A new approach of bipolar valued fuzzy set theory applied on semigroups
Pairote Yiarayong
Abstract
In this paper, we generalize the concept of interval-valued bipolar fuzzy sets (IVBFSs) and define interval-valued bipolar fuzzy subsemigroups (IVBF-subsemigroup) and interval-valued bipolar fuzzy left (right, two-sided) ideals (IVBF-left [right, two-sided] ideals) over semigroups, which is a generalization of the concept of an bipolar valued fuzzy set (BVFS) in a semigroup. The purpose of this paper is to deal with the algebraic structure of semigroups by applying IVBFS theory. We give characterizations of different classes of (intra-regular, left [right] regular, regular, semisimple) semigroups by the properties of their IVBF-ideals. We also characterize these classes in terms of IVBF-left ideals, IVBF-right ideals, and IVBF-two-sided ideals. In this respect, we prove that a semigroup is regular if and only if for every IVBF-right ideal A ˜ = μ A P ˜ , μ A N ˜ and every IVBF-left ideal B ˜ = μ B P ˜ , μ B N ˜ over S , we have A ˜ ∩ B ˜ = A ˜ ⊙ B ˜ . Further, we characterize intra-regular and regular semigroups and prove that a semigroup is intra-regular and regular if and only if for every IVBF-left ideal A ˜ = μ A P ˜ , μ A N ˜ and every IVBF-right ideal B ˜ = μ B P ˜ , μ B N ˜ over S we have A ˜ ∩ B ˜ ≼ A ˜ ⊙ B ˜ .