Abstract
Recently, fuzzy multi-sets have come to the forefront of scientists' interest and have been used in algebraic structures such as multi-groups, multi -rings, anti-fuzzy multigroup and (alpha, gamma)-anti-fuzzy subgroups. In this paper, we first summarize the knowledge about the algebraic structure of fuzzy multi-sets such as (alpha, gamma)-anti-multi-fuzzy subgroups. In a way, the notion of anti-fuzzy multigroup is an application of anti-fuzzy multi sets to the theory of group. The concept of anti-fuzzy multigroup is a complement of an algebraic structure of a fuzzy multi set that generalizes both the theories of classical group and fuzzy group. The aim of this paper is to highlight the connection between fuzzy multi-sets and algebraic structures from an anti-fuzzification point of view. Therefore, in this paper, we define (alpha, gamma)-anti-multi-fuzzy subgroups, (alpha, gamma)-anti-multi-fuzzy normal subgroups, (alpha, gamma)-anti-multi-fuzzy homomorphism on (alpha, gamma)-anti-multi-fuzzy subgroups and these been explicated some algebraic structures. Then, we introduce the concept (alpha, gamma)-anti-multi-fuzzy subgroups and (alpha, gamma)-anti-multi-fuzzy normal subgroups and of their properties. This new concept of homomorphism as a bridge among set theory, fuzzy set theory, anti-fuzzy multi sets theory and group theory and also shows the effect of anti-fuzzy multi sets on a group structure. Certain results that discuss the (alpha, gamma) cuts of anti-fuzzy multigroup are explored.