A Note on $(m, n)$-$\Gamma$-Ideals of Ordered $LA$-$\Gamma$-Semigroups

Abstract

In this paper, we investigate the notion of $(m, n)$-ideals in a non-associative algebraic structure, which we call an ordered $LA$-$\Gamma$-semigroup. We prove that if $(S, \Gamma, \cdot, \leq)$ is a unitary ordered $LA$-$\Gamma$-semigroup with zero and $S$ has the condition that it contains no non-zero nilpotent $(m, n)$-ideals and if $R(L)$ is a 0-minimal right (left) ideal of $S$, then either $(R\Gamma L]=\{0\}$ or $(R\Gamma L]$ is a 0-minimal $(m, n)$-ideal of $S$. Also, we prove that if $(S, \Gamma, \cdot, \leq)$ is a unitary ordered $LA$-$\Gamma$-semigroup; $A$ is an $(m, n)$-ideal of $S$ and $B$ is an $(m, n)$-ideal of $A$ such that $B$ is idempotent, then $B$ is an $(m, n)$-ideal of $S$.