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Ji, Zhulin; Ren, Xueming; Wang, Yanhui

Left quasi-abundant semigroups. (English) Zbl 1476.20057

J. Korean Math. Soc. 56, No. 5, 1159-1172 (2019).
The relations \(\tilde {\mathcal L}\), \(\tilde {\mathcal R}\) are defined on a semigroup \(S\) with the set of idempotents \(E\) by \((\forall e \in E)(ae = a \Leftrightarrow be = b)\), \(a\tilde {\mathcal R}b\) iff \((\forall e \in E)(ea = a \Leftrightarrow eb = b)\), \(a,b \in S\). \(S\) is weakly abundant if each \(\tilde {\mathcal L}\)-class and each \(\tilde {\mathcal R}\)-class contains an idempotent. A weakly abundant semigroup satisfies the congruence condition if \(\tilde {\mathcal L}\) and \(\tilde {\mathcal R}\) are a right congruence and a left congruence respectively; a weakly abundant semigroup is called an \(E\)-abundant if it has the congruence condition. An \(E\)-abundant (weakly abundant) semigroup is said to be (weakly) left quasi-abundant if its set of idempotents forms a left quasi-normal band. Here, necessary and sufficient conditions for the set of idempotents of a weakly abundant semigroup to be a left quasi-normal band are proved and a description of left quasi-abundant semigroups in terms of weak spined products is presented.
Reviewer: Jaak Henno (Tallinn)

MSC:

20M10 General structure theory for semigroups

Keywords:

weakly abundant semigroups; weak spined products; left quasi-abundant semigroups; type \(Q\) semigroups

Citations:

Zbl 1083.20053
PDFBibTeX XMLCite
\textit{Z. Ji} et al., J. Korean Math. Soc. 56, No. 5, 1159--1172 (2019; Zbl 1476.20057)
Full Text: DOI

References:

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