Semilattice Decomposition of Locally Associative AG**-groupoids

Abstract

We consider a locally associative AG **-groupoid S and show that it has associative powers. We define ρ on S as aρb if and only if bna = bn+1 and anb = an+1 for all a, b ∈ S, and show that S/ρ is a maximal separative homomorphic image of S. We show that every S can be uniquely expressible as a semilattice Y of locally associative Archimedean AG **-groupoids Sα(α ∈ Y), the semilattice Y is isomorphic to a maximal homomorphic image S/η of S, and Sα(α ∈ Y) are the equivalence classes of S mod η.