in the sequel. For a P-fuzzy set AM on A, let AP and AMP denote its collections of cut subsets and cut functions, respectively: AP := {Ap | p ∈ P};
AM P := {AM p | p ∈ P}:
% selja and Tepav%cevi*c [17]). Let AM : A → P be a P-valued fuzzy set on A. Then: Proposition 5 (Se% (1) if for P1 ⊆ P there is a supremum P1 in P, then (Ap | p ∈ P1 ) = A(p|p∈P1 ) ; (2)
(Ap | p ∈ P) = A;
(3) for every x ∈ A, (Ap | x ∈ Ap ) ∈ AP : Proof. (1) If the supremum of P1 exists, then M ¿ (p|p ∈ P1 ); x ∈ A(p|p∈P1 ) if and only if A(x) M if and only if A(x) ¿ p for all p ∈ P1 ; if and only if for all p ∈ P1 ; x ∈ Ap ; if and only if x ∈ (Ap | p ∈ P1 ):
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M = p ∈ P. Then, (2) Let x ∈ A and A(x) x ∈ Ap and x ∈ (Ap | p ∈ P): (3) For x ∈ A; x ∈ Ap if and only if AMp (x) = 1. Hence, (Ap | x ∈ Ap ) = (Ap | AM p (x) = 1) by (1) = A(p|AMp (x)=1) ∈ AP ; since by Proposition 1 that supremum always exists. Lattice-valued fuzzy sets have the above properties, and also some other. Namely, collection of cuts is closed under arbitrary intersections (not only under intersections connected with a particular element, as for P-valued sets). Moreover, this collection is a lattice. Finally, the intersection of cuts is again a cut, indexed by the join, which obviously always exists in lattices. % selja et al. [28]). Let AM : A → L be an L-fuzzy set on A. Then Proposition 6 (Se% (1) The collection AML = {Ap | p ∈ L} of cut subsets of AM is a Moore’s family of subsets of A, and a lattice under the set inclusion; (2) (Ap | p ∈ K ⊆ L) = A(p|p∈K) : We shall also need the following simple property of lattice-valued fuzzy sets. This is a special case of (2) in Proposition 6. Lemma 1. Let AM : A → L be an L-fuzzy set on A. Now, if Aq ⊆ Ap ; then Aq = Aq∨p : As it can be concluded from the above propositions and examples, not all the cuts of a P- (or L-) fuzzy set are diNerent. Hence every fuzzy set induces a partition of the poset P, as shown in the sequel. Let AM : S → P be a P-fuzzy set on S, and ≈ a binary relation on P, such that for p; q ∈ P p≈q
if and only if Ap = Aq :
≈ is obviously an equivalence relation on P. As usual, M M A(S) = {p ∈ P | p = A(x); for some x ∈ S}: We shall state the main properties of the above partition and of the ordering induced on it. To begin with, we shall describe ≈ in terms of the order in P.
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% selja and Tepav%cevi*c [17]). If AM is a P-fuzzy set on S and p; q ∈ P, then Proposition 7 (Se% p ≈q
M M if and only if ↑ p ∩ A(S) =↑ q ∩ A(S):
Proof. The relation p ≈ q holds if and only if A p = Aq if and only if for every x ∈ S M M A(x) ¿ p ⇔ A(x) ¿q if and only if M M {x ∈ S | A(x) ∈↑ p} = {x ∈ S | A(x) ∈↑ q} if and only if M M ↑ p ∩ A(S) =↑ q ∩ A(S): The values of the membership function are maximum elements of ≈-classes, as shown in the following proposition. Note that for p ∈ P, [p]≈ := {q ∈ P | p ≈ q}: M Lemma 2. Let AM : S → P be a P-fuzzy set. Now if for x ∈ S p = A(x), then p is the greatest element of the ≈-class to which it belongs. M Proof. For every q ∈ [p]≈ ; p = A(x)¿q, hence p is the greatest element in the class. The relation 6 in the poset P induces an order on the set of equivalence classes modulo ≈, i.e., on P=≈, in the following way: for p; q ∈ P, let [p]≈ 6 [q]≈
M M if and only if ↑ q ∩ A(S) ⊆↑ p ∩ A(S):
(1)
Obviously 6 is well deHned, and since it is constructed by the set inclusion, it is an ordering M as it can relation on P=≈. This order is anti-isomorphic with the set inclusion among cut sets of A, be seen from the following proposition. Proposition 8. If AM is a P-fuzzy set on S, then: [p]≈ 6 [q]≈
if and only if Aq ⊆ Ap :
Proof. [p]≈ 6 [q]≈ if and only if M M ↑ q ∩ A(S) ⊆↑ p ∩ A(S)
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if and only if M M {x ∈ S | A(x) ∈↑ q} ⊆ {x ∈ S | A(x) ∈↑ p} if and only if for x ∈ S M M A(x) ¿ q implies A(x) ¿p if and only if A q ⊆ Ap : We proceed with particular properties of ≈, for lattice-valued fuzzy sets. Note that for p ∈ L; [p]≈ denotes the supremum of the class [p]≈ (to which it, by Proposition 6(2), belongs): [p]≈ = (q | q ∈ [p]≈ ): We shall prove that ≈-classes behave precisely like the corresponding suprema. Proposition 9. If AM is an L-fuzzy set on S, then: M = [A(x)] M (a) for every x ∈ S; A(x) ≈; (b) for any p; q ∈ L, [p]≈ 6 [q]≈ [p]≈ 6 [q]≈ if and only if (the order on the right is the one from L). Proof. (a) Follows by Lemma 2, since the supremum of the class is its element. (b) Let [p]≈ 6[q]≈ . Then by Proposition 8 and by (a) it follows that A[q]≈ ⊆ A[p]≈ : Now, by Lemma 1, [q]≈ ∨ [p]≈ ≈ [q]≈ ; and hence [q]≈ ∨ [p]≈ ∈ [q]≈ . Thus, [q]≈ ∨ [p]≈ 6 [q]≈ , i.e., [p]≈ 6 [q]≈ . The converse follows by Propositions 3 and 8. It is easy to prove that for lattice-valued fuzzy sets the mapping p → [p]≈ (p ∈ L) is a closure operation on L. Then in lattice terms, L=≈ is a quotient in L. Hence, if ≈-classes are one element sets, it is obvious that L ∼ = L=≈. The above property of ≈-classes for lattices, is not generally satisHed for posets. Namely, it is not true that for one-element ≈-classes, P=≈ is under the order deHned by (1) isomorphic with P, for every poset P. This is illustrated by the following simple example.
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Fig. 1.
Example 1. Let S = {a; b}, and let P be the poset given in Fig. 1. The P-fuzzy set AM = ( pa br ) induces one element ≈-classes on P. However, it is easy to check that by (1), these classes are ordered in a (three-element) chain. Hence, P is not isomorphic with P=≈. The following theorem is a kind of a converse of Proposition 5 in which the main properties of P-fuzzy sets were formulated. It turns out that a collection of subsets of a given set determines its P-fuzzy subset, provided that this collection satisHes particular conditions concerning set intersections.
Theorem 1. Let P be a family of subsets of a nonempty set A, union of which is also A, and such that for every x ∈ A, (p ∈ P | x ∈ p) ∈ P: Let AM : A → P be de=ned by M A(x) := (p ∈ P | x ∈ p): Then, AM is a P-fuzzy set on A, where (P; 6) is anti-isomorphic with (P; ⊆), and for every p ∈ P, p = Ap . M Proof. P is a set partially ordered by the set inclusion. The mapping A(x) is well deHned, and it is a fuzzy set on A. It remains to prove that p = Ap (i.e., that the family of cut subsets of the deHned P-fuzzy set coincides with P). M Let x ∈ A. Then x ∈ Ap if and only if A(x)¿p if and only if (q ∈ P | x ∈ q)¿p if and only if (q ∈ P | x ∈ q) ⊆ p if and only if x ∈ p. As a consequence, we have the following proposition, describing a construction of lattice-valued fuzzy sets by means of subsets. Theorem 2. Let A be a nonempty set and F a family of its subsets closed under arbitrary intersections and containing A (the Moore’s family of subsets of A). Let also L be the lattice dual to (F; ⊆) and AM : A → L an L-valued subset of A de=ned by M A(x) := (f ∈ F | x ∈ f): Then, the lattice of cut subsets of AM is isomorphic with (F; ⊆).
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Fig. 2.
The above induced order 6 on P=≈ (by (1)) shows that the same collection of cut sets can be obtained by the decomposition of diNerent P-fuzzy sets. Moreover, the induced P=≈-fuzzy set (i.e., a mapping S → P=≈) has one element ≈-classes. The construction described in Theorem 1 actually gives a P-fuzzy set with such classes. Importance of this factor poset can be seen from the following proposition. Proposition 10. If AM is a P-fuzzy set on S, then the poset (P=≈; 6) is dually isomorphic with the M poset (AMP ; ⊆) of cut subsets of A. Proof. The dual isomorphism is given by f : [p]≈ → Ap . Indeed, f is obviously onto, and by the deHnition of ≈-classes it is an injection. By Proposition 8, both f and its inverse dually preserve the order, hence f is an anti-isomorphism. Example 2. Let (P; 6) be the poset given in Fig. 2 (left), and S = {a; b; c; d}. For the P-fuzzy set a b c d AM = u v r q cut subsets are Ap = ∅;
Aq = As = {d};
Ar = At = Aw = {c; d};
Au = {a; c; d};
Av = {b; d};
Ax = Ay = {a; b; c; d}: The partition of P into ≈-classes is represented in the same diagram. (P=≈; 6) is dually isomorM phic with the poset A(S) of cuts, which is ordered by the set inclusion. The corresponding diagram of cuts is given in Fig. 2, on the right.
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In case of lattice-valued fuzzy sets posets of cuts are lattices under the set inclusion (Proposition 6(1)), hence there is an immediate consequence of Proposition 10, concerning posets of ≈-classes. Corollary 2. Let AM : S → L be an L-fuzzy set on S. Then the poset (L=≈; 6) is a lattice, antiM isomorphic with the lattice (AML ; ⊆) of cuts of A. For lattice-valued fuzzy sets, there is a particular way to check whether all ≈-classes are diNerent, in which case L; L=≈ and the dual of AML are isomorphic lattices. Recall that an element p of the lattice L is meet-irreducible if b ∧ c = a implies b = a or c = a. Proposition 11. Let L be a lattice of =nite length, and AM : S → L an L-fuzzy set. Then all cuts of AM M are distinct if and only if all meet-irreducible elements of L di>erent from 1 are contained in A(S). M Proof. (⇒) Suppose p = 1 is a meet-irreducible element which is not in A(S). Since L is of HM M ∩ ↑q, and by nite length, there is a unique q ∈ L covering p. Then, obviously, A(S) ∩ ↑p = A(S) Proposition 7, AMp = AMq . (⇐) Suppose that p and q are distinct elements of L. Now there are two possibilities. Case 1: p = 1 or q = 1. Assume p = 1 = q. Then by the known lattice property (see, e.g., [2]) M ∩ ↑ q, but r is not an element there is a meet-irreducible element r such that q6r¡1. Thus, r ∈ A(S) M M M of A(S) ∩ ↑p, and so by Proposition 6, Ap = Aq . Case 2: p = 1 and q = 1. Then, as above,
p = {r ∈ L | p 6 r ¡ 1; r is meet-irreducible};
q = {s ∈ L | q 6 s ¡ 1; s is meet-irreducible}: M ∩ ↑ q = A(S) M ∩ ↑p, and by Proposition 6, AMp = AMq . Since p = q, this implies that A(S) The characterization of cut sets by meet-irreducible elements cannot be applied on real intervalvalued fuzzy sets, simply because the unite interval is not a lattice of Hnite length. However, if the lattice is a Hnite chain, then all its elements are meet-irreducible, which means that diNerent cuts could be obtained only if the set of values coincides with the set of all elements of the lattice distinct from 1. 4. Canonical representation. Completion In the previous section it was shown that the poset of cut subsets preserves basic properties of a fuzzy set. In fact, Theorems 1 and 2 say that for a given fuzzy set AM : A → P (where P is a poset M with exactly the same or a lattice), there is another function from A to the poset of cuts of A, structure of cut subsets. It also means that every fuzzy set has its inner representation, independent of the poset or the lattice which is the codomain of the mapping. This situation resembles to the one with homomorphisms and congruence relations in universal algebra: every homomorphic image
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of an algebraic structure is isomorphic with the structure factored by a suitable congruence relation, which is an inner representation of the homomorphic image. Let AM : A → P be a P-fuzzy set on A, and C the poset of cuts of AM with the order 6 dual to the set inclusion. The canonical representation of AM is a mapping kA : A → C; deHned as in Theorem 1: for x ∈ A, kA(x) = (f ∈ C | x ∈ f): The quoted theorem proves that AM and kA have isomorphic posets of cuts. We shall also prove that these two fuzzy sets have isomorphic posets of codomains, from which they take corresponding values under that isomorphism. Proposition 12. Let AM be a P-fuzzy set on A, and kA its canonical representation. Then (a) for every x ∈ A; kA(x) = AA(x) M ; (b) for f ∈ C; kAf = f (kAf is the f-cut of kA). (Observe that C is the set of cuts of AM and at the same time the codomain of kA.) Proof. (a) If y ∈ kA(x) = (Ap | x ∈ Ap ), then y belongs to every cut Ap which contains x. Hence, (since x ∈ AA(x) y ∈ AA(x) M M ). Thus, kA(x) ⊆ AA(x) M . On the other hand, if y ∈ AA(x) M , then y ∈ Ap for every p ∈ P such that AA(x) ⊆ A . Hence, M p M = (Ap | x ∈ Ap ), i.e., y ∈ kA(x). Hence, AA(x) y ∈ (Ap | AA(x) M ⊆ Ap ) = (Ap | A(x)¿p) M ⊆ kA(x), and the required equality is proved. (b) Let f = Ap , for p ∈ P, and let x ∈ kAAp . Then, kA(x)¿Ap in C, which means that AA(x) M ⊆ Ap , M and by Proposition 4(b), AA(x) and hence x ∈ Ap . On the other hand, if x ∈ Ap , then A(x)¿p, M ⊆ Ap , ⊆ Ap } = {y | kA(y) ⊆ Ap } = {y | kA(y)¿Ap } = kAAp . The equality of two sets and hence x ∈ {y | AA(y) M is thus proved. The above proposition proves the conjecture from the beginning of the section: a P-fuzzy set AM and its canonical representation kA have isomorphic posets of codomains, and identical posets of cuts. Example 3. The mapping a b c AM = p q r is an L-fuzzy set on A = {a; b; c}, where L is the lattice represented in Fig. 3(i). The cut subsets of AM are A1 = As = ∅;
Ar = {c};
Ap = {a};
Aq = {b; c};
A0 = {a; b; c}:
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Fig. 3.
Fig. 4.
The poset C of the above cuts is under the order dual to the set inclusion, a pentagon lattice (Fig. 3(ii)). The canonical representation of AM is the mapping kA : A → C, given by a b c kA = {a} {b; c} {c} (recall that kA(x) = (f ∈ C | x ∈ f)). Obviously, AM and kA have identical collections of cuts, and their posets of codomains are isomorphic (Fig. 4). We have already seen that some collections of subsets of a set A uniquely determine a P-( or L-) fuzzy set on A, in the sense of the above canonical representation. Precisely, the union of the collection should be the whole set, and particular closedness under set intersections should be fulHlled. However, we can start with an arbitrary collection C of subsets of A and look for a P-fuzzy M There is always such fuzzy set AM on A, such that C can be embedded into the poset of cuts of A. set obtained simply by taking the power set of A to be the collection of cuts. In the following, we show by a particular construction that, if the union of C is A, there is a P-fuzzy set whose poset of cuts is minimal among those that contain C as a subset of the poset of cuts. The following lemma says that adding the intersection of all sets containing one element, does not change intersections connected with other elements.
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Lemma 3. Let C be a collection of subsets of a nonempty set A. If a; b ∈ C and a ∈ {f ∈ C | b ∈ f}, then {f | a ∈ f} ⊆ {f | b ∈ f}: (2) Proof. Let x ∈ {f | a ∈ f}. This statement is equivalent with the following: (∀f ∈ C)(a ∈ f → x ∈ f). Similarly,by the assumption, (∀f ∈ C)(b ∈ f → a ∈ f). Hence, if b ∈ f then x ∈ f, which proves that x ∈ {f | b ∈ f}. Thus, (2) holds. Theorem 3. Let C be the collection of subsets of a nonempty set A, union of which is A. Then there is a P-fuzzy set AM on A with the minimal poset of cuts containing C (i.e., there is no fuzzy set BM on A, such that C ⊆ BP , and BP ⊆ AP ). Proof. Let P be the collection of subsets of A obtained by adding to C, for every x ∈ A, the intersection of all members of C to which x belongs. Since the union of C is A, every x is in some member of C, and this intersection can be uniquely determined. Obviously, C ⊆ P. By the previous lemma, P is a collection of subsets of A which satisHes conditions of Theorem 1, hence there is a P-fuzzy set AM having P as a poset of cuts, i.e., for which AP = P. Thus, AP contains precisely the elements of C and, for every x ∈ A, the intersection of all members of C to which it belongs. Moreover, no fuzzy set on A could have smaller (under ⊆) collection of cuts which includes C, since any such smaller collection would lack the intersection associated to some a ∈ A. Theorem 3 gives an algorithm for the construction of a fuzzy set AM on A, with a particular collection C as a subset of AP , provided that the union of C is A. In addition, AM has a minimal (however not generally unique) such collection of cuts. We shall say that AM is a P-completion of the collection C). Example 4. (a) Let C be the following collection of subsets of a set S = {a; b; c; d; e; f}: C = {{a; b; c; d; f}; {b; c; d; e}; {a; b}; {c}; {d; e}}: The union of C is S, and the P-completion of C, the collection P, is obtained by adding to C, for every element of S, the intersection of all members of C which contain this element: P = C ∪ {{b}; {d}}: Note that P is not closed under arbitrary intersections ({b; c; d} is missing). However, P is a poset under the order dual to the set inclusion (Fig. 5), and it satisHes conditions for the construction of a P-fuzzy set AM : S → P (Theorem 1): a b c d e f AM = : {a; b} {b} {c} {d} {d; e} {a; b; c; d; f} If C is an arbitrary (nonempty) collection of subsets of A, then the union of C might not be equal to A. In that case, it is not possible to apply the above algorithm. However, since there are P-fuzzy sets containing C as a subset of cuts, we shall deHne a completion of C in the following way.
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Fig. 5.
Let A be a nonempty set and C a collection of its subsets. Let C = C ∪ {A}: C satisHes conditions of Theorem 3. The P-fuzzy completion of C will be called a P-completion of the collection C. Thus, for every collection C of subsets of A, there is a unique P-fuzzy set AM on A, which is a P-completion of C. Summing up the above constructions, we can describe the P-completion CP of a collection C of subsets of A as follows: if C = A; C ∪ C CP := C ∪ {A} ∪ C else; where
C := Fx | x ∈ A and Fx = (f ∈ C ∪ {A} | x ∈ f) :
4.1. Lattice valued completions Situation with lattice-valued fuzzy sets, concerning possible completions, is slightly diNerent. Namely, we shall prove that for every collection of subsets of a set A, there is an L-fuzzy set with the unique minimal collection of cuts in the above sense. Let C be a collection of subsets of a set A, as before. A lattice-valued completion (L-completion) of C is a collection of subsets CL of A, deHned by CL := C ∪ C ∪ {A}; where
C := F | F = (f | f ∈ K ⊆ C)
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(K runs over all nonempty subsets of C). In other words, the L-completion of C is obtained by adding to C all the intersections of its members, and also the whole set A (if it is not already there). Lemma 4. The L-completion CL of an arbitrary collection C of subsets of a set A is a lattice under the set inclusion. Theorem 4. Let A be a set, C a nonempty collection of its subsets, and CL the L-completion of C. Then, the mapping AM : A → CL (CL is ordered dually to the set inclusion), de=ned by M A(x) = (f ∈ CL | x ∈ f); is an L-fuzzy set on A, whose poset of cuts is CL . In addition, AM is the L-valued subset of A with the smallest collection of cuts containing C. Proof. The collection CL satisHes conditions of Theorem 2, and hence AM is an L-fuzzy subset of A whose poset of cuts is CL , which proves the Hrst part. Further, we shall prove that CL is the subset of every collection of cuts of any L-fuzzy subset containing C. Let BM be an L-fuzzy subset of A whose poset of cuts H contains C. Obviously, H is closed under intersections and contains A. On the other hand, every element f of CL is either a subset from C, or it is the intersection of some subsets from C; it can also be A. In each of these cases, f also belongs to H. Hence, among the L-valued subsets of A; AM has the minimal collection of cuts containing C.
5. Completion of posets to complete lattices Here we use the procedure described above (to fuzzy sets) to make a completion of an arbitrary poset to a complete lattice. Every poset can be represented by an isomorphic collection of cuts of the particular fuzzy set, as follows. M = x, for all x ∈ P. Then, P is Let P be a partially ordered set. Let AM : P → P be deHned by A(x) anti-isomorphic with AP , under p → Ap . (Anti-isomorphism is isomorphism of P with AP ordered dually to inclusion.) A completion of the poset P (possible a lattice which is not complete), to a complete lattice by fuzzy sets, runs as follows. Firstly, we make a representation by cut sets, deHning a fuzzy set AM : P → P above. By Proposition M closed 5 the collection of cut sets of this fuzzy set is a family of subsets of the domain P of A, under componentwise intersections. Now, we complete this collection of subsets by all missing set intersections, and add the domain P to the collection. In such a way we obtain a lattice of subsets and a corresponding fuzzy set as in Theorem 3. This lattice is a lattice of cuts of the obtained fuzzy set, and a completion of poset P. The above algorithm by which every poset can be completed to a complete lattice is a completion by fuzzy sets.
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Fig. 6.
Fig. 7.
Example 5. Let P set representing P a b AM = a b
be a fuzzy set given by its Hasse diagram in Fig. 6. P = {a; b; c; d}, and a fuzzy is AM : P → P, c d : c d
The corresponding cut sets are: AMa = {a}; AMb = {b}; AMc = {a; b; c}; AMd = {a; b; d}. Family AMP is closed under componentwise intersections. In order to obtain a completion of this family, we add all intersections of elements from AMP : {a; b}; ∅ and set P. In this way we obtain the collection {∅; {a}; {b}; {a; b}; {a; b; c}; {a; b; d}; {a; b; c; d}} which is a lattice under inclusion. The obtained completion is the lattice L dual to the above collection lattice and it is represented in Fig. 7. In the following we prove that the completion of posets by fuzzy sets preserves all existing suprema and inHma. More precisely, we show that though the procedure is completely diNerent, the obtained lattice is the Dedekind–MacNeille completion. Theorem 5. Let P be a partially ordered set, and LP the lattice obtained by the completion of P by fuzzy sets. This completion coincides with the Dedekind–MacNeille completion. Proof. Let P be a partially ordered set. Let AM : A → P be a fuzzy set such that its family F of cut sets, is under inclusion the poset anti-isomorphic with P. Family F is closed under componentwise
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M Let L be a family of all intersections of all intersections, and its set union is the domain of A. subfamilies of F plus A: L= F1 | F1 ⊆ F ∪ {A}: Since L is a family of subsets of set A closed under intersections containing set A; L is a lattice under inclusion. Let L = (L; 6) be a lattice anti-isomorphic with (L; ⊆). This lattice is a completion of P, since F is anti-isomorphic with P. If the dual of L is a completion of the dual of P, than L is a completion of P, as well. We have that for a p ∈ P, cut Ap is equal to ↑p, i.e., in the dual of P, Ap = ↓p. Now we prove that L = DM (P). Recall that DM (P) = {X ⊆ P | X u‘ = X } and L=
{↓ a | a ∈ X ⊆ P} ∪ {P}:
Since {↓ x}u‘ = ↓ x, we have that ↓ x ∈ DM (P). To prove that L ⊆ DM (P), let X ⊆ P. An element of L is P or it is of the form {↓ a | a ∈ X }= {x ∈ P | (∀a ∈ X )x 6 a} = X ‘: u‘
Since P u‘ = P and X ‘ = X ‘ , it follows that in both cases we obtain an element from DM (P), hence L ⊆ DM (P). On the other hand, if X ⊆ P, such that X ∈ DM (P), i.e., X u‘ = X , then X u‘ = {↓ a | a ∈ X u } ∈ L: Hence, DM (P) ⊆ L, completing the proof. 6. Conclusion Fuzzy sets have been introduced as a tool for investigating imprecise reasoning, particularly to deal with problems which could hardly be handled by methods of classical mathematics. It turns out that the development of fuzzy logic and its techniques have inPuenced—as a feedback—many branches of contemporary mathematics. Our intention was to present a part of such inPuence in the theory of sets and ordered structures. Our investigations were mainly focused on the algebraic and ordertheoretic properties of the collection of cuts of a fuzzy structure. We point out the importance of the canonical representation of fuzzy structures, which makes this structure independent of the co-domain. In applications, our technique turn out to be very successful in completions of some classes of posets and lattices. Still, it remains to apply this cut approach in completions and structural analysis of posets appearing in theoretical computer science: algebraic and complete posets, continuous lattices, Boolean algebras, etc.
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A. Achache, Parties Flues Compactes d’un Ensemble, Portugal. Math. 42 (3) (1983–84) 311–315. G. BirkhoN, Lattice Theory, American Mathematical Society, Providence RI, 1967. B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1992. J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. A. Kaufmamm, J.G. Aluja, Selection of aTnities by means of fuzzy relations and Galois lattices, Fuzzy Systems Artif. Intell., Rep. Lett. 1 (1) (1992) 51–67. % selja, Constructing maximal block-codes by bisemilattice valued fuzzy sets, Novi Sad J. Math. V. Lazarevi*c, B. Se% 28 (2) (1998) 79–90. % selja, A. Tepav%cevi*c, Bisemilattice-valued fuzzy sets, Novi Sad J. Math. 28 (3) (1998) 105–114. V. Lazarevi*c, B. Se% V. Lazarevi*c, A. Tepav%cevi*c, Theorem of synthesis for bisemilattice-valued fuzzy sets, Math. Moravica 2 (1998) 75 –83. Y.M. Liu, M.K. Luo, Fuzzy Topology, in: Advances in Fuzzy Systems—Applications and Theory, vol. 9, World ScientiHc, Singapore, 1997. S. Mili*c, A. Tepav%cevi*c, On P-fuzzy correspondences and generalized associativity, Fuzzy Sets and Systems 96 (1998) 223–229. C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, in: Interdisciplinary Systems Research Series, vol. 11, Birkhauser, Basel, Stuttgart and Halsted Press, New York, 1975. C.V. Negoita, D.A. Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. D.A. Ralescu, A generalization of the representation theorem, Fuzzy Sets and Systems 51 (1992) 309–311. % selja, Lattice of partially ordered fuzzy subalgebras, Fuzzy Sets and Systems 81 (1996) 265–269. B. Se% % selja, Distributive lattices via fuzzy sets, Proc. Conf. IPMU, Madrid, 2000, pp. 1591–1594. B. Se% % B. Seselja, Homomorphisms of poset valued algebras, Fuzzy Sets and Systems 121 (2001) 333–340. % selja, A. Tepav%cevi*c, On a construction of codes by P-fuzzy sets, Rev. Res. Fac. Sci. Univ. Novi Sad 20 (2) B. Se% (1990) 71–80. % selja, A. Tepav%cevi*c, Relational valued fuzzy sets, Fuzzy Sets and Systems 52 (1992) 217–222. B. Se% % selja, A. Tepav%cevi*c, Fuzzy boolean algebras, Automated Reasoning, IFIP Transactions A-19, North-Holland B. Se% Publishing Company, Amsterdam, 1992, pp. 83–88. % selja, A. Tepav%cevi*c, On generalizations of fuzzy algebras and congruences, Fuzzy Sets and Systems 65 (1994) B. Se% 85–94. % selja, A. Tepav%cevi*c, Representation of lattices by fuzzy sets, Inform. Sci. 79 (3– 4) (1994) 171–180. B. Se% % selja, A. Tepav%cevi*c, Partially ordered and relational valued fuzzy relations I, Fuzzy Sets and Systems 72 B. Se% (1995) 205–213. % selja, A. Tepav%cevi*c, Fuzzy groups and collection of subgroups, Fuzzy Sets and Systems 83 (1996) 85–91. B. Se% % selja, A. Tepav%cevi*c, Representation of semilattices by meet-irreducibles, Indian J. Pure Appl. Math. 27 (11) B. Se% (1996) 1093–1100. % selja, A. Tepav%cevi*c, Semilattices and fuzzy sets, J. Fuzzy Math. 4 (1997) 899 –906. B. Se% % selja, A. Tepav%cevi*c, On a representation of posets by fuzzy sets, Fuzzy Sets and Systems 98 (1998) 127–132. B. Se% % B. Seselja, A. Tepav%cevi*c, On a completion of poset by fuzzy sets, Novi Sad J. Math. 28 (2) (1998) 71–77. % selja, A. Tepav%cevi*c, G. Vojvodi*c, L-fuzzy sets and codes, Fuzzy Sets and Systems 53 (1993) 217–222. B. Se% % selja, G. Vojvodi*c, Fuzzy sets on S as closure operations on P(S), Rev. Res. Fac. Sci. Univ. Novi Sad 14 (1) B. Se% (1984) 117–127. % selja, G. Vojvodi*c, A note on the lattice of weak L-valued congruences on the group of prime order, Fuzzy B. Se% Sets and Systems 22 (1987) 321–324. U.M. Swamy, D. Viswanadha Raju, Algebraic fuzzy systems, Fuzzy Sets and Systems 41 (1991) 187–194. A. Tepav%cevi*c, G. Trajkovski, L-fuzzy lattices: an introduction, Fuzzy Sets and Systems 123 (2001) 206–219. A. Tepav%cevi*c, A. Vuji*c, On an application of fuzzy relations in biogeography, Inform. Sci. 89 (1996) 77–94. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.
Completion of ordered structures by cuts of fuzzy sets: an overview % selja∗ , Andreja Tepav%cevi*c Branimir Se% Technical Faculty, Department of Mathematics, Institute of Mathematics, University of Novi Sad, Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia Received 23 July 2001; received in revised form 17 May 2002; accepted 9 July 2002
Abstract The aim of the paper is to present a role of fuzzy sets in the theory of ordered structures. Main algebraic properties of cuts of fuzzy sets are given, and a completion of partially ordered sets to complete lattices is described. It turns out that this completion is equivalent with the famous Dedekind–MacNeille completion, but the algorithm presented here is much simpler. c 2002 Elsevier Science B.V. All rights reserved. MSC: 04A72; 06A23 Keywords: Algebra; Partially ordered set; Fuzzy set; Cut; Canonical representation; Completion
1. Introduction One of the basic properties of fuzzy structures is that they are mappings, generalizations of characteristic functions in set theory. Another important property is algebraic: every fuzzy structure is uniquely determined by the collection of subsets of the support, known as cuts. These cuts can be ordered naturally: by inclusion, or by the reverse ordering. The obtained collection of sets is therefore an ordered structure. Algorithms for transitions from the function (fuzzy structure) to the ordered family of cuts and vice versa are based on closure operators, particular set-theoretic completions, and structure properties of ordered sets. Techniques and methods obtained for such purposes are useful tools for investigating fuzzy structures. On the other hand, these can be applied back in the classical theory of ordered structures: it is possible to represent ordered structure by cuts of a suitable fuzzy set , and consequently by the mapping itself. ∗
The research supported by Serbian Ministry of Science and Technology, Grant No. 1227. Corresponding author. % selja). E-mail address: [email protected] (B. Se%
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 6 5 - 2
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By the above, the aim of this article is to present systematically algebraic background and basic properties of cuts and ordering on the collection of these. To provide compact presentation, we introduce the canonical representation of fuzzy structures as mappings, which make them independent of the co-domain. By this approach, every fuzzy set : A → P, where P is an ordered structure (interval [0; 1], in Zadeh’s sense [34], lattice as introduced by Goguen [4], poset, etc., see [14,19]), is replaced by the suitable mapping from A into its power set. Still, main ordering properties of the original function are preserved. Next we introduce particular completion in the collection of subsets of a given set, in order to obtain a collection of cuts of the “nearest” fuzzy set. In this way we get the famous Dedekind–McNeille completion, best known as a procedure to construct reals out of rationals (see e.g., [3]). This overview is a result of investigations conducted by the authors over several years. Some of the presented results are known some are new. Our intention was to collect and unify these results and present them as an algebraic background for investigating fuzzy structures. Naturally, in our research we have been using relevant known results from the algebraic theory of fuzzy structures. Some of these are cited in references; we mention particularly the early work of Negoita and Ralescu [11,12,13], the paper [1] and as a new and signiHcant contribution, the book by Liu and Luo [9]. The rest of the papers listed in references are connected with generalizations of the characteristic function ([6,7,8,18,29]), with some applications in mathematics and other sciences ([5,10,15,21,24,25,26,27,33]) and also with particular developing of fuzzy algebraic structures ([16,19,20,23,30,31,32]). In addition to our papers, we added only those which are closely related to our investigations. Therefore the list should not be considered as complete in any sense. 2. Preliminaries Let (P; 6) be a partially ordered set (poset), denoted also by the underlying set P only. The bottom element of a poset, if it exists, is denoted by 0; similarly, the top element is denoted by 1. If S ⊆ P, the set of all upper bounds of S is denoted by S u , and the set of all lower bounds is denoted by S ‘ : S u := {x ∈ P | (∀s ∈ S)s 6 x} and S ‘ := {x ∈ P | (∀s ∈ S)s ¿ x}: The greatest lower bound (inHmum) of a set S ⊆ P, if it exists, is denoted by S and dually, the supremum (least upper bound) by S. The inHmum of x and y, if it exists, is denoted by x ∧ y, and is said to be their meet. Similarly, supremum of x and y, if it exists, is their join, x ∨ y. If x ∈ P, then the ideal generated by x is denoted by ↓ x: ↓ x := {y ∈ P | y 6 x}: Dually, the Hlter generated by x is denoted by ↑ x: ↑ x := {y ∈ P | y ¿ x}:
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If supremum and inHmum exist for each pair x; y of elements in a poset, then this poset is a lattice and it is denoted by L. Recall that lattice L is complete if for every S ⊆ L, there exist supremum and inHmum. If P is a poset, L a complete lattice and ’ : P → L an order embedding (i.e., an injection preserving the order), than L is a completion of P. If (P; 6) is a poset, then DM (P) := {Q ⊆ P | Qu‘ = Q} is a complete lattice under inclusion, and it is Dedekind–MacNeille completion of P. In this case ’(x) =↓ x:
DM preserves all suprema and inHma existing in P, i.e., if Q ⊆ P and Q exists in P, then (P) ’( Q) = ’(Q), and analogously for inHmum (note that ’(Q) = {’(q) | q ∈ Q}). 3. Cuts The two-element poset ({0; 1}; 6) as the codomain of characteristic functions, is the starting point for all deHnitions of fuzzy sets. Zadeh’s deHnition follows the fact that 0 and 1 are numbers; thus, fuzzy sets as mappings from a nonempty set to the unit interval [0; 1] enable the use of real functions in this generalization of set theory. However, this interval is still a poset under the usual ordering, moreover it is a bounded distributive lattice. It turns out that some very important notions concerning fuzziness (such as cut functions and their structure) depend only on the fact that the codomain of the membership function is a poset. Moreover, some algebraic methods, particularly from the lattice theory, can be successfully used in applications of fuzzy structures, provided that this codomain is some special poset or a lattice. This motivates the following main deHnition. If A is a nonempty set and (P; 6) a partially ordered one, then a function AM : A → P is a partially ordered fuzzy set (P-fuzzy set) on A, or a fuzzy subset of A. M for p ∈ P, is a mapping A cut function (p-cut, cut) of A, AM p : A → {0; 1}; such that for x ∈ A, AM p (x) = 1
M if and only if A(x) ¿ p:
Obviously, AMp is the characteristic function of the following subset of A, called also a cut (p-cut, M cut subset) of A: M ¿ p}: Ap = {x ∈ A | A(x) If the partially ordered set is a complete lattice L with the bottom element 0 and the top element 1, then the corresponding fuzzy set AM : A → L is said to be lattice-valued (L-valued). Main properties of fuzzy sets are consequences of two basic facts: (a) they are functions, and (b) they can be uniquely represented by collections of cut subsets. More details are given in the sequel.
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All the statements formulated for partially ordered fuzzy sets are also fulHlled for the lattice-valued ones. Particular statements for L-fuzzy sets are given in cases when they diNer from the properties of P-valued ones. The Hrst proposition shows that for a given P-fuzzy set, every element of the supporting set provides the existence of a particular supremum in P. Proposition 1. If AM : A → P is a P-fuzzy set on A, then for every x ∈ A M A(x) = (p ∈ P | AM p (x) = 1); M i.e., the supremum of the set {p ∈ P | AMp (x) = 1} exists and is equal to A(x). M M = r ∈ P. Then, AMr (x) = 1. If for p ∈ P, AMp (x) = 1, then A(x)¿p, i.e., r¿p. Proof. For x ∈ A, let A(x) M Since r ∈ {p ∈ P | Ap (x) = 1}; r is the supremum of that set. It follows that M A(x) = r = {p | AM p (x) = 1}: In the case of lattice-valued fuzzy sets, the above property is also satisHed, however it can be formulated in terms of lattice operations. In that way, the synthesis of a fuzzy set is obtained. The following proposition is a direct consequence of Proposition 1. Proposition 2. If AM is an L-fuzzy set on A, then for every x ∈ A M p ◦ AM p (x); A(x) = p ∈L
where
p ◦ AM p (x) =
p
if AM p (x) = 1;
0
otherwise:
The inHnite join appearing in the formulation of the preceding proposition always exists, as proved in Proposition 1. However, the lattice L is usually taken to be complete, since most of the operations on fuzzy algebraic structures require arbitrary meets (see, e.g., [22]). One of the best known properties of cut subsets is their connection with the order in P, precisely the fact that smaller cuts (under the set inclusion) correspond to greater elements in P. The following proposition can be easily proved. Proposition 3. Let AM : A → P be a P-fuzzy set on A. Now, if p; q ∈ P and p 6 q; then Aq ⊆ Ap : It is easy to check that the opposite implication is not true in general. Conditions under which it is satisHed, and some other, more detailed descriptions of cuts are given in the sequel. Actually, elements of a poset behave exactly like cuts of a fuzzy set, provided that these elements are values of the fuzzy set.
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Proposition 4. Let AM : A → P be a P-valued fuzzy set on A. Then M = A(y) M (a) for x; y ∈ A; A(x) if and only if AA(x) M = AA(y) M ; M (b) for p ∈ P and x ∈ A; A(x)¿p if and only if AA(x) M ⊆ Ap . M = A(y). M Proof. (a) Let A(x) Then obviously AA(x) M = AA(y) M . M = A(y), M M ¿A(y), M M 6A(y). M M M On the other hand, if A(x) then A(x) or A(x) In both cases, {z | A(z)¿ A(x)} M M ={z | A(z)¿A(y)}, since x or y does not belong to both sets. (b) ⇒ Obvious, by Proposition 3. M ¿p, then x ∈ M M ⇐ If A(x) = Ap , and hence AA(x) (obviously, A(x)¿ A(x)). M ⊆ Ap , since x ∈ AA(x) M Corollary 1. Let AM : A → P be a P-valued fuzzy set on A. Then, for x; y ∈ A, M M ⊆ AA(y) A(y) 6 A(x) if and only if AA(x) M M : M Proof. Straightforward by Proposition 4(b), for p = A(y). General properties of P-fuzzy sets are also connected with cut subsets. The poset of cuts is closed under particular intersections: for every element, the intersection of all cuts containing it, is again a cut set. Moreover, the union of all cuts is the whole set. This is proved in the sequel. For a P-fuzzy set AM on A, let AP and AMP denote its collections of cut subsets and cut functions, respectively: AP := {Ap | p ∈ P};
AM P := {AM p | p ∈ P}:
% selja and Tepav%cevi*c [17]). Let AM : A → P be a P-valued fuzzy set on A. Then: Proposition 5 (Se% (1) if for P1 ⊆ P there is a supremum P1 in P, then (Ap | p ∈ P1 ) = A(p|p∈P1 ) ; (2)
(Ap | p ∈ P) = A;
(3) for every x ∈ A, (Ap | x ∈ Ap ) ∈ AP : Proof. (1) If the supremum of P1 exists, then M ¿ (p|p ∈ P1 ); x ∈ A(p|p∈P1 ) if and only if A(x) M if and only if A(x) ¿ p for all p ∈ P1 ; if and only if for all p ∈ P1 ; x ∈ Ap ; if and only if x ∈ (Ap | p ∈ P1 ):
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M = p ∈ P. Then, (2) Let x ∈ A and A(x) x ∈ Ap and x ∈ (Ap | p ∈ P): (3) For x ∈ A; x ∈ Ap if and only if AMp (x) = 1. Hence, (Ap | x ∈ Ap ) = (Ap | AM p (x) = 1) by (1) = A(p|AMp (x)=1) ∈ AP ; since by Proposition 1 that supremum always exists. Lattice-valued fuzzy sets have the above properties, and also some other. Namely, collection of cuts is closed under arbitrary intersections (not only under intersections connected with a particular element, as for P-valued sets). Moreover, this collection is a lattice. Finally, the intersection of cuts is again a cut, indexed by the join, which obviously always exists in lattices. % selja et al. [28]). Let AM : A → L be an L-fuzzy set on A. Then Proposition 6 (Se% (1) The collection AML = {Ap | p ∈ L} of cut subsets of AM is a Moore’s family of subsets of A, and a lattice under the set inclusion; (2) (Ap | p ∈ K ⊆ L) = A(p|p∈K) : We shall also need the following simple property of lattice-valued fuzzy sets. This is a special case of (2) in Proposition 6. Lemma 1. Let AM : A → L be an L-fuzzy set on A. Now, if Aq ⊆ Ap ; then Aq = Aq∨p : As it can be concluded from the above propositions and examples, not all the cuts of a P- (or L-) fuzzy set are diNerent. Hence every fuzzy set induces a partition of the poset P, as shown in the sequel. Let AM : S → P be a P-fuzzy set on S, and ≈ a binary relation on P, such that for p; q ∈ P p≈q
if and only if Ap = Aq :
≈ is obviously an equivalence relation on P. As usual, M M A(S) = {p ∈ P | p = A(x); for some x ∈ S}: We shall state the main properties of the above partition and of the ordering induced on it. To begin with, we shall describe ≈ in terms of the order in P.
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% selja and Tepav%cevi*c [17]). If AM is a P-fuzzy set on S and p; q ∈ P, then Proposition 7 (Se% p ≈q
M M if and only if ↑ p ∩ A(S) =↑ q ∩ A(S):
Proof. The relation p ≈ q holds if and only if A p = Aq if and only if for every x ∈ S M M A(x) ¿ p ⇔ A(x) ¿q if and only if M M {x ∈ S | A(x) ∈↑ p} = {x ∈ S | A(x) ∈↑ q} if and only if M M ↑ p ∩ A(S) =↑ q ∩ A(S): The values of the membership function are maximum elements of ≈-classes, as shown in the following proposition. Note that for p ∈ P, [p]≈ := {q ∈ P | p ≈ q}: M Lemma 2. Let AM : S → P be a P-fuzzy set. Now if for x ∈ S p = A(x), then p is the greatest element of the ≈-class to which it belongs. M Proof. For every q ∈ [p]≈ ; p = A(x)¿q, hence p is the greatest element in the class. The relation 6 in the poset P induces an order on the set of equivalence classes modulo ≈, i.e., on P=≈, in the following way: for p; q ∈ P, let [p]≈ 6 [q]≈
M M if and only if ↑ q ∩ A(S) ⊆↑ p ∩ A(S):
(1)
Obviously 6 is well deHned, and since it is constructed by the set inclusion, it is an ordering M as it can relation on P=≈. This order is anti-isomorphic with the set inclusion among cut sets of A, be seen from the following proposition. Proposition 8. If AM is a P-fuzzy set on S, then: [p]≈ 6 [q]≈
if and only if Aq ⊆ Ap :
Proof. [p]≈ 6 [q]≈ if and only if M M ↑ q ∩ A(S) ⊆↑ p ∩ A(S)
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if and only if M M {x ∈ S | A(x) ∈↑ q} ⊆ {x ∈ S | A(x) ∈↑ p} if and only if for x ∈ S M M A(x) ¿ q implies A(x) ¿p if and only if A q ⊆ Ap : We proceed with particular properties of ≈, for lattice-valued fuzzy sets. Note that for p ∈ L; [p]≈ denotes the supremum of the class [p]≈ (to which it, by Proposition 6(2), belongs): [p]≈ = (q | q ∈ [p]≈ ): We shall prove that ≈-classes behave precisely like the corresponding suprema. Proposition 9. If AM is an L-fuzzy set on S, then: M = [A(x)] M (a) for every x ∈ S; A(x) ≈; (b) for any p; q ∈ L, [p]≈ 6 [q]≈ [p]≈ 6 [q]≈ if and only if (the order on the right is the one from L). Proof. (a) Follows by Lemma 2, since the supremum of the class is its element. (b) Let [p]≈ 6[q]≈ . Then by Proposition 8 and by (a) it follows that A[q]≈ ⊆ A[p]≈ : Now, by Lemma 1, [q]≈ ∨ [p]≈ ≈ [q]≈ ; and hence [q]≈ ∨ [p]≈ ∈ [q]≈ . Thus, [q]≈ ∨ [p]≈ 6 [q]≈ , i.e., [p]≈ 6 [q]≈ . The converse follows by Propositions 3 and 8. It is easy to prove that for lattice-valued fuzzy sets the mapping p → [p]≈ (p ∈ L) is a closure operation on L. Then in lattice terms, L=≈ is a quotient in L. Hence, if ≈-classes are one element sets, it is obvious that L ∼ = L=≈. The above property of ≈-classes for lattices, is not generally satisHed for posets. Namely, it is not true that for one-element ≈-classes, P=≈ is under the order deHned by (1) isomorphic with P, for every poset P. This is illustrated by the following simple example.
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Fig. 1.
Example 1. Let S = {a; b}, and let P be the poset given in Fig. 1. The P-fuzzy set AM = ( pa br ) induces one element ≈-classes on P. However, it is easy to check that by (1), these classes are ordered in a (three-element) chain. Hence, P is not isomorphic with P=≈. The following theorem is a kind of a converse of Proposition 5 in which the main properties of P-fuzzy sets were formulated. It turns out that a collection of subsets of a given set determines its P-fuzzy subset, provided that this collection satisHes particular conditions concerning set intersections.
Theorem 1. Let P be a family of subsets of a nonempty set A, union of which is also A, and such that for every x ∈ A, (p ∈ P | x ∈ p) ∈ P: Let AM : A → P be de=ned by M A(x) := (p ∈ P | x ∈ p): Then, AM is a P-fuzzy set on A, where (P; 6) is anti-isomorphic with (P; ⊆), and for every p ∈ P, p = Ap . M Proof. P is a set partially ordered by the set inclusion. The mapping A(x) is well deHned, and it is a fuzzy set on A. It remains to prove that p = Ap (i.e., that the family of cut subsets of the deHned P-fuzzy set coincides with P). M Let x ∈ A. Then x ∈ Ap if and only if A(x)¿p if and only if (q ∈ P | x ∈ q)¿p if and only if (q ∈ P | x ∈ q) ⊆ p if and only if x ∈ p. As a consequence, we have the following proposition, describing a construction of lattice-valued fuzzy sets by means of subsets. Theorem 2. Let A be a nonempty set and F a family of its subsets closed under arbitrary intersections and containing A (the Moore’s family of subsets of A). Let also L be the lattice dual to (F; ⊆) and AM : A → L an L-valued subset of A de=ned by M A(x) := (f ∈ F | x ∈ f): Then, the lattice of cut subsets of AM is isomorphic with (F; ⊆).
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Fig. 2.
The above induced order 6 on P=≈ (by (1)) shows that the same collection of cut sets can be obtained by the decomposition of diNerent P-fuzzy sets. Moreover, the induced P=≈-fuzzy set (i.e., a mapping S → P=≈) has one element ≈-classes. The construction described in Theorem 1 actually gives a P-fuzzy set with such classes. Importance of this factor poset can be seen from the following proposition. Proposition 10. If AM is a P-fuzzy set on S, then the poset (P=≈; 6) is dually isomorphic with the M poset (AMP ; ⊆) of cut subsets of A. Proof. The dual isomorphism is given by f : [p]≈ → Ap . Indeed, f is obviously onto, and by the deHnition of ≈-classes it is an injection. By Proposition 8, both f and its inverse dually preserve the order, hence f is an anti-isomorphism. Example 2. Let (P; 6) be the poset given in Fig. 2 (left), and S = {a; b; c; d}. For the P-fuzzy set a b c d AM = u v r q cut subsets are Ap = ∅;
Aq = As = {d};
Ar = At = Aw = {c; d};
Au = {a; c; d};
Av = {b; d};
Ax = Ay = {a; b; c; d}: The partition of P into ≈-classes is represented in the same diagram. (P=≈; 6) is dually isomorM phic with the poset A(S) of cuts, which is ordered by the set inclusion. The corresponding diagram of cuts is given in Fig. 2, on the right.
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In case of lattice-valued fuzzy sets posets of cuts are lattices under the set inclusion (Proposition 6(1)), hence there is an immediate consequence of Proposition 10, concerning posets of ≈-classes. Corollary 2. Let AM : S → L be an L-fuzzy set on S. Then the poset (L=≈; 6) is a lattice, antiM isomorphic with the lattice (AML ; ⊆) of cuts of A. For lattice-valued fuzzy sets, there is a particular way to check whether all ≈-classes are diNerent, in which case L; L=≈ and the dual of AML are isomorphic lattices. Recall that an element p of the lattice L is meet-irreducible if b ∧ c = a implies b = a or c = a. Proposition 11. Let L be a lattice of =nite length, and AM : S → L an L-fuzzy set. Then all cuts of AM M are distinct if and only if all meet-irreducible elements of L di>erent from 1 are contained in A(S). M Proof. (⇒) Suppose p = 1 is a meet-irreducible element which is not in A(S). Since L is of HM M ∩ ↑q, and by nite length, there is a unique q ∈ L covering p. Then, obviously, A(S) ∩ ↑p = A(S) Proposition 7, AMp = AMq . (⇐) Suppose that p and q are distinct elements of L. Now there are two possibilities. Case 1: p = 1 or q = 1. Assume p = 1 = q. Then by the known lattice property (see, e.g., [2]) M ∩ ↑ q, but r is not an element there is a meet-irreducible element r such that q6r¡1. Thus, r ∈ A(S) M M M of A(S) ∩ ↑p, and so by Proposition 6, Ap = Aq . Case 2: p = 1 and q = 1. Then, as above,
p = {r ∈ L | p 6 r ¡ 1; r is meet-irreducible};
q = {s ∈ L | q 6 s ¡ 1; s is meet-irreducible}: M ∩ ↑ q = A(S) M ∩ ↑p, and by Proposition 6, AMp = AMq . Since p = q, this implies that A(S) The characterization of cut sets by meet-irreducible elements cannot be applied on real intervalvalued fuzzy sets, simply because the unite interval is not a lattice of Hnite length. However, if the lattice is a Hnite chain, then all its elements are meet-irreducible, which means that diNerent cuts could be obtained only if the set of values coincides with the set of all elements of the lattice distinct from 1. 4. Canonical representation. Completion In the previous section it was shown that the poset of cut subsets preserves basic properties of a fuzzy set. In fact, Theorems 1 and 2 say that for a given fuzzy set AM : A → P (where P is a poset M with exactly the same or a lattice), there is another function from A to the poset of cuts of A, structure of cut subsets. It also means that every fuzzy set has its inner representation, independent of the poset or the lattice which is the codomain of the mapping. This situation resembles to the one with homomorphisms and congruence relations in universal algebra: every homomorphic image
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of an algebraic structure is isomorphic with the structure factored by a suitable congruence relation, which is an inner representation of the homomorphic image. Let AM : A → P be a P-fuzzy set on A, and C the poset of cuts of AM with the order 6 dual to the set inclusion. The canonical representation of AM is a mapping kA : A → C; deHned as in Theorem 1: for x ∈ A, kA(x) = (f ∈ C | x ∈ f): The quoted theorem proves that AM and kA have isomorphic posets of cuts. We shall also prove that these two fuzzy sets have isomorphic posets of codomains, from which they take corresponding values under that isomorphism. Proposition 12. Let AM be a P-fuzzy set on A, and kA its canonical representation. Then (a) for every x ∈ A; kA(x) = AA(x) M ; (b) for f ∈ C; kAf = f (kAf is the f-cut of kA). (Observe that C is the set of cuts of AM and at the same time the codomain of kA.) Proof. (a) If y ∈ kA(x) = (Ap | x ∈ Ap ), then y belongs to every cut Ap which contains x. Hence, (since x ∈ AA(x) y ∈ AA(x) M M ). Thus, kA(x) ⊆ AA(x) M . On the other hand, if y ∈ AA(x) M , then y ∈ Ap for every p ∈ P such that AA(x) ⊆ A . Hence, M p M = (Ap | x ∈ Ap ), i.e., y ∈ kA(x). Hence, AA(x) y ∈ (Ap | AA(x) M ⊆ Ap ) = (Ap | A(x)¿p) M ⊆ kA(x), and the required equality is proved. (b) Let f = Ap , for p ∈ P, and let x ∈ kAAp . Then, kA(x)¿Ap in C, which means that AA(x) M ⊆ Ap , M and by Proposition 4(b), AA(x) and hence x ∈ Ap . On the other hand, if x ∈ Ap , then A(x)¿p, M ⊆ Ap , ⊆ Ap } = {y | kA(y) ⊆ Ap } = {y | kA(y)¿Ap } = kAAp . The equality of two sets and hence x ∈ {y | AA(y) M is thus proved. The above proposition proves the conjecture from the beginning of the section: a P-fuzzy set AM and its canonical representation kA have isomorphic posets of codomains, and identical posets of cuts. Example 3. The mapping a b c AM = p q r is an L-fuzzy set on A = {a; b; c}, where L is the lattice represented in Fig. 3(i). The cut subsets of AM are A1 = As = ∅;
Ar = {c};
Ap = {a};
Aq = {b; c};
A0 = {a; b; c}:
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Fig. 3.
Fig. 4.
The poset C of the above cuts is under the order dual to the set inclusion, a pentagon lattice (Fig. 3(ii)). The canonical representation of AM is the mapping kA : A → C, given by a b c kA = {a} {b; c} {c} (recall that kA(x) = (f ∈ C | x ∈ f)). Obviously, AM and kA have identical collections of cuts, and their posets of codomains are isomorphic (Fig. 4). We have already seen that some collections of subsets of a set A uniquely determine a P-( or L-) fuzzy set on A, in the sense of the above canonical representation. Precisely, the union of the collection should be the whole set, and particular closedness under set intersections should be fulHlled. However, we can start with an arbitrary collection C of subsets of A and look for a P-fuzzy M There is always such fuzzy set AM on A, such that C can be embedded into the poset of cuts of A. set obtained simply by taking the power set of A to be the collection of cuts. In the following, we show by a particular construction that, if the union of C is A, there is a P-fuzzy set whose poset of cuts is minimal among those that contain C as a subset of the poset of cuts. The following lemma says that adding the intersection of all sets containing one element, does not change intersections connected with other elements.
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Lemma 3. Let C be a collection of subsets of a nonempty set A. If a; b ∈ C and a ∈ {f ∈ C | b ∈ f}, then {f | a ∈ f} ⊆ {f | b ∈ f}: (2) Proof. Let x ∈ {f | a ∈ f}. This statement is equivalent with the following: (∀f ∈ C)(a ∈ f → x ∈ f). Similarly,by the assumption, (∀f ∈ C)(b ∈ f → a ∈ f). Hence, if b ∈ f then x ∈ f, which proves that x ∈ {f | b ∈ f}. Thus, (2) holds. Theorem 3. Let C be the collection of subsets of a nonempty set A, union of which is A. Then there is a P-fuzzy set AM on A with the minimal poset of cuts containing C (i.e., there is no fuzzy set BM on A, such that C ⊆ BP , and BP ⊆ AP ). Proof. Let P be the collection of subsets of A obtained by adding to C, for every x ∈ A, the intersection of all members of C to which x belongs. Since the union of C is A, every x is in some member of C, and this intersection can be uniquely determined. Obviously, C ⊆ P. By the previous lemma, P is a collection of subsets of A which satisHes conditions of Theorem 1, hence there is a P-fuzzy set AM having P as a poset of cuts, i.e., for which AP = P. Thus, AP contains precisely the elements of C and, for every x ∈ A, the intersection of all members of C to which it belongs. Moreover, no fuzzy set on A could have smaller (under ⊆) collection of cuts which includes C, since any such smaller collection would lack the intersection associated to some a ∈ A. Theorem 3 gives an algorithm for the construction of a fuzzy set AM on A, with a particular collection C as a subset of AP , provided that the union of C is A. In addition, AM has a minimal (however not generally unique) such collection of cuts. We shall say that AM is a P-completion of the collection C). Example 4. (a) Let C be the following collection of subsets of a set S = {a; b; c; d; e; f}: C = {{a; b; c; d; f}; {b; c; d; e}; {a; b}; {c}; {d; e}}: The union of C is S, and the P-completion of C, the collection P, is obtained by adding to C, for every element of S, the intersection of all members of C which contain this element: P = C ∪ {{b}; {d}}: Note that P is not closed under arbitrary intersections ({b; c; d} is missing). However, P is a poset under the order dual to the set inclusion (Fig. 5), and it satisHes conditions for the construction of a P-fuzzy set AM : S → P (Theorem 1): a b c d e f AM = : {a; b} {b} {c} {d} {d; e} {a; b; c; d; f} If C is an arbitrary (nonempty) collection of subsets of A, then the union of C might not be equal to A. In that case, it is not possible to apply the above algorithm. However, since there are P-fuzzy sets containing C as a subset of cuts, we shall deHne a completion of C in the following way.
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Fig. 5.
Let A be a nonempty set and C a collection of its subsets. Let C = C ∪ {A}: C satisHes conditions of Theorem 3. The P-fuzzy completion of C will be called a P-completion of the collection C. Thus, for every collection C of subsets of A, there is a unique P-fuzzy set AM on A, which is a P-completion of C. Summing up the above constructions, we can describe the P-completion CP of a collection C of subsets of A as follows: if C = A; C ∪ C CP := C ∪ {A} ∪ C else; where
C := Fx | x ∈ A and Fx = (f ∈ C ∪ {A} | x ∈ f) :
4.1. Lattice valued completions Situation with lattice-valued fuzzy sets, concerning possible completions, is slightly diNerent. Namely, we shall prove that for every collection of subsets of a set A, there is an L-fuzzy set with the unique minimal collection of cuts in the above sense. Let C be a collection of subsets of a set A, as before. A lattice-valued completion (L-completion) of C is a collection of subsets CL of A, deHned by CL := C ∪ C ∪ {A}; where
C := F | F = (f | f ∈ K ⊆ C)
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(K runs over all nonempty subsets of C). In other words, the L-completion of C is obtained by adding to C all the intersections of its members, and also the whole set A (if it is not already there). Lemma 4. The L-completion CL of an arbitrary collection C of subsets of a set A is a lattice under the set inclusion. Theorem 4. Let A be a set, C a nonempty collection of its subsets, and CL the L-completion of C. Then, the mapping AM : A → CL (CL is ordered dually to the set inclusion), de=ned by M A(x) = (f ∈ CL | x ∈ f); is an L-fuzzy set on A, whose poset of cuts is CL . In addition, AM is the L-valued subset of A with the smallest collection of cuts containing C. Proof. The collection CL satisHes conditions of Theorem 2, and hence AM is an L-fuzzy subset of A whose poset of cuts is CL , which proves the Hrst part. Further, we shall prove that CL is the subset of every collection of cuts of any L-fuzzy subset containing C. Let BM be an L-fuzzy subset of A whose poset of cuts H contains C. Obviously, H is closed under intersections and contains A. On the other hand, every element f of CL is either a subset from C, or it is the intersection of some subsets from C; it can also be A. In each of these cases, f also belongs to H. Hence, among the L-valued subsets of A; AM has the minimal collection of cuts containing C.
5. Completion of posets to complete lattices Here we use the procedure described above (to fuzzy sets) to make a completion of an arbitrary poset to a complete lattice. Every poset can be represented by an isomorphic collection of cuts of the particular fuzzy set, as follows. M = x, for all x ∈ P. Then, P is Let P be a partially ordered set. Let AM : P → P be deHned by A(x) anti-isomorphic with AP , under p → Ap . (Anti-isomorphism is isomorphism of P with AP ordered dually to inclusion.) A completion of the poset P (possible a lattice which is not complete), to a complete lattice by fuzzy sets, runs as follows. Firstly, we make a representation by cut sets, deHning a fuzzy set AM : P → P above. By Proposition M closed 5 the collection of cut sets of this fuzzy set is a family of subsets of the domain P of A, under componentwise intersections. Now, we complete this collection of subsets by all missing set intersections, and add the domain P to the collection. In such a way we obtain a lattice of subsets and a corresponding fuzzy set as in Theorem 3. This lattice is a lattice of cuts of the obtained fuzzy set, and a completion of poset P. The above algorithm by which every poset can be completed to a complete lattice is a completion by fuzzy sets.
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Fig. 6.
Fig. 7.
Example 5. Let P set representing P a b AM = a b
be a fuzzy set given by its Hasse diagram in Fig. 6. P = {a; b; c; d}, and a fuzzy is AM : P → P, c d : c d
The corresponding cut sets are: AMa = {a}; AMb = {b}; AMc = {a; b; c}; AMd = {a; b; d}. Family AMP is closed under componentwise intersections. In order to obtain a completion of this family, we add all intersections of elements from AMP : {a; b}; ∅ and set P. In this way we obtain the collection {∅; {a}; {b}; {a; b}; {a; b; c}; {a; b; d}; {a; b; c; d}} which is a lattice under inclusion. The obtained completion is the lattice L dual to the above collection lattice and it is represented in Fig. 7. In the following we prove that the completion of posets by fuzzy sets preserves all existing suprema and inHma. More precisely, we show that though the procedure is completely diNerent, the obtained lattice is the Dedekind–MacNeille completion. Theorem 5. Let P be a partially ordered set, and LP the lattice obtained by the completion of P by fuzzy sets. This completion coincides with the Dedekind–MacNeille completion. Proof. Let P be a partially ordered set. Let AM : A → P be a fuzzy set such that its family F of cut sets, is under inclusion the poset anti-isomorphic with P. Family F is closed under componentwise
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M Let L be a family of all intersections of all intersections, and its set union is the domain of A. subfamilies of F plus A: L= F1 | F1 ⊆ F ∪ {A}: Since L is a family of subsets of set A closed under intersections containing set A; L is a lattice under inclusion. Let L = (L; 6) be a lattice anti-isomorphic with (L; ⊆). This lattice is a completion of P, since F is anti-isomorphic with P. If the dual of L is a completion of the dual of P, than L is a completion of P, as well. We have that for a p ∈ P, cut Ap is equal to ↑p, i.e., in the dual of P, Ap = ↓p. Now we prove that L = DM (P). Recall that DM (P) = {X ⊆ P | X u‘ = X } and L=
{↓ a | a ∈ X ⊆ P} ∪ {P}:
Since {↓ x}u‘ = ↓ x, we have that ↓ x ∈ DM (P). To prove that L ⊆ DM (P), let X ⊆ P. An element of L is P or it is of the form {↓ a | a ∈ X }= {x ∈ P | (∀a ∈ X )x 6 a} = X ‘: u‘
Since P u‘ = P and X ‘ = X ‘ , it follows that in both cases we obtain an element from DM (P), hence L ⊆ DM (P). On the other hand, if X ⊆ P, such that X ∈ DM (P), i.e., X u‘ = X , then X u‘ = {↓ a | a ∈ X u } ∈ L: Hence, DM (P) ⊆ L, completing the proof. 6. Conclusion Fuzzy sets have been introduced as a tool for investigating imprecise reasoning, particularly to deal with problems which could hardly be handled by methods of classical mathematics. It turns out that the development of fuzzy logic and its techniques have inPuenced—as a feedback—many branches of contemporary mathematics. Our intention was to present a part of such inPuence in the theory of sets and ordered structures. Our investigations were mainly focused on the algebraic and ordertheoretic properties of the collection of cuts of a fuzzy structure. We point out the importance of the canonical representation of fuzzy structures, which makes this structure independent of the co-domain. In applications, our technique turn out to be very successful in completions of some classes of posets and lattices. Still, it remains to apply this cut approach in completions and structural analysis of posets appearing in theoretical computer science: algebraic and complete posets, continuous lattices, Boolean algebras, etc.
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