Structures of bipolar fuzzy Γ-hyperideals in Γ-semihypergroups

Abstract

The theory of bipolar fuzzy sets introduced by Lee [9] has been applied to many branches of mathematics. In this paper, we initiate a study on bipolar fuzzy sets in Γ-semihypergroups. We define bipolar fuzzy left (right, bi-, interior, (1,2)-) Γ-hyperideals and explore some related properties. We use these bipolar fuzzy Γ-hyperideals to characterize some classes of Γ-semihypergroups. We consider the Γ-semihypergroup $\underline{\mathcal{H}}$ of the bipolar fuzzy points of a Γ-semihypergroup H to discuss the relation between the bipolar fuzzy sub Γ-semihypergroup (left, right, bi-, interior, (1,2)-) Γ-hyperideal and the subsets of $\underline{\mathcal{H}}$ in a (regular) Γ-semihypergroup. In the end, we discuss in detail a number of results on homomorphic images and preimages of bipolar fuzzy Γ-hyperideals.