An Analogy of Clifford Decomposition Theorem for Abel-Grassmann Groupoids

Abstract

Let S be an inverse AG-groupoid (Abel-Grassmann groupoid) and define a relation γ on S by a γ b if and only if there exist some positive integers n and m such that bm ∈ (Sa)S and an ∈ (Sb)S. We prove that S/γ is a maximal semilattice homomorphic image of S. Thus, every inverse AG-groupoid S is uniquely expressible as a semilattice Y of some Archimedean inverse AG-groupoids Sα (α ∈ Y). Our result can be regarded as an analogy of the well known Clifford theorem in semigroups for AG-groupoids.