Finite {\bf $Γ$-AG}-groupoids with Left Identities and Left Zeros (European Journal of Pure and Applied Mathematics, 2025)
Let Γ be a nonempty set. A nonempty set A is called a Γ-AG-groupoid if there is a function f of A × Γ × A into A, customary denoted aγb for f(a, γ, b), satisfying the identity (aγb)βc = (cγb)βa for any a, b, c ∈ A and γ, β ∈ Γ. For each γ ∈ Γ, an operation on A associated to γ is given by ab = aγb. Suppose further that A is finite, contains a left identity and a left zero a0. The objective of this paper is to provide sufficient conditions under which the set A \ {a0} is a commutative group under the operation on A determined by γ for all γ ∈ Γ.