Properties Of Fuzzy Relations Research Articles

On the Aggregation of Comonotone or Countermonotone Fuzzy Relations

The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min/max and commonly used increasing/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified.

Bounded Rational Reciprocal Preference Relation for Decision Making

Fuzzy relations have been widely applied in decision making process. However, the application process requires people to have a high level of ability to compute and infer information. As people usually have limited ability of computing and inferring, the fuzzy relation needs to be adapted to fit the abilities of people. The bounded rationality theory holding the view that people have limited rationality in terms of computing and inferring meets such a requirement, so we try to combine the fuzzy relation with the bounded rationality theory in this study. To do this, first of all, we investigate four properties of fuzzy relations (i.e. reflexivity, symmetry, transitivity and reciprocity) within the bounded rationality context and find that these properties are not compatible with the bounded rationality theory. Afterwards, we study a new property called the bounded rational reciprocity of fuzzy relations, to make it possible to combine a fuzzy relation with the bounded rationality theory. Based on the bounded rational reciprocity, the bounded rational reciprocal preference relation is then introduced. A rationality visualization technique is proposed to intuitively display the rationality of experts. Finally, a bounded rationality net-flow-based ranking method is presented to solve real decision-making problems with bounded rational reciprocal preference relations, and a numerical example with comparative analysis is given to demonstrate the advantages of the proposed methods.

Properties of fuzzy relations and aggregation process in decision making

In this contribution connections between input fuzzy relations R1, . . . ,Rn on a set X and the output fuzzy relationRF = F(R1, . . . ,Rn) are studied. F is a function of the form F : [0, 1]n → [0, 1] and RF is called an aggregated fuzzyrelation. In the literature the problem of preservation, by a function F, diverse types of properties of fuzzy relationsR1, . . . ,Rn is examined. Here, it is considered the converse approach. Namely, fuzzy relation RF = F(R1, . . . ,Rn) isassumed to have a given property and then it is checked if fuzzy relations R1, . . . ,Rn have this property. Moreover, adiscussion on the mentioned two approaches is provided. The properties, which are examined in this paper, depend ontheir notions on binary operations B : [0, 1]2 → [0, 1]. By incorporating operation B these properties are generalizedversions of known properties of fuzzy relations.

A Revisit to Traces in Characterizing Properties of Fuzzy Relations

In this paper, we focus on the characterizations of some properties of fuzzy relations by using the notions of traces. The investigation is two-fold. On one hand, we attempt to reformulate the results obtained by Fodor under weakened conditions. On the other hand, we present some new characterizations of properties, mainly T-asymmetry, T-S-semitransitivity and T-S-Ferrers property, in terms of traces.

Similarity issues of confluence of fuzzy relations

We study similarity issues of confluence and related properties of fuzzy relations. The ordinary notions of divergence, convergence, convertibility, and confluence are essential properties of relations which appear in abstract rewriting systems. In a graded setting, we can introduce analogous notions related to the idea of approximate rewriting. This paper is a continuation of our previous paper (Belohlavek et al. 2010), where we have introduced such notions using residuated structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure of truth degrees is the two-valued Boolean algebra. In this paper, we focus on similarity issues of confluence and present formulas which provide approximations of degrees of confluence and related properties based on similarity of the original fuzzy relations.

Do exact shapes of fuzzy sets matter?

An often raised objection to fuzzy logic can be articulated as follows. Fuzzy sets are intended to represent vague collections of objects. On the one hand, boundaries of such collections are inherently imprecise. On the other hand, fuzzy sets are defined by membership degrees which are precise. There is, therefore, an obvious discrepancy between the imprecision of a collection and the precision of a fuzzy set representing the collection. The objection leads to a question posed in the title of this paper: do exact shapes of fuzzy sets matter? That is, do the exact values of membership degrees of fuzzy sets matter? This is a fundamental question, particularly when fuzzy sets are supplied as an input to further processing. This paper presents a particular answer to this question. We focus on the case when the fuzzy sets in question are part of the input I from which the output O is produced in a way which can be described by logical formulas. This case covers several widely used fuzzy logical models including extension principle, products of fuzzy relations, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. We present formulas which say how similar is the original output O to a new output O′ when the original input I is replaced by a new input I′. The presented formulas provide us with an analytical tool which enables us to answer the question of whether (and to what extent) the exact shapes of fuzzy sets in a particular fuzzy logical model matter. We present application of our formulas to selected topics including the above-mentioned extension principle, fuzzy relational products, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. Based on the analysis, we argue that in several applications of fuzzy logic, the exact shapes of fuzzy sets do not really matter.

Fuzzy sets and their operations, II

This paper investigates the algebraic properties of fuzzy sets under the new operations “drastic product≓ and “drastic sum≓ introduced by Dubois in 1979, and the algebraic properties in the case where these new operations are combined with the well-known operations for fuzzy sets. The properties of fuzzy relations are also shown under a new composition of fuzzy relations which is defined by using the drastic product.