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Zbl 1203.20065
Mushtaq, Qaiser; Khan, Madad
M-systems in LA-semigroups. (English)
[J] Southeast Asian Bull. Math. 33, No. 2, 321-327 (2009). ISSN 0129-2021; ISSN 0219-175X/e

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 $(AB)^n$ is an ideal and $(AB)^n=A^nB^n$, 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 2000:: *20N02 Sets with a single binary operation (groupoids)
20M99 Semigroups
20M12 Ideal theory of semigroups

Keywords: LA-semigroups; invertive law; AG-groupoids; medial law; M-systems; P-systems; prime ideals; quasi-prime ideals; semiprime ideals; left almost semigroups

Cited in: Zbl 1217.06012

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