The main motivation behind this paper is to study some structural properties of a non-associative structure ( LA -semigroup) in terms of fuzzy sets as it hasn’t attracted much attention compared to associative structures. An LA -semigroup can be referred to as a non-associative semigroup, as the main difference between a semigroup and an 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 LA -semigroup. We give a new and alternate definition for a strongly regular element of a unitary LA -semigroup and show novel conditions under which a strongly regular LA -semigroup becomes an 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 LA -semigroup in terms of fuzzy one-sided ideals and fuzzy bi-ideals. Finally we introduce and investigate the properties of fuzzy inner-unitary LA -subsemigroups in an LA * * -semigroup. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

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Some studies in fuzzy non-associative semigroups