Generalized fuzzy equivalent relations

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> is a equivalent relation.