Decomposition of locally associative $Gamma$-AG-groupoids

Tariq Shah, Inayatur-Rehman

It is well known that power associativity and congruences play an important role in the decomposition of semigroups and $\Gamma$-semigroups. In this study, we discuss these notions for a non-associative and non-commutative algebraic structure, known as $\Gamma$-AG-groupoids. Specifically, we show that for a locally associative $\Gamma$-AG-groupoid $S$ with a left identity, $S/\rho$ is the maximal weakly separative homomorphic image of $S$, where $\rho$ is a relation on S defined by: $a\rho b$ if and only if a $b\gamma n^n=b^{n+1}$ and $b\gamma a^n=a^{n+1}$ for some positive integer $n$ and for all $\gamma\in \Gamma$, where $a,b\in S$.