Generalized fuzzy equivalent relations

Abstract

In the paper, by the use of the neighborhood relations between fuzzy points and fuzzy relations, the concept of (β̅,α̅)-fuzzy equivalent relation is presented. Firstly, we derived that the acceptable non-trivial concepts obtained in this manner are the (∈, ∈)-fuzzy equivalent relation, (∈, ∈∨q)-fuzzy equivalent relation and (∈̅, ∈̅ ∨q)-fuzzy equivalent relation. Secondly, we generalized that the (∈, ∈)-fuzzy equivalent relation, the (∈, ∈q)-fuzzy equivalent relation and the (∈̅, ∈̅ ∨q̅)-fuzzy equivalent relation to the (λ, µ]-fuzzy equivalent relation. We proved that R is a (λ, µ]-fuzzy equivalent relation if and only if, for any t ∈ (λ, µ], the cut relation Rt is a equivalent relation.