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Non-standard cut classification of fuzzy sets
Information Sciences 177 (2007) 161–169 www.elsevier.com/locate/ins
Non-standard cut classification of fuzzy sets Vladimı´r Janisˇ
a,1
, Branimir Sˇesˇelja b, Andreja Tepavcˇevic´
b,*,2
a
b
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovske´ho 40, SK-974 01 Banska´ Bystrica, Slovak Republic Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovic´a 4, 21000 Novi Sad, Serbia Received 26 October 2005; received in revised form 10 May 2006; accepted 22 May 2006
Abstract Several important non-standard cut sets of lattice-valued fuzzy sets are investigated. These are strong cuts, ‘‘not less’’ and ‘‘neither less nor equal’’ cuts. In each case it is proved that collection of all cuts of any lattice-valued fuzzy set form a complete lattice under inclusion. Decomposition theorem (representation by cuts) is proved for ‘‘neither less nor equal’’ cuts. Necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain have equal families of corresponding cut sets are given. 2006 Elsevier Inc. All rights reserved. MSC: Primary 03B52; 03E72; Secondary 06A15 Keywords: Lattice-valued fuzzy set; Cut; Closure; Equivalent fuzzy sets
1. Introduction In investigations of fuzzy structures, collections of cut sets are one of the most important tools. Cut sets (also called levels) witness one side of the dual nature of fuzzy sets. Indeed, 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 classified according to the transferability to cut sets; these are known as ‘‘cutworthy’’ properties (for the notion of cutworthiness see e.g., the book [7] by Klir and Yuan). Particular interest for cut sets appeared in investigation of lattice-valued structures; this fact was noticed even in the early period of fuzzy set theory (e.g., the book [11] by Negoita and Ralescu in 1975). In order to explain the importance of the present study, let us mention one of the basic properties of every fuzzy structure, known as Representation Theorem. It states that every fuzzy structure is uniquely determined by its collection of cut structures. This theorem and its generalizations (concerning different co-domains, or *
1 2
Corresponding author. E-mail addresses: [email protected] (V. Janisˇ), [email protected] (B. Sˇesˇelja), [email protected] (A. Tepavcˇevic´). The research supported by Grant No. 1/2002/05, Slovak Grant Agency VEGA. The research supported by Serbian Ministry of Science and Technology, Grant No. 144011.
0020-0255/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.05.001
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different crisp (sub)sets) serve as a bridge from fuzziness to crisp objects, and is often used as a powerful tool in investigation of fuzzy structures. It has always been posed as one of the first problems to be solved, when some new approach to fuzziness was settled. More explicit example of the above mentioned importance of the present study is fuzzy topology. Namely, different kinds of levels (so called stratifications) are the most essential characteristic of fuzzy topological structures. These establish the most important relation between fuzzy structures and ordinary sets. As pointed out in the comment related to the book [13], ‘‘it is just level structure itself which makes fuzzy topological spaces possesses more abundant properties, making the relation between fuzzy topology and other branches of classical mathematics closer’’. Not only classical levels, but also some non-standard cut-sets play an important role in fuzzy topologies (see e.g., the book of Liu and Luo [8], Chapters 2 and 4, and the list of references given there). As the third important aspect of the present study we mention fuzzy algebraic structures (fuzzy groups, semigroups, and other fuzzy algebras). Classical levels are known to be crisp algebras of the same type and are used as the main tool in investigation of the fuzzy ones. For non-standard levels this is an important open problem. An extensive list of articles supporting the above claims can be found in the books Liu and Luo [8], Klir and Yuan [7], Bandemer and Gottwald [2] and in others containing theoretical background of fuzziness; we also present some relevant papers in References (in particular the overview articles [17,18,6] and references in these). Observe that the terminology concerning cuts of fuzzy sets is not unified. Ordinary cuts are sometimes called strong cuts [2], sometimes levels or stratifications [8]; strong cuts (as defined here) are also called strict cuts, while other non-standards cuts (not less, not less nor equal) are only denoted by a particular expression. In the present article we investigate three known types of non-standard cuts for lattice-valued fuzzy sets: strong cuts, ‘‘not less’’ cuts and ‘‘neither less nor equal’’ cut sets. In each case we prove that the family of corresponding cut sets constitutes a complete lattice under inclusion. Moreover, we prove Decomposition theorem for lattice-valued fuzzy sets, by means of ‘‘neither less nor equal’’ cuts, and we show that such a representation does not hold for other two types of cuts. Observe that Decomposition theorems are also known as Representation theorems by cuts (e.g., Negoita and Ralescu [11], for classical cuts). In the second part we investigate the class of all lattice-valued fuzzy sets with the same domain and codomain. We present classifications of these fuzzy sets, according to the equality of their collections of nonstandard cut sets (in each of the three cases). Systematic investigation of the analogue equality for ordinary cut sets (in finite case) can be found in Murali and Makamba [10] and in some previous investigation (as cited in [10]) in Makamba [9] and Alkhamees [1]. The problem has been completely solved in Sˇesˇelja and Tepavcˇevic´ [19] (in case of lattice-valued fuzzy sets) and by the same authors in [20] in most general settings (with a poset as a co-domain). Here we give necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain and co-domain have equal collections of these non-standard cut sets. 2. Preliminaries In this part, notation and basic facts about lattices and lattice-valued (L-valued) fuzzy sets are given; for more details, we refer to the survey papers [17,18]. If (L, 6) is a partially ordered set (poset), i.e., a non-empty set L equipped with a reflexive, antisymmetric and transitive relation 6, then (L, 6) is a complete lattice if every subset of L has a least upper bound (lub, supremum) and a greatest lower bound (glb, infimum). As usual, we denote by a _ b and a ^ b the supremum and the infimum of a, b 2 L, respectively; similarly, if M L, then the supremum and the infimum of M are denoted by ¤M and §M, respectively. Consequently, a complete lattice L contains a top (1 = ¤L) and a bottom element (0 = §L). Important examples of complete lattices are the real interval [0, 1] ordered by the usual ordering of real numbers and the power set (collection of all subsets) of an arbitrary set ordered by inclusion. In most cases we denote a lattice (L, 6) simply by L. We use the following well known facts about complete lattices (see e.g. [3] or [4]). Lemma 1. Any poset with the greatest (smallest) element, closed under arbitrary infima (suprema) is a complete lattice.
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In particular, a collection of subset of a non-empty set A closed under intersections and containing A is (by the above lemma) a complete lattice, called a closure system over A. If L ¼ ðL; 6Þ is a poset (lattice), then L ¼ ðL; PÞ is a dual poset or lattice to the lattice (L, 6), where P is a dual ordering relation defined by x P y if and only if y 6 x. A semi-open interval in a lattice L is defined by ða; b :¼ fx 2 Lja < x 6 bg; where x < y if and only if x 6 y and x 5 y. All other intervals are defined analogously, by inter-changing signs < and 6 accordingly. A fuzzy set l : X ! L is a mapping from a non-empty set X (domain) into a complete lattice L (co-domain). According to the original definition L is the unit interval [0, 1] of real numbers (which is a complete lattice under 6). However, we consider a complete lattice L in a more general setting and sometimes we use the term L-fuzzy set, or lattice-valued fuzzy set [5]. By l(X) we denote the set of images of l lðX Þ ¼ fp 2 Ljp ¼ lðxÞ; for some x 2 X g: In the following we provide definitions of four types of cut sets of a fuzzy set l : X ! L. We use the terminology from [7]. The most commonly used are ordinary cut sets. If l : X ! L is a fuzzy set on a set X then for p 2 L, the set lp :¼ fx 2 X jlðxÞ P pg is a p-cut, or a cut set, (cut) of l. The definitions of other types of cut sets follow3: If l : X ! L is a fuzzy set on a set X then for p 2 L, the set l> p :¼ fx 2 X jlðxÞ > pg is a strong p-cut, or a strong cut set or (strong cut) of l. For a fuzzy l, and p 2 P, not less cut is the following subset of X lp¥ :¼ fx 2 X jlðxÞ ¥ pg; where a 6< b if and only if not (a < b). It is straightforward to see that in the case when a lattice L is the real interval [0, 1], a ‘‘not less’’ cut set coincides with an ordinary cut set. However, this is not the case in general. Finally, for a fuzzy set l, and p 2 P, neither less nor equal cut is a subset of X, defined by li p :¼ fx 2 X jlðxÞipg; where a i b if and only if not (a 6 b). The collection of all ordinary (P) cuts of l is denoted by lL, that is, lL :¼ flp jp 2 Lg: Throughout the text, the collections of all strong cuts, ‘‘not less’’ cuts and ‘‘neither less nor equal’’ cuts of l ¥ i are denoted by l> L , lL and lL , respectively. If l : X ! L is a lattice-valued fuzzy set on X, then its collection {lpjp 2 L} of (ordinary) cuts constitutes a closure system over X; therefore it is a complete lattice under inclusion. This property is known as a decomposition of l into cut sets. This decomposition into cuts as indexed subsets of the domain uniquely determines the fuzzy set l. Therefore, the following holds. For every x 2 X _ lðxÞ ¼ fp 2 Ljx 2 lp g: ð1Þ 3
Notation and names are not unique; we adopt the most common ones.
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Formula (1) is called Decomposition theorem for lattice-valued fuzzy sets (see e.g. [8,15,16]). This Theorem is also called Representation theorem (see [12,14], etc.). The dual formula is also valid: For every x 2 X ^ lðxÞ ¼ fp 2 LjlðxÞ 6 pg: ð2Þ Observe that the unique determination of a fuzzy set by cuts essentially depends on the fact that cuts are indexed by elements of the domain. On the other hand, if cuts are considered only as subsets of the domain (without indices), then different fuzzy sets may have equal such collections. For standard cuts this problem was addressed in [19,20], and for non-standard cuts the problem is examined in Section 5 of the present paper. 3. Decomposition into cuts Here we prove that the collections of (strong, ‘‘not less’’, ‘‘neither less nor equal’’) cut sets of a lattice-valued fuzzy set constitute complete lattices under inclusion. Although the collection of all standard cuts of a lattice-valued fuzzy set is a closure system on X (i.e., closed under all set intersections), other foregoing collections of cut sets do not possess the closeness property. In other words, these families of sets are not closed under intersections in general. 3.1. Strong cuts In the following we introduce particular collections of subsets of images of fuzzy sets, which serve as a useful tool for investigation of families of strong cuts. For l 2 FL ðX Þ, let L> l :¼ ðfðp; 1 \ lðX Þjp 2 Lg; Þ: By the definition, L> l consists of particular collections of images of l in L and is a poset under inclusion. Let l> be the collection of strong cut sets of l: L > l> L :¼ flp jp 2 Lg:
First we prove the connection of strong cuts and elements of L> l: Proposition 1. If l is a fuzzy set on X and p, q 2 L, then > l> p lq if and only if ðp; 1 \ lðX Þ ðq; 1 \ lðX Þ: > Proof. Let for p, q 2 X, l> p lq . This is equivalent with
for every x 2 X ; if lðxÞ > p then lðxÞ > q; which is further equivalent with for every x 2 X ; if lðxÞ 2 ðp; 1 then lðxÞ 2 ðq; 1: The last statement is, finally, equivalent with ðp; 1 \ lðX Þ ðq; 1 \ lðX Þ:
We state the following lemma in order to prove that the collection l> L of strong cuts of l is a lattice under inclusion. Lemma 2. If X L, then the poset ðfða; 1 \ X ja 2 Lg; Þ is a complete lattice. Proof. In the collection {(a, 1] \ Xja 2 L} ordered by inclusion, the set (0, 1] \ X is the top element.
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In the sequel we prove that every family {(ai, 1] \ Xji 2 I} has the infimum, and by Lemma 1 it will follow that the required collection is a complete lattice. In case ¤{aiji 2 I} (in the sequel shortly denoted by ¤ai) is equal to some element ak of the family {aiji 2 I}, obviously \ ^ fðai ; 1 \ X ji 2 Ig ¼ fðai ; 1 \ X ji 2 Ig ¼ ðak ; 1 \ X ; i.e., the required infimum is exactly the set intersection. We now assume that ¤ai is not equal to some element of the family {aiji 2 I}. First, observe that in this case h_ i \ fðai ; 1 \ X ji 2 Ig ¼ ai ; 1 \ X :
ð3Þ
Now we analyze two cases. (i) ¤ai 62 X. Then by (3), (¤ ai, 1] \ X is the set intersection, and thus the required infimum. (ii) ¤ai 2 X. If [¤ai, 1] \ X is the required infimum, we are done; suppose it is not. We prove that in such a case the infimum is (¤ai, 1] \ X. First we note that (¤ai, 1] \ X is a lower bound for the family {(ai, 1] \ Xji 2 I}. Let (c, 1] \ X be another lower bound. Then, (c, 1] \ X [¤ai, 1] \ X. We have to prove that (c, 1] \ X (¤ai, 1] \ X. Suppose that this is not the case and that (c, 1] \ X 6 (¤ai, 1] \ X. This means that there is an element in the first set that does not belong to the second. By (c, 1] \ X [¤ai, 1] \ X, the only such element could be ¤ai. Indeed, sets (¤ai, 1] and [¤ai, 1] differs only in ¤ai. Thus, ¤ai 2 (c, 1] \ X and hence c 6 ¤ai. Therefore, [¤ai, 1] \ X (c, 1] \ X and [¤ai, 1] \ X = (c, 1] \ X. Hence, [¤ai, 1] \ X is one of the members from the starting family, and being the set intersection (by (3)), it would be its infimum. This is a contradiction, because, by assumption, this element is not the infimum. Thus, _ i ^ fðai ; 1 \ X ji 2 Ig ¼ ai ; 1 \ X ; which completes the proof.
h
As an immediate consequence of the previous lemma, we get the following. Corollary 1. L> l is a complete lattice. Theorem 1. If l : X ! L is a fuzzy set on X, then the poset l> L of strong cuts of l is a lattice isomorphic with the lattice L> l. > Proof. We consider the correspondence f : l> p 7! ðp; 1 \ lðX Þ which maps the poset lL of cuts of l onto the > lattice Ll . As a direct consequence of Proposition 1 and by the construction, we obtain that this correspondence is a bijection which is compatible with inclusion in both directions. Hence, f is an isomorphism. h
The above theorem provides a decomposition of a fuzzy set into strong cuts, which, by this theorem, constitute a complete lattice. 3.2. ‘‘Not less’’ cuts The following theorem deals with the structure of the collection of ‘‘not less cuts’’. Theorem 2. Let l : X ! L be a fuzzy set. Then its family of ‘‘not less’’ cuts lL¥ ¼ flp¥ :¼ fx 2 X j lðxÞ ¥ pgjp 2 Lg is a lattice under inclusion.
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Proof. We consider a fuzzy set l : X ! L, where L is a dual lattice to the lattice L, defined by lðxÞ :¼ lðxÞ. We prove that ‘‘not less’’ cuts of fuzzy set l are exactly the complements of strong cuts of l. By Theorem 1, the family of all strong cuts is a complete lattice under inclusion. It is clear that the family of complements of strong cuts is also a complete lattice, which is in fact the lattice of all ‘‘not less’’ cuts of the fuzzy set l. h 3.3. ‘‘Neither less nor equal’’ cuts Next we consider the ordered collection of ‘‘neither less nor equal’’ cuts of a lattice-valued fuzzy set. The following lemma is straightforward. Lemma 3. If l : X ! L is a lattice-valued fuzzy set and 1 is the top element of the lattice L, then li 1 ¼ ;. Proposition 2. The collection of ‘‘neither less nor equal’’ cuts of a fuzzy set l : X ! L is closed under set unions. Proof. Let fli pi ji 2 Ig be an arbitrary collection of ‘‘neither less nor equal’’ cuts of l, where {piji 2 I} is a family of elements from L. Then, for an x 2 X S x 2 fli pi ji 2 Ig if and only if ð9iÞðx 2 li pi Þ if and only if ($i) not (l(x) 6 pi) if and only if not ("i)(l(x) 6 pi) if and only if l(x) i §{pij 2 I} if and only if V x 2 li . fp ji2Ig i
Hence,
S
i V fli pi ji 2 Ig ¼ l fp ji2Ig , which proves the proposition.
h
i
Theorem 3. The collection of ‘‘neither less nor equal’’ cut sets of a fuzzy set l : X ! L is a complete lattice under inclusion. Proof. By Proposition 2, this collection ordered by the set inclusion is closed under arbitrary unions, and by Lemma 3 it contains the smallest element, the empty set. Hence, it is a complete lattice, by Lemma 1. h 4. Decomposition theorem We refer again to the formula (1) in Preliminaries. This formula is true because the decomposition into cuts as indexed subsets of X uniquely determines the fuzzy set l. Here we investigate the analogue problems for three non-standard types of cut sets. The following example shows that lattice-valued fuzzy sets are not in general uniquely determined by strong cuts, neither by ‘‘not less’’ ones. Example 1. Let X = {x, y} and let the lattice L be as in Fig. 1.
Fig. 1. Lattice codomain of fuzzy sets.
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Further, let l, m be two fuzzy sets given by x y x y l¼ m¼ : p q p r Then, it is straightforward to check that ordinary cuts of these fuzzy sets are: l1 ¼ lr ¼ ;;
lp ¼ fxg;
lq ¼ fyg;
m1 ¼ mq ¼ ;;
mp ¼ fxg;
mr ¼ fyg;
l0 ¼ fx; yg; m0 ¼ fx; yg:
Hence, these two indexed families are different (cuts with the same index do not coincide as subsets of X). On the other hand, the collections of strong cuts for l and m, as well as the collection of ‘‘not less’’ cuts for these two fuzzy sets coincide. Indeed, strong cuts in both cases are empty sets with the exception of 0-cut, which is X. ‘‘Not less’’ cuts also coincide; 1-cut is in both cases empty set, and others are X. Finally, ‘‘neither less nor equal’’ cuts do not coincide: li 1 ¼ ;;
li p ¼ fyg;
li q ¼ fxg;
mi 1 ¼ ;;
mi p ¼ fyg;
mi r ¼ fxg;
i li r ¼ l0 ¼ fx; yg; i mi q ¼ m0 ¼ fx; yg:
By the above example, it is not possible to get a fuzzy set by a synthesis of strong cuts or ‘‘not less’’ ones, as it is the case for ordinary cut sets (formula (1)). Next we give the positive answer for the third case. Namely, the following is Decomposition theorem of a fuzzy set by ‘‘neither less nor equal’’ cuts. Theorem 4. Let l : X ! L be a fuzzy set. Then, for every x 2 X ^ lðxÞ ¼ fp 2 Ljx 62 li p g: Proof. The proof is straightforward using formula (2). h 5. Classification of fuzzy sets by collections of cuts In this part a natural classification of fuzzy sets with the same domain and co-domain, according to equality of collections of non-standard cuts is done. As we have already mentioned, the classification according to the ordinary cuts has been completed in [19]. Again, let L be a complete lattice, X a non-empty set, and FL ðX Þ the collection of all fuzzy sets on X whose co-domain is L. As before, first we deal with strong cuts. If l and m are two fuzzy sets from FL ðX Þ, then we define the function f : lðX Þ ! PðmðX ÞÞðPðmðX ÞÞ is a collection of all subsets of m(X)), by f ðpÞ ¼ fq 2 mðX Þjð9x 2 X Þðp ¼ lðxÞ and q ¼ mðxÞÞg:
Theorem 5. Let l and m be two fuzzy sets from FL ðX Þ. Then l and m have equal collections of strong cut sets if > and only if there is a lattice isomorphism F from the lattice L> l to the lattice Lm , such that F ðða; 1 \ lðX ÞÞ ¼ ðb; 1 \ mðX Þ; where ðb; 1 \ mðX Þ ¼ fq 2 f ðpÞj for p 2 ða; 1 \ lðX Þg: Proof. First we suppose that l and m have equal collections of strong cuts. Let p 2 L and let us consider > (p, 1] \ l(X). For every p, there is q 2 L such that l> p ¼ mq . Therefore, for every x 2 X, l(x) > p if and only if m(x) > q. Hence, the mapping defined by F((p, 1] \ l(X)) = (q, 1] \ m(X) is well defined and
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ðq; 1 \ mðX Þ ¼ fr 2 f ðsÞj for s 2 ðp; 1 \ lðX Þg: > We have to prove that F is an isomorphism from L> l to Lm . Obviously (i.e., by the way it is defined) it is a surjection. We prove that it is compatible with the order (set inclusion) in both directions. Suppose that (p1, 1] \ l(X) (p2, 1] \ l(X). By the definition of f, we have that F((p1, 1] \ l(X)) F((p2, 1] \ l(X)). The reverse inclusion we obtain as follows. Let F((p1, 1] \ l(X)) = (q1, 1] \ m(X) and F((p2, 1] \ l(X)) = (q2, 1] \ m(X). Suppose s 2 (p1, 1] \ l(X). Then s P p1 and s = l(x) for some x 2 X. Let r = m(x). Then r 2 (q1, 1] \ m(X) (q2, 1] \ m(X). In addition, r 2 f(s) and r 2 (q2, 1] \ m(X). Therefore s 2 (p2, 1] \ l(X). This also proves injectivity of F. On the other hand, suppose that there is a required lattice isomorphism F from the lattice L> l to the lattice > Lm . That is, for each p 2 P, there is q 2 P such that
F ððp; 1 \ lðX ÞÞ ¼ ðq; 1 \ mðX Þ and ðq; 1 \ mðX Þ ¼ fr 2 f ðsÞ for s 2 ðp; 1 \ lðX Þg: > It is straightforward then that l> p ¼ mq .
h
Next we investigate equivalence of fuzzy sets from FL ðX Þ according to equal collections of ‘‘not less’’ cuts. Recall that L is the lattice dual to L, and for a fuzzy set l : X ! L, we define a fuzzy set l : X ! L by lðxÞ :¼ lðxÞ. Theorem 6. Fuzzy sets l : X ! L and m : X ! L have equal collections of ‘‘not less’’ cut sets if and only if fuzzy sets l : X ! L and m : X ! L have equal collections of strong cut sets. Proof. This is the consequence of the already proved (in Theorem 2) fact that ‘‘not less’’ cuts of fuzzy set l are exactly the set complements of strong cuts of l. h Corollary 2. Fuzzy sets l : X ! L and m : X ! L have the same families of ‘‘not less’’ cut sets if and only if there > is a lattice isomorphism F from the lattice L> l to the lattice Lm , such that F ðða; 1 \ lðX ÞÞ :¼ ðb; 1 \ mðX Þ; where ðb; 1 \ mðX Þ ¼ fq 2 f ðpÞ for p 2 ða; 1 \ lðX Þg and f is defined by f(p) = {q 2 m(X)j($x 2 X)(p = l(x) ^ q = m(x)). Finally, we deal with equivalence of fuzzy sets according to collections of ‘‘neither less nor equal’’ cuts. Similarly to the previous case, it is straightforward that ‘‘neither less nor equal’’ cut sets of l are set complements of ordinary cuts of l. Therefore, we have the following. Theorem 7. Fuzzy sets l : X ! L and m : X ! L have equal collections of ‘‘neither less nor equal’’ cut sets if and only if fuzzy sets l : X ! L and m : X ! L have equal collections of ordinary cut sets. Using the result concerning ordinary cut sets proved in [19], one can directly deduce from the above theorem an analogue proposition to Corollary 2. 6. Conclusion In the present article we investigate the ‘‘crisp part’’ of lattice-valued fuzzy structures, i.e., we deal with different collections of non-standard cut sets, appearing in the literature. We present basic properties of these, namely theorems of decomposition and equivalence according to equality of their collections. Among several possible applications of the foregoing results, we mention fuzzy algebraic and ordered structures. It is known that for fuzzy algebras, ordinary cuts are precisely crisp subalgebras. Now, by the proven properties, non-standard cuts also uniquely represent these structures, but their algebraic properties should be
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investigated. Next, representation problems for ordered structures can be successfully investigated by cuts. Non-standard cuts are a new tool in this field. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Y. Alkhamees, Fuzzy cyclic subgroups and fuzzy cyclic p-subgroups, J. Fuzzy Math. 3 (4) (1995) 911–919. H. Bandemer, S. Gottwald, Fuzzy Sets Fuzzy Methods Fuzzy Logic with Applications, John Wiley Sons, London, 1995. G. Birkhoff, Lattice Theory, Amer. Math. Soc., Providence, RI, 1967. B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1992. J.A. Goguen, L-fuzzy Sets, J. Math. Anal. Appl. 18 (1967) 145–174. V. Janisˇ, A. Tepavcˇevic´, Metric in a Space with Fuzzy Compatibility, Indian J. Pure Appl. Math. 35 (6) (2004) 737–745. G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-Hall PTR, 1995. Ying-Ming Liu, Mao-Kang Luo, Fuzzy Topology, World Scientific, Singapor, 1997. B.B. Makamba, Studies in fuzzy groups, Ph.D. Thesis, Rhodes University, Grahamstown, 1992. V. Murali, B.B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets Syst. 123 (2001) 259–264. C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to System Analysis, Birkha¨user Verlag, Basel, 1975. C.V. Negoita, D. A Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. N. Palaniappan, Fuzzy Topology, Narosa Publishing House, 2005. 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, 1979, pp. 77–91. B. Sˇesˇelja, A. Tepavcˇevic´, Relational valued fuzzy sets, Fuzzy Sets Syst. 52 (1992) 217–222. B. Sˇesˇelja, A. Tepavcˇevic´, Representation of lattices by fuzzy sets, Inform. Sci. 79 (1994) 171–180. B. Sˇeselja, A. Tepavcˇevic´, Completion of ordered structures by cuts of fuzzy sets, an overview, Fuzzy Sets Syst. 136 (2003) 1–19. B. Sˇeselja, A. Tepavcˇevic´, Representing ordered structures by fuzzy sets, an overview, Fuzzy Sets Syst. 136 (2003) 21–39. B. Sˇesˇelja, A. Tepavcˇevic´, A note on natural equivalence relation on fuzzy power set, Fuzzy Sets Syst. 148 (2) (2004) 201–210. B. Sˇesˇelja, A. Tepavcˇevic´, Equivalent fuzzy sets, Kybernetika 41 (2005) 115–128.
Non-standard cut classification of fuzzy sets Vladimı´r Janisˇ
a,1
, Branimir Sˇesˇelja b, Andreja Tepavcˇevic´
b,*,2
a
b
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovske´ho 40, SK-974 01 Banska´ Bystrica, Slovak Republic Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovic´a 4, 21000 Novi Sad, Serbia Received 26 October 2005; received in revised form 10 May 2006; accepted 22 May 2006
Abstract Several important non-standard cut sets of lattice-valued fuzzy sets are investigated. These are strong cuts, ‘‘not less’’ and ‘‘neither less nor equal’’ cuts. In each case it is proved that collection of all cuts of any lattice-valued fuzzy set form a complete lattice under inclusion. Decomposition theorem (representation by cuts) is proved for ‘‘neither less nor equal’’ cuts. Necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain have equal families of corresponding cut sets are given. 2006 Elsevier Inc. All rights reserved. MSC: Primary 03B52; 03E72; Secondary 06A15 Keywords: Lattice-valued fuzzy set; Cut; Closure; Equivalent fuzzy sets
1. Introduction In investigations of fuzzy structures, collections of cut sets are one of the most important tools. Cut sets (also called levels) witness one side of the dual nature of fuzzy sets. Indeed, 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 classified according to the transferability to cut sets; these are known as ‘‘cutworthy’’ properties (for the notion of cutworthiness see e.g., the book [7] by Klir and Yuan). Particular interest for cut sets appeared in investigation of lattice-valued structures; this fact was noticed even in the early period of fuzzy set theory (e.g., the book [11] by Negoita and Ralescu in 1975). In order to explain the importance of the present study, let us mention one of the basic properties of every fuzzy structure, known as Representation Theorem. It states that every fuzzy structure is uniquely determined by its collection of cut structures. This theorem and its generalizations (concerning different co-domains, or *
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Corresponding author. E-mail addresses: [email protected] (V. Janisˇ), [email protected] (B. Sˇesˇelja), [email protected] (A. Tepavcˇevic´). The research supported by Grant No. 1/2002/05, Slovak Grant Agency VEGA. The research supported by Serbian Ministry of Science and Technology, Grant No. 144011.
0020-0255/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.05.001
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different crisp (sub)sets) serve as a bridge from fuzziness to crisp objects, and is often used as a powerful tool in investigation of fuzzy structures. It has always been posed as one of the first problems to be solved, when some new approach to fuzziness was settled. More explicit example of the above mentioned importance of the present study is fuzzy topology. Namely, different kinds of levels (so called stratifications) are the most essential characteristic of fuzzy topological structures. These establish the most important relation between fuzzy structures and ordinary sets. As pointed out in the comment related to the book [13], ‘‘it is just level structure itself which makes fuzzy topological spaces possesses more abundant properties, making the relation between fuzzy topology and other branches of classical mathematics closer’’. Not only classical levels, but also some non-standard cut-sets play an important role in fuzzy topologies (see e.g., the book of Liu and Luo [8], Chapters 2 and 4, and the list of references given there). As the third important aspect of the present study we mention fuzzy algebraic structures (fuzzy groups, semigroups, and other fuzzy algebras). Classical levels are known to be crisp algebras of the same type and are used as the main tool in investigation of the fuzzy ones. For non-standard levels this is an important open problem. An extensive list of articles supporting the above claims can be found in the books Liu and Luo [8], Klir and Yuan [7], Bandemer and Gottwald [2] and in others containing theoretical background of fuzziness; we also present some relevant papers in References (in particular the overview articles [17,18,6] and references in these). Observe that the terminology concerning cuts of fuzzy sets is not unified. Ordinary cuts are sometimes called strong cuts [2], sometimes levels or stratifications [8]; strong cuts (as defined here) are also called strict cuts, while other non-standards cuts (not less, not less nor equal) are only denoted by a particular expression. In the present article we investigate three known types of non-standard cuts for lattice-valued fuzzy sets: strong cuts, ‘‘not less’’ cuts and ‘‘neither less nor equal’’ cut sets. In each case we prove that the family of corresponding cut sets constitutes a complete lattice under inclusion. Moreover, we prove Decomposition theorem for lattice-valued fuzzy sets, by means of ‘‘neither less nor equal’’ cuts, and we show that such a representation does not hold for other two types of cuts. Observe that Decomposition theorems are also known as Representation theorems by cuts (e.g., Negoita and Ralescu [11], for classical cuts). In the second part we investigate the class of all lattice-valued fuzzy sets with the same domain and codomain. We present classifications of these fuzzy sets, according to the equality of their collections of nonstandard cut sets (in each of the three cases). Systematic investigation of the analogue equality for ordinary cut sets (in finite case) can be found in Murali and Makamba [10] and in some previous investigation (as cited in [10]) in Makamba [9] and Alkhamees [1]. The problem has been completely solved in Sˇesˇelja and Tepavcˇevic´ [19] (in case of lattice-valued fuzzy sets) and by the same authors in [20] in most general settings (with a poset as a co-domain). Here we give necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain and co-domain have equal collections of these non-standard cut sets. 2. Preliminaries In this part, notation and basic facts about lattices and lattice-valued (L-valued) fuzzy sets are given; for more details, we refer to the survey papers [17,18]. If (L, 6) is a partially ordered set (poset), i.e., a non-empty set L equipped with a reflexive, antisymmetric and transitive relation 6, then (L, 6) is a complete lattice if every subset of L has a least upper bound (lub, supremum) and a greatest lower bound (glb, infimum). As usual, we denote by a _ b and a ^ b the supremum and the infimum of a, b 2 L, respectively; similarly, if M L, then the supremum and the infimum of M are denoted by ¤M and §M, respectively. Consequently, a complete lattice L contains a top (1 = ¤L) and a bottom element (0 = §L). Important examples of complete lattices are the real interval [0, 1] ordered by the usual ordering of real numbers and the power set (collection of all subsets) of an arbitrary set ordered by inclusion. In most cases we denote a lattice (L, 6) simply by L. We use the following well known facts about complete lattices (see e.g. [3] or [4]). Lemma 1. Any poset with the greatest (smallest) element, closed under arbitrary infima (suprema) is a complete lattice.
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In particular, a collection of subset of a non-empty set A closed under intersections and containing A is (by the above lemma) a complete lattice, called a closure system over A. If L ¼ ðL; 6Þ is a poset (lattice), then L ¼ ðL; PÞ is a dual poset or lattice to the lattice (L, 6), where P is a dual ordering relation defined by x P y if and only if y 6 x. A semi-open interval in a lattice L is defined by ða; b :¼ fx 2 Lja < x 6 bg; where x < y if and only if x 6 y and x 5 y. All other intervals are defined analogously, by inter-changing signs < and 6 accordingly. A fuzzy set l : X ! L is a mapping from a non-empty set X (domain) into a complete lattice L (co-domain). According to the original definition L is the unit interval [0, 1] of real numbers (which is a complete lattice under 6). However, we consider a complete lattice L in a more general setting and sometimes we use the term L-fuzzy set, or lattice-valued fuzzy set [5]. By l(X) we denote the set of images of l lðX Þ ¼ fp 2 Ljp ¼ lðxÞ; for some x 2 X g: In the following we provide definitions of four types of cut sets of a fuzzy set l : X ! L. We use the terminology from [7]. The most commonly used are ordinary cut sets. If l : X ! L is a fuzzy set on a set X then for p 2 L, the set lp :¼ fx 2 X jlðxÞ P pg is a p-cut, or a cut set, (cut) of l. The definitions of other types of cut sets follow3: If l : X ! L is a fuzzy set on a set X then for p 2 L, the set l> p :¼ fx 2 X jlðxÞ > pg is a strong p-cut, or a strong cut set or (strong cut) of l. For a fuzzy l, and p 2 P, not less cut is the following subset of X lp¥ :¼ fx 2 X jlðxÞ ¥ pg; where a 6< b if and only if not (a < b). It is straightforward to see that in the case when a lattice L is the real interval [0, 1], a ‘‘not less’’ cut set coincides with an ordinary cut set. However, this is not the case in general. Finally, for a fuzzy set l, and p 2 P, neither less nor equal cut is a subset of X, defined by li p :¼ fx 2 X jlðxÞipg; where a i b if and only if not (a 6 b). The collection of all ordinary (P) cuts of l is denoted by lL, that is, lL :¼ flp jp 2 Lg: Throughout the text, the collections of all strong cuts, ‘‘not less’’ cuts and ‘‘neither less nor equal’’ cuts of l ¥ i are denoted by l> L , lL and lL , respectively. If l : X ! L is a lattice-valued fuzzy set on X, then its collection {lpjp 2 L} of (ordinary) cuts constitutes a closure system over X; therefore it is a complete lattice under inclusion. This property is known as a decomposition of l into cut sets. This decomposition into cuts as indexed subsets of the domain uniquely determines the fuzzy set l. Therefore, the following holds. For every x 2 X _ lðxÞ ¼ fp 2 Ljx 2 lp g: ð1Þ 3
Notation and names are not unique; we adopt the most common ones.
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Formula (1) is called Decomposition theorem for lattice-valued fuzzy sets (see e.g. [8,15,16]). This Theorem is also called Representation theorem (see [12,14], etc.). The dual formula is also valid: For every x 2 X ^ lðxÞ ¼ fp 2 LjlðxÞ 6 pg: ð2Þ Observe that the unique determination of a fuzzy set by cuts essentially depends on the fact that cuts are indexed by elements of the domain. On the other hand, if cuts are considered only as subsets of the domain (without indices), then different fuzzy sets may have equal such collections. For standard cuts this problem was addressed in [19,20], and for non-standard cuts the problem is examined in Section 5 of the present paper. 3. Decomposition into cuts Here we prove that the collections of (strong, ‘‘not less’’, ‘‘neither less nor equal’’) cut sets of a lattice-valued fuzzy set constitute complete lattices under inclusion. Although the collection of all standard cuts of a lattice-valued fuzzy set is a closure system on X (i.e., closed under all set intersections), other foregoing collections of cut sets do not possess the closeness property. In other words, these families of sets are not closed under intersections in general. 3.1. Strong cuts In the following we introduce particular collections of subsets of images of fuzzy sets, which serve as a useful tool for investigation of families of strong cuts. For l 2 FL ðX Þ, let L> l :¼ ðfðp; 1 \ lðX Þjp 2 Lg; Þ: By the definition, L> l consists of particular collections of images of l in L and is a poset under inclusion. Let l> be the collection of strong cut sets of l: L > l> L :¼ flp jp 2 Lg:
First we prove the connection of strong cuts and elements of L> l: Proposition 1. If l is a fuzzy set on X and p, q 2 L, then > l> p lq if and only if ðp; 1 \ lðX Þ ðq; 1 \ lðX Þ: > Proof. Let for p, q 2 X, l> p lq . This is equivalent with
for every x 2 X ; if lðxÞ > p then lðxÞ > q; which is further equivalent with for every x 2 X ; if lðxÞ 2 ðp; 1 then lðxÞ 2 ðq; 1: The last statement is, finally, equivalent with ðp; 1 \ lðX Þ ðq; 1 \ lðX Þ:
We state the following lemma in order to prove that the collection l> L of strong cuts of l is a lattice under inclusion. Lemma 2. If X L, then the poset ðfða; 1 \ X ja 2 Lg; Þ is a complete lattice. Proof. In the collection {(a, 1] \ Xja 2 L} ordered by inclusion, the set (0, 1] \ X is the top element.
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In the sequel we prove that every family {(ai, 1] \ Xji 2 I} has the infimum, and by Lemma 1 it will follow that the required collection is a complete lattice. In case ¤{aiji 2 I} (in the sequel shortly denoted by ¤ai) is equal to some element ak of the family {aiji 2 I}, obviously \ ^ fðai ; 1 \ X ji 2 Ig ¼ fðai ; 1 \ X ji 2 Ig ¼ ðak ; 1 \ X ; i.e., the required infimum is exactly the set intersection. We now assume that ¤ai is not equal to some element of the family {aiji 2 I}. First, observe that in this case h_ i \ fðai ; 1 \ X ji 2 Ig ¼ ai ; 1 \ X :
ð3Þ
Now we analyze two cases. (i) ¤ai 62 X. Then by (3), (¤ ai, 1] \ X is the set intersection, and thus the required infimum. (ii) ¤ai 2 X. If [¤ai, 1] \ X is the required infimum, we are done; suppose it is not. We prove that in such a case the infimum is (¤ai, 1] \ X. First we note that (¤ai, 1] \ X is a lower bound for the family {(ai, 1] \ Xji 2 I}. Let (c, 1] \ X be another lower bound. Then, (c, 1] \ X [¤ai, 1] \ X. We have to prove that (c, 1] \ X (¤ai, 1] \ X. Suppose that this is not the case and that (c, 1] \ X 6 (¤ai, 1] \ X. This means that there is an element in the first set that does not belong to the second. By (c, 1] \ X [¤ai, 1] \ X, the only such element could be ¤ai. Indeed, sets (¤ai, 1] and [¤ai, 1] differs only in ¤ai. Thus, ¤ai 2 (c, 1] \ X and hence c 6 ¤ai. Therefore, [¤ai, 1] \ X (c, 1] \ X and [¤ai, 1] \ X = (c, 1] \ X. Hence, [¤ai, 1] \ X is one of the members from the starting family, and being the set intersection (by (3)), it would be its infimum. This is a contradiction, because, by assumption, this element is not the infimum. Thus, _ i ^ fðai ; 1 \ X ji 2 Ig ¼ ai ; 1 \ X ; which completes the proof.
h
As an immediate consequence of the previous lemma, we get the following. Corollary 1. L> l is a complete lattice. Theorem 1. If l : X ! L is a fuzzy set on X, then the poset l> L of strong cuts of l is a lattice isomorphic with the lattice L> l. > Proof. We consider the correspondence f : l> p 7! ðp; 1 \ lðX Þ which maps the poset lL of cuts of l onto the > lattice Ll . As a direct consequence of Proposition 1 and by the construction, we obtain that this correspondence is a bijection which is compatible with inclusion in both directions. Hence, f is an isomorphism. h
The above theorem provides a decomposition of a fuzzy set into strong cuts, which, by this theorem, constitute a complete lattice. 3.2. ‘‘Not less’’ cuts The following theorem deals with the structure of the collection of ‘‘not less cuts’’. Theorem 2. Let l : X ! L be a fuzzy set. Then its family of ‘‘not less’’ cuts lL¥ ¼ flp¥ :¼ fx 2 X j lðxÞ ¥ pgjp 2 Lg is a lattice under inclusion.
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Proof. We consider a fuzzy set l : X ! L, where L is a dual lattice to the lattice L, defined by lðxÞ :¼ lðxÞ. We prove that ‘‘not less’’ cuts of fuzzy set l are exactly the complements of strong cuts of l. By Theorem 1, the family of all strong cuts is a complete lattice under inclusion. It is clear that the family of complements of strong cuts is also a complete lattice, which is in fact the lattice of all ‘‘not less’’ cuts of the fuzzy set l. h 3.3. ‘‘Neither less nor equal’’ cuts Next we consider the ordered collection of ‘‘neither less nor equal’’ cuts of a lattice-valued fuzzy set. The following lemma is straightforward. Lemma 3. If l : X ! L is a lattice-valued fuzzy set and 1 is the top element of the lattice L, then li 1 ¼ ;. Proposition 2. The collection of ‘‘neither less nor equal’’ cuts of a fuzzy set l : X ! L is closed under set unions. Proof. Let fli pi ji 2 Ig be an arbitrary collection of ‘‘neither less nor equal’’ cuts of l, where {piji 2 I} is a family of elements from L. Then, for an x 2 X S x 2 fli pi ji 2 Ig if and only if ð9iÞðx 2 li pi Þ if and only if ($i) not (l(x) 6 pi) if and only if not ("i)(l(x) 6 pi) if and only if l(x) i §{pij 2 I} if and only if V x 2 li . fp ji2Ig i
Hence,
S
i V fli pi ji 2 Ig ¼ l fp ji2Ig , which proves the proposition.
h
i
Theorem 3. The collection of ‘‘neither less nor equal’’ cut sets of a fuzzy set l : X ! L is a complete lattice under inclusion. Proof. By Proposition 2, this collection ordered by the set inclusion is closed under arbitrary unions, and by Lemma 3 it contains the smallest element, the empty set. Hence, it is a complete lattice, by Lemma 1. h 4. Decomposition theorem We refer again to the formula (1) in Preliminaries. This formula is true because the decomposition into cuts as indexed subsets of X uniquely determines the fuzzy set l. Here we investigate the analogue problems for three non-standard types of cut sets. The following example shows that lattice-valued fuzzy sets are not in general uniquely determined by strong cuts, neither by ‘‘not less’’ ones. Example 1. Let X = {x, y} and let the lattice L be as in Fig. 1.
Fig. 1. Lattice codomain of fuzzy sets.
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Further, let l, m be two fuzzy sets given by x y x y l¼ m¼ : p q p r Then, it is straightforward to check that ordinary cuts of these fuzzy sets are: l1 ¼ lr ¼ ;;
lp ¼ fxg;
lq ¼ fyg;
m1 ¼ mq ¼ ;;
mp ¼ fxg;
mr ¼ fyg;
l0 ¼ fx; yg; m0 ¼ fx; yg:
Hence, these two indexed families are different (cuts with the same index do not coincide as subsets of X). On the other hand, the collections of strong cuts for l and m, as well as the collection of ‘‘not less’’ cuts for these two fuzzy sets coincide. Indeed, strong cuts in both cases are empty sets with the exception of 0-cut, which is X. ‘‘Not less’’ cuts also coincide; 1-cut is in both cases empty set, and others are X. Finally, ‘‘neither less nor equal’’ cuts do not coincide: li 1 ¼ ;;
li p ¼ fyg;
li q ¼ fxg;
mi 1 ¼ ;;
mi p ¼ fyg;
mi r ¼ fxg;
i li r ¼ l0 ¼ fx; yg; i mi q ¼ m0 ¼ fx; yg:
By the above example, it is not possible to get a fuzzy set by a synthesis of strong cuts or ‘‘not less’’ ones, as it is the case for ordinary cut sets (formula (1)). Next we give the positive answer for the third case. Namely, the following is Decomposition theorem of a fuzzy set by ‘‘neither less nor equal’’ cuts. Theorem 4. Let l : X ! L be a fuzzy set. Then, for every x 2 X ^ lðxÞ ¼ fp 2 Ljx 62 li p g: Proof. The proof is straightforward using formula (2). h 5. Classification of fuzzy sets by collections of cuts In this part a natural classification of fuzzy sets with the same domain and co-domain, according to equality of collections of non-standard cuts is done. As we have already mentioned, the classification according to the ordinary cuts has been completed in [19]. Again, let L be a complete lattice, X a non-empty set, and FL ðX Þ the collection of all fuzzy sets on X whose co-domain is L. As before, first we deal with strong cuts. If l and m are two fuzzy sets from FL ðX Þ, then we define the function f : lðX Þ ! PðmðX ÞÞðPðmðX ÞÞ is a collection of all subsets of m(X)), by f ðpÞ ¼ fq 2 mðX Þjð9x 2 X Þðp ¼ lðxÞ and q ¼ mðxÞÞg:
Theorem 5. Let l and m be two fuzzy sets from FL ðX Þ. Then l and m have equal collections of strong cut sets if > and only if there is a lattice isomorphism F from the lattice L> l to the lattice Lm , such that F ðða; 1 \ lðX ÞÞ ¼ ðb; 1 \ mðX Þ; where ðb; 1 \ mðX Þ ¼ fq 2 f ðpÞj for p 2 ða; 1 \ lðX Þg: Proof. First we suppose that l and m have equal collections of strong cuts. Let p 2 L and let us consider > (p, 1] \ l(X). For every p, there is q 2 L such that l> p ¼ mq . Therefore, for every x 2 X, l(x) > p if and only if m(x) > q. Hence, the mapping defined by F((p, 1] \ l(X)) = (q, 1] \ m(X) is well defined and
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ðq; 1 \ mðX Þ ¼ fr 2 f ðsÞj for s 2 ðp; 1 \ lðX Þg: > We have to prove that F is an isomorphism from L> l to Lm . Obviously (i.e., by the way it is defined) it is a surjection. We prove that it is compatible with the order (set inclusion) in both directions. Suppose that (p1, 1] \ l(X) (p2, 1] \ l(X). By the definition of f, we have that F((p1, 1] \ l(X)) F((p2, 1] \ l(X)). The reverse inclusion we obtain as follows. Let F((p1, 1] \ l(X)) = (q1, 1] \ m(X) and F((p2, 1] \ l(X)) = (q2, 1] \ m(X). Suppose s 2 (p1, 1] \ l(X). Then s P p1 and s = l(x) for some x 2 X. Let r = m(x). Then r 2 (q1, 1] \ m(X) (q2, 1] \ m(X). In addition, r 2 f(s) and r 2 (q2, 1] \ m(X). Therefore s 2 (p2, 1] \ l(X). This also proves injectivity of F. On the other hand, suppose that there is a required lattice isomorphism F from the lattice L> l to the lattice > Lm . That is, for each p 2 P, there is q 2 P such that
F ððp; 1 \ lðX ÞÞ ¼ ðq; 1 \ mðX Þ and ðq; 1 \ mðX Þ ¼ fr 2 f ðsÞ for s 2 ðp; 1 \ lðX Þg: > It is straightforward then that l> p ¼ mq .
h
Next we investigate equivalence of fuzzy sets from FL ðX Þ according to equal collections of ‘‘not less’’ cuts. Recall that L is the lattice dual to L, and for a fuzzy set l : X ! L, we define a fuzzy set l : X ! L by lðxÞ :¼ lðxÞ. Theorem 6. Fuzzy sets l : X ! L and m : X ! L have equal collections of ‘‘not less’’ cut sets if and only if fuzzy sets l : X ! L and m : X ! L have equal collections of strong cut sets. Proof. This is the consequence of the already proved (in Theorem 2) fact that ‘‘not less’’ cuts of fuzzy set l are exactly the set complements of strong cuts of l. h Corollary 2. Fuzzy sets l : X ! L and m : X ! L have the same families of ‘‘not less’’ cut sets if and only if there > is a lattice isomorphism F from the lattice L> l to the lattice Lm , such that F ðða; 1 \ lðX ÞÞ :¼ ðb; 1 \ mðX Þ; where ðb; 1 \ mðX Þ ¼ fq 2 f ðpÞ for p 2 ða; 1 \ lðX Þg and f is defined by f(p) = {q 2 m(X)j($x 2 X)(p = l(x) ^ q = m(x)). Finally, we deal with equivalence of fuzzy sets according to collections of ‘‘neither less nor equal’’ cuts. Similarly to the previous case, it is straightforward that ‘‘neither less nor equal’’ cut sets of l are set complements of ordinary cuts of l. Therefore, we have the following. Theorem 7. Fuzzy sets l : X ! L and m : X ! L have equal collections of ‘‘neither less nor equal’’ cut sets if and only if fuzzy sets l : X ! L and m : X ! L have equal collections of ordinary cut sets. Using the result concerning ordinary cut sets proved in [19], one can directly deduce from the above theorem an analogue proposition to Corollary 2. 6. Conclusion In the present article we investigate the ‘‘crisp part’’ of lattice-valued fuzzy structures, i.e., we deal with different collections of non-standard cut sets, appearing in the literature. We present basic properties of these, namely theorems of decomposition and equivalence according to equality of their collections. Among several possible applications of the foregoing results, we mention fuzzy algebraic and ordered structures. It is known that for fuzzy algebras, ordinary cuts are precisely crisp subalgebras. Now, by the proven properties, non-standard cuts also uniquely represent these structures, but their algebraic properties should be
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investigated. Next, representation problems for ordered structures can be successfully investigated by cuts. Non-standard cuts are a new tool in this field. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Y. Alkhamees, Fuzzy cyclic subgroups and fuzzy cyclic p-subgroups, J. Fuzzy Math. 3 (4) (1995) 911–919. H. Bandemer, S. Gottwald, Fuzzy Sets Fuzzy Methods Fuzzy Logic with Applications, John Wiley Sons, London, 1995. G. Birkhoff, Lattice Theory, Amer. Math. Soc., Providence, RI, 1967. B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1992. J.A. Goguen, L-fuzzy Sets, J. Math. Anal. Appl. 18 (1967) 145–174. V. Janisˇ, A. Tepavcˇevic´, Metric in a Space with Fuzzy Compatibility, Indian J. Pure Appl. Math. 35 (6) (2004) 737–745. G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-Hall PTR, 1995. Ying-Ming Liu, Mao-Kang Luo, Fuzzy Topology, World Scientific, Singapor, 1997. B.B. Makamba, Studies in fuzzy groups, Ph.D. Thesis, Rhodes University, Grahamstown, 1992. V. Murali, B.B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets Syst. 123 (2001) 259–264. C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to System Analysis, Birkha¨user Verlag, Basel, 1975. C.V. Negoita, D. A Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. N. Palaniappan, Fuzzy Topology, Narosa Publishing House, 2005. 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, 1979, pp. 77–91. B. Sˇesˇelja, A. Tepavcˇevic´, Relational valued fuzzy sets, Fuzzy Sets Syst. 52 (1992) 217–222. B. Sˇesˇelja, A. Tepavcˇevic´, Representation of lattices by fuzzy sets, Inform. Sci. 79 (1994) 171–180. B. Sˇeselja, A. Tepavcˇevic´, Completion of ordered structures by cuts of fuzzy sets, an overview, Fuzzy Sets Syst. 136 (2003) 1–19. B. Sˇeselja, A. Tepavcˇevic´, Representing ordered structures by fuzzy sets, an overview, Fuzzy Sets Syst. 136 (2003) 21–39. B. Sˇesˇelja, A. Tepavcˇevic´, A note on natural equivalence relation on fuzzy power set, Fuzzy Sets Syst. 148 (2) (2004) 201–210. B. Sˇesˇelja, A. Tepavcˇevic´, Equivalent fuzzy sets, Kybernetika 41 (2005) 115–128.