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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 n2, is an ideal for each I in Y. Also (AB) n is an ideal and (AB) 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 SP is an M-system. A left ideal I of S with left identity is quasi-semiprime if and only if SI 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:
20N02Sets with a single binary operation (groupoids)
20M99Semigroups
20M12Ideal theory of semigroups