Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1203.20065
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

Highlights
Master Server