Strong semilattices of completely simple semigroups
type: Item
instance of: scholarly article
title: Strong semilattices of completely simple semigroups
zbMATH Open document ID: 0877.20042
author: O. Diekmann
published in: Southeast Asian Bulletin of Mathematics
publication date: 1997-12-07T00:00:00Z
review text: A construction of normal bands of groups \(S\) (i.e., strong semilattices \(Y\) of completely simple semigroups \(S_\alpha\), \(\alpha\in Y\)) is given. An essential ingredient in it is the natural partial order \(\leq_N\) on \(S\) which in this case is compatible with multiplication (on both sides) and has the property that the intersection of any principal order ideal of \((S,\leq_N)\) with an arbitrary component \(S_\alpha\) of \(S\) is either empty or consists of a single element. These properties are used to define on a disjoint union \(S\) of completely simple semigroups \(S_\alpha\) (\(\alpha\in Y\)) a multiplication \(*\), which makes \(S\) a normal band of groups. It is interesting to note that the arbitrary partial ordering \(\leq\), which is used in this construction and satisfies the above conditions, is then the natural partial order \(\leq_N\) on \((S,*)\). Also, it is shown that a completely regular semigroup \(S\) (i.e., a semilattice \(Y\) of completely simple semigroups \(S_\alpha\), \(\alpha\in Y\)) is a normal band of groups (i.e., is strong) if and only if for every \(a\in S_\alpha\) and every \(\beta\leq\alpha\) in \(Y\) the intersection of any principal order ideal \((a]\) of \((S,\leq_N)\) with \(S_\beta\subseteq S\) consists of a single element.
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 1022780
zbMATH Keywords: normal bands of groups
zbMATH Keywords: completely simple semigroups
zbMATH Keywords: natural partial orders
zbMATH Keywords: completely regular semigroups
reviewed by: H. Mitsch
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1022780