Semigroups for which the continuum congruences form finite chains
type: Item
instance of: scholarly article
title: Semigroups for which the continuum congruences form finite chains
zbMATH Open document ID: 0533.20037
DOI: 10.1007/BF02573451
author: jun. Kenneth D. Magill
published in: Semigroup Forum
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134717
review text: S(X) denotes the semigroup of all continuous selfmaps of the topological space X. Let \({\mathcal A}\) be a nonempty family of subcontinua of X with the property that if \(A\in {\mathcal A}\), \(f\in S(X)\) and f maps A homeomorphically onto B, then \(B\in {\mathcal A}\). We associate with each such family \({\mathcal A}\), a congruence \(\sigma\) (\({\mathcal A})\) on S(X) by identifying two functions if whenever one of them is injective on any \(A\in {\mathcal A}\), then the two functions agree on A. Any congruence of the form \(\sigma\) (\({\mathcal A})\) is referred to as a continuum congruence and the partially ordered family of all continuum congruences on S(X) is denoted by \(Con_{{\mathcal K}}(S(X))\). Various conditions are obtained which are all equivalent to the condition that \(Con_{{\mathcal K}}(S(X))\) forms a finite chain. In particular, those spaces X for which this is the case are characterized within the Peano continua and although there are infinitely many such spaces there are essentially only five different types of such spaces.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 3845852
zbMATH Keywords: semigroup of continuous selfmaps
zbMATH Keywords: continuum congruences
zbMATH Keywords: Peano continua
MaRDI profile type: Publication
cites work: Q5532996
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description: scientific article; zbMATH DE number 3845852