Some studies in fuzzy non-associative semigroups

Abstract

The main motivation behind this paper is to study some structural properties of a non-associative structure ($\mathcal{LA}$-semigroup) in terms of fuzzy sets as it hasn’t attracted much attention compared to associative structures. An $\mathcal{LA}$-semigroup can be referred to as a non-associative semigroup, as the main difference between a semigroup and an $\mathcal{LA}$-semigroup is the switching of an associative law. In this paper, we introduce and characterize fuzzy (0, 2)-ideals, fuzzy (0, 2)-bi-ideals and fuzzy (1, 2)-ideals of an $\mathcal{LA}$-semigroup. We give a new and alternate definition for a strongly regular element of a unitary $\mathcal{LA}$-semigroup and show novel conditions under which a strongly regular $\mathcal{LA}$-semigroup becomes an ${\mathcal{LA}}^{**}$-semigroup. These characterizations do not exist in literature before this work. As an application of our results we get characterizations of a strongly regular $\mathcal{LA}$-semigroup in terms of fuzzy one-sided ideals and fuzzy bi-ideals. Finally we introduce and investigate the properties of fuzzy inner-unitary $\mathcal{LA}$-subsemigroups in an ${\mathcal{LA}}^{**}$-semigroup. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.