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A Clone-Based Representation of the Fuzzy Tolerance Or Equivalence Relations a Strict Order Relation Is Compatible With

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A Clone-Based Representation of the Fuzzy Tolerance Or Equivalence Relations a Strict Order Relation Is Compatible With Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems 296 (2016) 35–50 www.elsevier.com/locate/fss A clone-based representation of the fuzzy tolerance or equivalence relations a strict order relation is compatible with ∗ Bernard De Baets a, , Lemnaouar Zedam b, Azzedine Kheniche b a KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium b Laboratory of Pure and Applied Mathematics, Department of Mathematics, Med Boudiaf University – Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria Received 17 May 2015; received in revised form 14 August 2015; accepted 16 September 2015 Available online 30 September 2015 Abstract We show that although there exists no non-trivial (fuzzy) tolerance relation a partial order relation is compatible with (in the sense of Belohlávek),ˇ the situation is quite different when considering its strict part. More specifically, we provide a representation of all fuzzy tolerance (and, in particular, all fuzzy equivalence) relations a strict order relation is compatible with. To that end, we introduce the notion of clone relation associated with a partially ordered set and discuss its basic properties. The mentioned representation is intimately connected with this clone relation. © 2015 Elsevier B.V. All rights reserved. Keywords: Clone relation; Compatibility; Equivalence relation; Fuzzy relation; Order relation; Tolerance relation 1. Introduction Order relations and equivalence relations are basic mathematical concepts that are fundamental to numerous math- ematical and computational disciplines. Not surprisingly, these notions have been generalized to the setting of fuzzy sets in the early days of fuzzy set theory [18], and have been the subject of many studies since, with new ones still appearing at a regular pace. One such relatively new notion is that of compatibility of a fuzzy relation R with a fuzzy equivalence relation E, stating that R(x1,y1) ∗ E(x1,x2) ∗ E(y1,y2) ≤ R(x2,y2), for any x1, x2, y1, y2 in the universe of discourse X, introduced by Belohlávekˇ [1]. It expresses that elements that are similar to related elements are related as well. Actually, this notion is equivalent to the older notion of extensionality introduced by Höhle and Blanchard [13]. This compatibility notion is witnessing increased attention. It appears, * Corresponding author. E-mail address: [email protected] (B. De Baets). http://dx.doi.org/10.1016/j.fss.2015.09.015 0165-0114/© 2015 Elsevier B.V. All rights reserved. 36 B. De Baets et al. / Fuzzy Sets and Systems 296 (2016) 35–50 among others, in the study of fuzzy lattices [1,10,14,17,19], in the study of fuzzy functions [6,9,15], in the definition of strict fuzzy orderings by Bodenhofer and Demirci [5], and so on. Furthermore, it is key to the fuzzy approach to concept lattices of Belohlávekˇ [2]. In several of the mentioned works, one encounters the implicit standing assumption freely stated as: “Consider a fuzzy order relation R that is compatible with a given fuzzy equivalence relation E”. Having in mind the ever lurking danger of voidness results [3,11], in this paper we would like to challenge the even more basic assumption “Consider an order relation that is compatible with a given fuzzy equivalence relation E”. We will show that this is indeed a void assumption, but can be circumvented by only considering the strict part of the order relation. This enquiry turns out to stumble upon some previously unexplored notions in the theory of partially ordered sets. This paper is organized as follows. In Section 2, we recall some basic concepts of partially ordered sets and lattices, as well as the basic properties of lattice-valued fuzzy relations of relevance to our work. In Section 3, we recall the compatibility notion of Belohlávekˇ and show the negative result that the only reflexive fuzzy relation an order relation can be compatible with is the crisp equality. The notion of clone relation is introduced in full detail in Section 4. Whereas the clone relation is a tolerance relation in general, we show that it is intransitive for comparable clones, while it is transitive for incomparable clones. In order to provide deeper insights, we also study the clone relation of the union and linear sum of posets in Section 5. The main results are included in Section 6, in which we characterize the compatibility of a strict order relation with a fuzzy tolerance (resp. fuzzy equivalence) relation in terms of the clone relation introduced in Section 4. Moreover, where we lay bare a representation of the fuzzy tolerance (resp. equivalence) relations a strict order relation is compatible with in terms of two fuzzy tolerance (resp. equivalence) relations, one restricted to comparable clones, the other to incomparable clones. We conclude with some hints of future research in Section 7. 2. Basic definitions This section serves an introductory purpose. First, we recall some basic definitions and properties of partially ordered sets and lattices (for more details, see [7,16]). Second, we recall the definition of L-relations and their basic properties. 2.1. Partially ordered sets An order relation on a set X is a binary relation on X that is reflexive (i.e., x x, for any x ∈ X), antisymmetric (i.e., x y and y x implies x = y, for any x, y ∈ X) and transitive (i.e., x y and y z implies x z, for any x, y, z ∈ X). A set X equipped with an order relation is called a partially ordered set (poset, for short), denoted (X, ). A poset is called bounded if it has a smallest and a greatest element, respectively denoted by 0 and 1. A strict order relation < on a set X is a binary relation on X that is irreflexive (i.e., x<xdoes not hold for any x ∈ X) and transitive, implying that it is asymmetric (i.e., x<yimplies ¬(y < x), for any x, y ∈ X). To any order relation corresponds a strict order relation < (its irreflexive kernel): x<yif x y and x = y. Conversely, to any strict order relation < corresponds an order relation (its reflexive closure): x y if x<yor x = y. Two elements x and y of a poset (X, ) are called incomparable, denoted x y, if ¬(x y) and ¬(y x); otherwise, they are called comparable, denoted x ∦ y. The covering relation of a poset (X, ) is the binary relation on X defined by: x y if x<yand there exists no z ∈ X such that x<z<y. A poset can be conveniently represented by a Hasse diagram, displaying the covering relation . Note that x<yif there is a sequence of connected lines upwards from x to y. 2.2. Lattices A poset (L, ≤) is called a lattice if any two elements x and y have a greatest lower bound, denoted x ∧ y and called the meet (infimum) of x and y, as well as a smallest upper bound, denoted x ∨ y and called the join (supremum) of x and y. A lattice can also be defined as an algebraic structure: a set L equipped with two binary operations ∧ and ∨ that are idempotent, commutative and associative, and satisfy the absorption laws (x ∧ (x ∨ y) = x and x ∨ (x ∧ y) = x, for any x, y ∈ L). The order relation and the meet and join operations are then related as follows: x ≤ y if and only if x ∧ y = x; x ≤ y if and only if x ∨ y = y. Often, the notation (L, ≤, ∧, ∨) is used. Download English Version: https://daneshyari.com/en/article/389142 Download Persian Version: https://daneshyari.com/article/389142 Daneshyari.com.
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