On Γ-Ideals and Γ-Bi-Ideals in Γ-AG-Groupoids

In this paper we introduce Γ-ideals and Γ-bi-ideals of Γ-AG- groupoids which are in fact a generalization of ideals and bi-ideals of AG- groupoids. we study some characteristics of Γ-ideals and Γ-bi-ideals of Γ-AG- groupoids. Specifically, we show that a Γ-AG-groupoid S with left identity e is fully Γ-prime if and only if every Γ-ideal in S is Γ-idempotent and the set of Γ-ideals of S is totally ordered under inclusion. We also prove the equivalent conditions for Γ-bi-ideals of S that is (1) every Γ-bi-ideal of S is Γ-idempotent, (2) H ∩ K = HΓK, where H and K are any Γ-bi-ideals of S and (3) the Γ- ideals of S form a semilattice (LS, ∧), where H ∧K = HΓK. Also we show that every Γ-bi-ideal of a Γ-AG-groupoid S with left identity e is a Γ-prime if and only if it is Γ-idempotent and the set of Γ-bi-ideals of S is totally ordered under inclusion. In the end we prove that every Γ-ideal in a regular Γ-AG-groupoid S is Γ-prime if and only if it is strongly irreducible.