Generalized fuzzy equivalent relations

In the paper, by the use of the neighborhood relations between fuzzy points and fuzzy relations, the concept of (betamacr,alphamacr)-fuzzy equivalent relation is presented. Firstly, we derived that the acceptable non-trivial concepts obtained in this manner are the (isin, isin)-fuzzy equivalent relation, (isin, isin Vq)-fuzzy equivalent relation and (isinmacr, isinmacr Vq)-fuzzy equivalent relation. Secondly, we generalized that the (isin, isin)-fuzzy equivalent relation, the (isin, isin Vq)-fuzzy equivalent relation and the (isinmacr, isinmacr V qmacr)-fuzzy equivalent relation to the (lambda, mu]-fuzzy equivalent relation. We proved that R is a (lambda, mu]-fuzzy equivalent relation if and only if, for any t isin (lambda, mu], the cut relation R_t is a equivalent relation.