Bipolar Anti Fuzzy HX Group and its Lower Level Sub HX Groups

Abstract

In this paper, we define a new algebraic structure of a bipolar fuzzy HX group and a bipolar anti fuzzy sub HX group of a HX group and lower level sub HX group of a HX group and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and group theory in bipolar anti fuzzy sub HX group of a HX group. Characterizations of lower level subsets of a bipolar anti fuzzy sub HX group of a HX group are given. We also discussed the relation between a given a bipolar anti fuzzy sub HX group of a HX group and its lower level sub HX groups and investigate the conditions under which a given HX group has a properly inclusive chain of sub HX groups. In particular, we formulate how to structure a bipolar anti fuzzy sub HX group of a HX group by a given chain of sub HX groups. We also establish the relation between bipolar fuzzy HX group and bipolar anti fuzzy HX group of a HX group.