Semilattice decomposition of locally associative $\Gamma$-AG$^{**}$-groupoids

M. Khan, S. Anis, K. Hila

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