Structure and characteristics of left Clifford semigroups
type: Item
instance of: scholarly article
title: Structure and characteristics of left Clifford semigroups
zbMATH Open document ID: 0781.20035
author: Pinyu Zhu
author: Y. Q. Guo
author: K. P. Shum
published in: Science in China. Series A
publication date: 1993-01-17T00:00:00Z
review text: A regular semigroup \(S\) is called left Clifford semigroup (left \(C\)- semigroup) if \(aS \subseteq Sa\) for all \(a\in S\). The \(\xi\)-product \(S = T\times_ \xi I\) of semigroups \(T\) and \(I\) is defined as particular product defined on \(S = \bigcup_{\alpha\in Y}S_ \alpha\), where \(Y\) is a semilattice, \(S_ \alpha = T_ \alpha \times I_ \alpha\), \(T = \bigcup_{\alpha \in Y} T_ \alpha\) is a strong semilattice of semigroups \(T_ \alpha\), \(I = \bigcup_{\alpha \in Y} I_ \alpha\) is a semilattice of left zero bands \(I_{\alpha}\), and \(\xi\) is a special map from \(S\) to the transformation semigroup over \(I\). It is proved that the \(\xi\)-product \(T \times_ \xi I\), where \(T\) is a strong semilattice of groups \(G_ \alpha\), is a left \(C\)-semigroup and conversely any left \(C\)-semigroup is precisely the \(\xi\)-product of the above semigroups. Moreover, some characteristics for left \(C\)-semigroups are obtained, some special classes of left \(C\)-semigroups are studied, and questions how the \(\xi\)-product becomes a spined product and the semilattice decomposition becomes a strong semilattice decomposition are discussed.
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M17
zbMATH DE Number: 98074
zbMATH Keywords: left \(C\)-semigroup
zbMATH Keywords: regular semigroup
zbMATH Keywords: left Clifford semigroup
zbMATH Keywords: strong semilattice of semigroups
zbMATH Keywords: left zero bands
zbMATH Keywords: strong semilattice decomposition
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 98074