Semigroups whose regular \({\mathcal J}\)-classes form finite lattices
type: Item
instance of: scholarly article
title: Semigroups whose regular \({\mathcal J}\)-classes form finite lattices
zbMATH Open document ID: 0876.20042
DOI: 10.1007/BF02573524
published in: Semigroup Forum
publication date: 1995-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/135385
review text: A \(\mathcal J\)-class of a semigroup \(S\) is defined to be regular if it contains at least one regular element. The family of all regular \(\mathcal J\)-classes of a semigroup \(S\) is denoted by \({\mathcal J}_R(S)\) and is partially ordered in the usual manner. That is, \(J_a\leq J_b\) if \(S^1aS^1\subseteq S^1bS^1\). The symbol \(S(X)\) denotes the semigroup of all continuous selfmaps of the topological space \(X\) where the binary operation is composition. In this paper, we characterize, among the dendrites, those spaces \(X\) for which \({\mathcal J}_R(S(X))\) forms a finite lattice. Specifically, what we do is to exhibit one particular space and eight infinite classes of spaces and we show that, among the dendrites, the subcontinua of these spaces are exactly those spaces with the property that the partially ordered family of all regular \(\mathcal J\)-classes of their semigroups are finite lattices. All this is made precise in Theorem 3.6, the last result of the paper.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 763793
zbMATH Keywords: semigroups of continuous selfmaps
zbMATH Keywords: regular \(\mathcal J\)-classes
zbMATH Keywords: dendrites
zbMATH Keywords: finite lattices
author: jun. Kenneth D. Magill
author: Prabudh R. Misra
MaRDI profile type: Publication
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OpenAlex ID: W1977161763
description: scientific article; zbMATH DE number 763793