Abstract

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 $a_{\Gamma}^{n}\subseteq b\Gamma S$.

Keywords: $\Gamma$-AG-groupoid; $\Gamma$-left invertive law; $\Gamma$-medial law; $\Gamma$-congruences.

MSC: 20M10, 20N99

Pages:  273$-$280     

Volume  72 ,  Issue  3 ,  2020