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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 \(n\geq 2\), 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:

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