SEMILATTICE DECOMPOSITION OF LOCALLY ASSOCIATIVE Γ-AG∗∗-GROUPOIDS

Abstract

In this paper, we have shown that a locally associative $\mathrm{\Gamma }$-AG${}^{\ast \ast }$-groupoid $S$ has associative powers and $S/{\rho }_{\mathrm{\Gamma }}$ is a maximal separative homomorphic image of $S$, where $a{\rho }_{\mathrm{\Gamma }}b$ implies that $a\mathrm{\Gamma }{b}_{\mathrm{\Gamma }}^n=b_{\mathrm{\Gamma }}^{n+1},b\mathrm{\Gamma }{a}_{\mathrm{\Gamma }}^n=a_{\mathrm{\Gamma }}^{n+1},\mathrm{\forall }a,b\in S$. The relation ${\eta }_{\mathrm{\Gamma }}$ is the least left zero semilattice congruence on $S$, where ${\eta }_{\mathrm{\Gamma }}$ is defined on $S$ as $a{\eta }_{\mathrm{\Gamma }}b$ if and only if there exist some positive integers $m,n$ such that $b_{\mathrm{\Gamma }}^m\subseteq a\mathrm{\Gamma }S$ and $a_{\mathrm{\Gamma }}^n\subseteq b\mathrm{\Gamma }S$.