in a recent paper, de baets et al. have characterized the fuzzytolerance and fuzzy equivalence relations that a given strict order relation iscompatible with. in this paper, we generalize this characterization by consideringan arbitrary (crisp) relation instead of a strict order relation, while payingattention to the particular cases of a reflexive or irreflexive relation. the reasoninglargely ...

In a recent paper, De Baets et al. have characterized the fuzzytolerance and fuzzy equivalence relations that a given strict order relation iscompatible with. In this paper, we generalize this characterization by consideringan arbitrary (crisp) relation instead of a strict order relation, while payingattention to the particular cases of a reflexive or irreflexive relation. The reasoninglargely ...

Remark 1 In this text, we always work between 0 and 1 with 1/2 playing the rôle of the midpoint. A simple rescaling allows to work between −1 and 1 with 0 playing the rôle of a midpoint. • A valued relation T is said to be reflexive if T (x, x) = 1, for all x ∈ X. A valued relation T on X such that T (x, y) ∈ {0, 1}, for all x, y ∈ X, is said to be crisp. As is usual, we write x T y instead of ...

Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space.

The starting point of the paper is the (well-known)observation that the “classical” Rough Set Theory as introduced by PAWLAK is equivalent to the S5 Propositional Modal Logic where the reachability relation is an equivalence relation. By replacing this equivalence relation by an arbitrary binary relation (satisfying certain properties, for instance, reflexivity and transitivity) we shall obtain...

We present a fuzzy version of the crisp spatial logic developed by Randell et al., which takes the single relation connected−with as primitive. Membership functions are defined for each spatial relation defined in the crisp theory. Furthermore, principles are presented for defining linguistic variables whose linguistic values are spatial relations. The work reported here addresses spatial reaso...

In this paper we consider a decision maker who shows his/her preferences for different alternatives through a finite set of ordinal values. We analyze the problem of consistency taking into account some transitivity properties within this framework. These properties are based on the very general class of conjunctors on the set of ordinal values. Each reciprocal preference relation on a finite o...

Galois connection in crisp binary relations has proved to be useful for several applications in computer science. Unfortunately, data is not always presented as a crisp binary relation but may be composed of fuzzy values, thus forming a fuzzy binary relation. This paper aims at defining the notion of fuzzy galois connection corresponding to a fuzzy binary relation in two steps: firstly by defin...

We focus on the Ferrers property of fuzzy preference relations. We study the connection between the Ferrers property and fuzzy interval orders. A crisp total interval order is characterized by the Ferrers property of its strict preference relation. Also, a crisp preference structure is a total interval order if and only if its large preference relation satisfies the Ferrers property. For fuzzy ...

In this paper ∗–compatible extensions of fuzzy relations are studied, generalizing some results obtained by Duggan in case of crisp relations. From this general result are obtained as particular cases fuzzy versions of some important extension theorems for crisp relations (Szpilrajn, Hansson, Suzumura). Two notions of consistent closure of a fuzzy relation are introduced.