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

Authors : Abul BASAR

Pages : 107-111

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Publication Date : 2019-04-15

Article Type : Research Paper

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