On Semilattice Decomposition of an Abel–Grassmann’s Groupoid

In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as: aγb if and only if there exist positive integers m and n such that bm ∈ ( Sa)S and a n ∈ ( Sb)S for all a and b in S. We have proved that S/γ is a maximal separative semilattice homomorphic image of S. Every AG-groupoid S is uniquely expressible as a semilattice Y of archimedean AG-groupoids Sα (α∈ Y ). The semilattice Y is isomorphic to S/γ and the S α (α∈ Y ) are the equivalence classes of S mod γ.