Decomposition of properties and definition of fuzzy set

Abstract

A mathematical model of fuzzy set is suggested in this paper. It is based on the understanding of a fuzzy set as a collection of objects showing some common property. This property itself is defined as decomposable into elementary properties. The degree to which an object shows a decomposable property, depends on the size, or more generally, importance of the collection of its elementary properties, that is evaluated by means of a set function that we refer to as pseudomeasure. With each pseudomeasure on a class of collections of elementary properties we associate a pseudomeasure that is complementary to it, and with each decomposable property, a complementary decomposable property interpreted as not showing the original property. We consider products of decomposable properties corresponding to the traditional operations of intersection and union of fuzzy sets. The model of a fuzzy set introduced leads to formulating compositions of decomposable properties (and associated fuzzy sets) that are not confined only to those based on operations of intersection and union, although they include them.