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M-systems in LA-semigroups. (English) Zbl 1203.20065
Summary: We study M-systems, P-systems and ideals in LA-semigroups. It is proved that if $S$ is an LA-semigroup with left identity $e$, then the set of all ideals $K$ forms an LA-semigroup. If $S$ is fully idempotent, then $K$ is a locally associative LA-semigroup. It is shown that ${I}^{n}$, for $n\ge 2$, is an ideal for each $I$ in $Y$. Also ${\left(AB\right)}^{n}$ is an ideal and ${\left(AB\right)}^{n}={A}^{n}{B}^{n}$, for all ideals $A,B$ in $Y$, where $Y$ is the set of ideals and $K$ is a locally associative LA-semigroup. We prove that a left ideal $P$ of an LA-semigroup $S$ with left identity is quasi-prime if and only if $S\setminus P$ is an M-system. A left ideal $I$ of $S$ with left identity is quasi-semiprime if and only if $S\setminus I$ is a P-system. In particular, we prove that every right ideal is an M-system and every M-system is a P-system.

MSC:
 20N02 Sets with a single binary operation (groupoids) 20M99 Semigroups 20M12 Ideal theory of semigroups