On 2-absorbing bipolar fuzzy ideals over LA -semigroups

Abstract

The aim of this manuscript is to apply bipolar fuzzy sets for dealing with several kinds of theories in $\mathcal{LA}$ -semigroups. To begin with, we introduce the concept of 2-absorbing (quasi-2-absorbing) bipolar fuzzy ideals and strongly 2-absorbing (quasi-strongly 2-absorbing) bipolar fuzzy ideals in $\mathcal{LA}$ -semigroups, and investigate several related properties. In particular, we show that a bipolar fuzzy set $\mathcal{A}=\left({\mu }_{\mathcal{A}}^{\mathcal{P}},{\mu }_{\mathcal{A}}^{\mathcal{N}}\right)$ over an $\mathcal{LA}$ -semigroup S is weakly 2-absorbing if and only if $\left[\mathcal{B}\odot \mathcal{C}\right]\odot \mathcal{D}⪯\mathcal{A}$ implies $\mathcal{B}\odot \mathcal{C}⪯\mathcal{A}$ or $\mathcal{C}\odot \mathcal{D}⪯\mathcal{A}$ or $\mathcal{B}\odot \mathcal{D}⪯\mathcal{A}$ for any bipolar fuzzy sets $\mathcal{B}=\left({\mu }_{\mathcal{B}}^{\mathcal{P}},{\mu }_{\mathcal{B}}^{\mathcal{N}}\right),\mathcal{C}=\left({\mu }_{\mathcal{C}}^{\mathcal{P}},{\mu }_{\mathcal{C}}^{\mathcal{N}}\right)$ and $\mathcal{D}=\left({\mu }_{\mathcal{D}}^{\mathcal{P}},{\mu }_{\mathcal{D}}^{\mathcal{N}}\right)$ . Also, we give some characterizations of quasi-strongly 2-absorbing bipolar fuzzy ideals over an $\mathcal{LA}$ -semigroup S by bipolar fuzzy points. In conclusion of this paper we prove that the relationship between quasi-strongly 2-absorbing bipolar fuzzy ideals over an $\mathcal{LA}$ -semigroup S and quasi-2-absorbing bipolar fuzzy ideals over S.