The structure of \(\mathcal L^*\)-inverse semigroups.
type: Item
instance of: scholarly article
title: The structure of \(\mathcal L^*\)-inverse semigroups.
zbMATH Open document ID: 1117.20043
published in: Science in China. Series A
publication date: 2007-01-24T00:00:00Z
review text: A semigroup is `abundant' if each class of the `potential' Green's relations \({\mathcal L}^*\) and \({\mathcal R}^*\) contains an idempotent. As common generalizations of regular semigroups and cancellative monoids, they have attracted considerable effort devoted to generalizing theorems on particular classes of regular semigroups to appropriate classes of abundant semigroups. For instance, the class of `type A' (now frequently termed `ample') semigroups extends the class of inverse semigroups appropriately, although a somewhat technical condition known as idempotent-connectedness is frequently needed for explicit generalizations of particular theorems. The aim of this paper is to generalize a theorem of \textit{M. Yamada} [Semigroup Forum 6, 113-128 (1973; Zbl 0255.20038)] on left inverse semigroups (regular semigroups in which the idempotents form a left regular band). The \({\mathcal L}^*\)-inverse semigroups are defined to be the abundant semigroups that are idempotent-connected and in which the idempotents once again form a left regular band. The concept of a left wreath product of a left regular band and a type A semigroup is defined and it is proven that the \({\mathcal L}^*\)-inverse semigroups correspond precisely to the class of all such wreath products.
reviewed by: Peter R. Jones
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 5118315
zbMATH Keywords: abundant semigroups
zbMATH Keywords: wreath products
zbMATH Keywords: inverse semigroups
zbMATH Keywords: type A semigroups
zbMATH Keywords: left regular bands
zbMATH Keywords: idempotents
author: Xueming Ren
author: K. P. Shum
MaRDI profile type: Publication
full work available at URL: s11425-006-1065-x
OpenAlex ID: W2228822624
cites work: Orthodox semigroups
cites work: Q4126563
cites work: Abundant Semigroups
cites work: Idempotent-connected abundant semigroups
cites work: Quasi-adequate semigroups
cites work: Adequate Semigroups
cites work: Abundant left C-lpp proper semigroups
cites work: PERFECTrppSEMIGROUPS
cites work: A structure theorem for right pp-semigroups with left central idempotents
cites work: The structure of superabundant semigroups.
cites work: Q4938151
cites work: The structure of type A semigroups
cites work: THE STRUCTURE OF TYPE A SEMIGROUPS
cites work: Right inverse semigroups
cites work: Right (left) inverse semigroups
cites work: Orthodox semigroups whose idempotents satisfy a certain identity
cites work: Semidirect products of semigroups
cites work: The natural partial order on an abundant semigroup
cites work: \({\mathcal L}^*\)-unipotent semigroups
DOI: 10.1007/S11425-006-1065-X
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description: scientific article; zbMATH DE number 5118315