Twofold fuzzy sets: An approach to the representation of sets with fuzzy boundaries based on possibility and necessity measures.

*(English)*Zbl 0562.54011Domains with fuzzy boundaries are considered. The first part deals with intervals of the real line bounded by two fuzzy numbers. The definitions of a closed interval and of an open interval are extended to fuzzy bounds. The closed interval corresponds to the fuzzy set of numbers possibly between or equal to the bounds and the open one to the fuzzy set of numbers necessarily strictly between the bounds. Various noticeable properties are established and an application to the problem of ranking fuzzy numbers is proposed. In the second part we present a generalization to abstract sets of the idea of representing a domain with a fuzzy boundary by a pair of nested fuzzy sets such that one of the fuzzy sets is included in the core of the other: such a pair is called a twofold fuzzy set. A twofold fuzzy set may be viewed as the interior and the closure of a domain with a fuzzy boundary. In this framework, a new definition of the concept of a fuzzy partition is proposed. Then, twofold fuzzy relations are introduced, their composition is defined, and their use in approximate reasoning is discussed.

##### MSC:

54A40 | Fuzzy topology |