On LA-Gamma-semihypergroups containing two-sided bases

Abstract

Ideal theory and base theory of algebraic structures play an important role in studied. In this study, we focus on studying bases of ideals in algebraic structures. The algebraic structure LA-Γ-semihypergroups is very interesting to study. This structure is generalization of many algebraic structures, for example, commutative semigroups, commutative Γ-semigroups, LA-semigroups, LA-Γ-semigroups, commutative semihypergroups, etc. Many results in these algebraic structures were extended to results in LA-Γ-semihypergroups. On the other hand, the results in these structures are the special cases of results in LA-Γ-semihypergroups. The main purpose of this paper is to introduce the notion of two-sided base of an LA-Γ-semihypergroup with pure left identity. We prove that every two-sided base of an LA-Γ-semihypergroup with pure left identity is one-element base. Finally, we prove that the compliment of union of all two-sided base of an LA-Γ-semihypergroup with pure left identity is maximal proper Γ-hyperideal.