One-to-one correspondences between [formula omitted]-partitions, [formula omitted]-equivalences and [formula omitted]-pseudometrics

Abstract

The generalization of the concept of a classical (or crisp) partition to that of a fuzzy partition, requires, in our researches, the choice of a family of fuzzy subsets which preserves the conditions characterizing crisp partitions; however, the usual constraints for the union and intersection of subsets are very limiting and seem to be far from the “fuzzy spirit”; so that the union of subsets in the partitions is forced to be “close to” the universe of discourse and the intersection of any two subsets in the partition is forced to be “close to” the empty set. Furthermore, since the classical definition of partition is very related to the union and intersection, the generalization of this concept has to be applicable for any extension of these operations for fuzzy sets. Apart from this, we need to redefine the concept of fuzzy partition to establish a one-to-one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations. The different definitions that we found in the literature did not exactly verify these requirements, although some of them were more appropriate than others. Finally, we have given a definition of fuzzy partition of any fuzzy subset of the universe, which is a generalization of crisp partitions and of most of the different definitions of fuzzy partitions, which preserves the classical ideas of covering and disjointness and such that it is appropriate in our researches (Classification Problems, Fuzzy Questionnaires, Fuzzy Information Theory). Once our fuzzy partitions ( ε-partitions) are defined, it is necessary to find the fuzzy binary relation which generates and characterizes the fuzzy partitions. This fuzzy relation has to be, by analogy with the crisp case, a fuzzy “equivalence” relation (two points in the same fuzzy subset of the partition will be “related”, and two points in different subsets will not). In relation with this “equivalence” relation, it is also possible to obtain a way to measure the “distance” between two points, by means of the degree of relation that there exists between them, and therefore, by means of the membership degree to a same subset in the fuzzy partition. The definitions of these concepts and the relationships between them are the purpose of this paper.