On a theorem of C-wrpp semigroups. (English) Zbl 0879.20030

Tang, Xiangdong

Commun. Algebra 25, No. 5, 1499-1504 (1997).

A semigroup \(S\) is called an rpp semigroup if every principal right ideal \(aS'\) (\(a\in S\)) considered as a right \(S^1\)-system is projective. Guo, Shum and Zhu called a semigroup C-rpp if it is an rpp semigroup and its idempotents are central. J. B. Fountain showed that a semigroup \(S\) is a C-rpp semigroup if and only if \(S\) is a strong semilattice of left cancellative monoids. In this paper, the author generalizes the result of Fountain to C-wrpp semigroups, that is, the semigroups in which every \({\mathcal L}^{**}\)-class of \(S\) contains an idempotent and all idempotents of \(S\) are central. By the relation \({\mathcal L}^{**}\) on \(S\), he means \((a,b)\in{\mathcal L}^{**}\) if and only if (for all \(x,y\in S^1\)), \((ax,ay)\in{\mathcal R}\Leftrightarrow (bx,by)\in{\mathcal R}\). It is proved that a semigroup \(S\) is a C-rpp semigroup if and only if \(S\) is a strong semilattice of \({\mathcal R}\)-left cancellative monoids.

Reviewer: K.-P.Shum (Hongkong)

MSC:

20M10	General structure theory for semigroups
20M50	Connections of semigroups with homological algebra and category theory

Keywords:

rpp semigroups; principal right ideals; right \(S\)-systems; idempotents; strong semilattices of left cancellative monoids; C-wrpp semigroups

References:

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