Fuzzy identities with application to fuzzy semigroups

Abstract

The notion of a fuzzy identity is introduced on a fuzzy subalgebra in the framework of fuzzy equalities instead of the crisp one. A fuzzy identity may be satisfied by a fuzzy subalgebra, while the underlying crisp algebra need not satisfy the analogous crisp identity. Among other properties, it is shown that for every fuzzy subalgebra μ of A there is a least fuzzy equality such that a fuzzy identity holds on μ. As an application, fuzzy semigroups are introduced with respect to a fuzzy equality. In particular, fuzzy semilattices, cancellative and regular fuzzy semigroups are investigated.