Fuzzy antisymmetry and order
TL;DR: Fuzzy antisymmetry and various types of ordering are defined and theorems relating to them proved and some of their properties investigated.
Abstract: Fuzzy antisymmetry and various types of ordering are defined and theorems relating to them proved. A few constants associated with a fuzzy relation are introduced and some of their properties investigated.
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TL;DR: This paper examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices and shows that a fuzzy partial order in the sense of Fan and Zhang and an L-order in thesense of Belohlavek are equivalent to each other.
Abstract: This paper deals with quantitative domain theory via fuzzy sets. It examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices. First, we show that a fuzzy partial order in the sense of Fan and Zhang and an L-order in the sense of Belohlavek are equivalent to each other. Then we redefine the concepts of fuzzy directed subsets and (continuous) fuzzy dcpos. We also define and study fuzzy Galois connections on fuzzy posets. We investigate some properties of (continuous) fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy-double-downward-arrow-operator has a right adjoint. We define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation.
95 citations
Cites background from "Fuzzy antisymmetry and order"
...Besides B-fpos, fuzzy partial orders are also defined and studied in [8,9,15,32, 41] by using different notions of antisymmetry....
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TL;DR: A fuzzy behavior of preference is defined which allows us to build up two determinant fuzzy coalitions which will be the base of the planner's requirement and the link between pessimistic results and optimistic ones, and May's theorem of majority choice is defined.
Abstract: The basic purpose of this paper is to link both theorems of impossibility and existence by introducing fuzzy relations of preference and an exogeneous requirement, the planner's one, and then proving the fundamental part played by the extremist agents, leximin and leximax. In other words, to bring out the link between the planner's requirement and the difficulty of the transition from individual to collective, as well as the theoric relation between this requirement and the extremist agents, we define a fuzzy behavior of preference which allows us to build up two determinant fuzzy coalitions. These coalitions will be the base of the planner's requirement and the link between pessimistic results (Arrow's impossibility) and optimistic ones (May's theorem of majority choice).
27 citations
01 Jan 2015
TL;DR: The presentation is uniform and integrated in the sense that every concept is developed on fuzzy set as base and is organized from the angle of foundational aspects of fuzzy set theory.
Abstract: In this paper, some earlier results dealing with fuzzy relations on fuzzy sets by one of the present authors are recapitulated to clarify the motivational background of the present study. Then concepts like fuzzy homorelation, correlation, cohomorelation, fuzzy function and fuzzy morphism are introduced and discussed. The presentation is uniform and integrated in the sense that every concept is developed on fuzzy set as base and is organized from the angle of foundational aspects of fuzzy set theory.
17 citations
Cites background from "Fuzzy antisymmetry and order"
...Some of the results from Chakraborty and Das (1983a,b); Chakraborty and Sarkar (1987); Chakraborty et al. (1985) are presented below....
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...Theorem 5 (Chakraborty and Sarkar 1987) Let R̃ be an antisymmetric and transitive relation on à and R̃1, R̃2 be the components of R̃ obtained by max–min resolution....
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...Proposition 2 (Chakraborty and Sarkar 1987) Let R̃1, R̃2 be two perfect antisymmetric relations....
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...A few constants, viz., order of antisymmetry, measure of antisymmetry are introduced in Chakraborty and Sarkar (1987)....
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...Definition 8 (Chakraborty and Sarkar 1987) (i) For a set {a, b, c} of three elements each of the following sets of ordered pairs is called an almost cyclic arrangement....
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03 Nov 2019
TL;DR: The concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation are introduced and it is proved that the set of all bipolar fuzzy equivalence classes is aipolar fuzzy partition and that the bipolar fuzziness equivalence relation is induced by a bipolar fuzzier partition.
Abstract: We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.
17 citations
01 Jan 1996
TL;DR: Fuzzy analysis was specifically designed to incorporate the subjectivity associated with human responses and questions in order to help characterize both the disease component as well as the risk factor aspects of epidemiology.
Abstract: Inherent in much of epidemiology is the concept of ambiguity of expression. Both the data gathered and the questions asked may have a subjective element in them. By applying traditional statistical methods to such information, overly precise statements may be made. Fuzzy analysis, on the other hand, was specifically designed to incorporate the subjectivity associated with human responses and questions. In this paper, fuzzy analysis is briefly described. Some of its methods are applied to several application areas of epidemiology to help characterize both the disease component as well as the risk factor aspects of epidemiology. The appropriateness of the methods are discussed and further research areas are
7 citations
Cites background from "Fuzzy antisymmetry and order"
...Chakraborty and Sarkar (1987) considered various types of ordering such as fuzzy quasi-order, fuzzy partial order, fuzzy weak ordering, and fuzzy linear ordering. They proved several theorems regarding these orderings and detailed several properties. Ovchinnikov (1989) generalized the Baas-Kwakernaak index which induces a...
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...Chakraborty and Sarkar (1987) considered various types of ordering such as fuzzy quasi-order, fuzzy partial order, fuzzy weak ordering, and fuzzy linear ordering....
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References
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TL;DR: An extended version of Szpilrajn's theorem is proved and various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, a fuzzy preordering is investigated which is reflexive and antisymmetric.
Abstract: The notion of ''similarity'' as defined in this paper is essentially a generalization of the notion of equivalence. In the same vein, a fuzzy ordering is a generalization of the concept of ordering. For example, the relation x @? y (x is much larger than y) is a fuzzy linear ordering in the set of real numbers. More concretely, a similarity relation, S, is a fuzzy relation which is reflexive, symmetric, and transitive. Thus, let x, y be elements of a set X and @m"s(x,y) denote the grade of membership of the ordered pair (x,y) in S. Then S is a similarity relation in X if and only if, for all x, y, z in X, @m"s(x,x) = 1 (reflexivity), @m"s(x,y) = @m"s(y,x) (symmetry), and @m"s(x,z) >= @? (@m"s(x,y) A @m"s(y,z)) (transitivity), where @? and A denote max and min, respectively. ^y A fuzzy ordering is a fuzzy relation which is transitive. In particular, a fuzzy partial ordering, P, is a fuzzy ordering which is reflexive and antisymmetric, that is, (@m"P(x,y) > 0 and x y) @? @m"P(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x y @? @m"s(x,y) > 0 or @m"s(y,x) > 0. A fuzzy preordering is a fuzzy ordering which is reflexive. A fuzzy weak ordering is a fuzzy preordering in which x y @? @m"s(x,y) > 0 or @m"s(y,x) > 0. Various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, an extended version of Szpilrajn's theorem is proved.
2,524 citations
01 Jan 1975
TL;DR: In this article, the concept of similarity relation introduced by L. A. Zadeh is derivable in much the same way as equivalence relation, and the resolution identity is brought out quite naturally.
Abstract: Publisher Summary This chapter discusses the basic terminologies and notations regarding fuzzy relations and fuzzy graphs. The the concept of similarity relation introduced by L. A. Zadeh is derivable in much the same way as equivalence relation. Moreover, through this derivation, the resolution identity is brought out quite naturally. The chapter analyzes fuzzy graphs from the connectedness viewpoint and the presents the application of results to clustering analysis and modeling of information networks. The usual graph-theoretical approaches to clustering analysis involve first obtaining a threshold graph from a fuzzy graph and then applying various techniques to obtain clusters as maximal components under different connectivity considerations. These methods have a common weakness, namely, the weight of edges are not treated fairly because any weight greater (less) than the threshold is treated as 1(0).
170 citations
TL;DR: F fuzzy reflexive, symmetric and transitive relations on fuzzy subsets are studied and the possible fuzzy partitioning of a fuzzy subset is investigated.
Abstract: Fuzzy reflexive, symmetric and transitive relations on fuzzy subsets are studied. Various types of reflexivities and corresponding equivalences are considered. The possible fuzzy partitioning of a fuzzy subset is investigated.
51 citations
TL;DR: In this paper a generalisation of fuzzy relations is introduced - fuzzy relations are defined on fuzzy subsets and properties like ordered reflexivity, symmetry, transitity, transitive closures of such generalised relations and operations on them are discussed.
Abstract: In this paper a generalisation of fuzzy relations is introduced - fuzzy relations are defined on fuzzy subsets. Properties like ordered reflexivity, symmetry, transitity, transitive closures of such generalised relations and operations on them are discussed.
42 citations
22 citations