FUZZY BETWEENNESS RELATIONS

The aim of this paper is to introduce the idea of fuzzy betweenness relation on a set X. This is done by generalizing the definition of betweenness relation proposed by Menger. It is proved that a separating T-indistinguishability operator on X (with T a strict archimedean t-norm) generates a fuzzy betweenness relation on X and reciprocally, every fuzzy betweenness relation on X defines a separating T-indistinguishability operator on X. Moreover, it is proved that the crisp part of a fuzzy betweenness relation is a classical (metric) betweenness relation.