Compact semitopological semigroups: An intrinsic theory
type: Item
instance of: scholarly article
title: Compact semitopological semigroups: An intrinsic theory
zbMATH Open document ID: 0606.22001
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Compact semigroups arise in three different types: The first has a jointly continuous multiplication (x,y)\(\mapsto xy:\) \(S\times S\to S\); the second has continuous translation \(x\mapsto sx, xs:\) \(S\to S\) for all \(x\in S\), and the third has continuous translation on ONE side, say, \(x\mapsto sx:\) \(S\to S\) for all \(s\in S\). Semigroups of the first type are called topological, those of the second type semitopological, and with the third type, the terminological trouble starts. Some authors call these semigroups left topological when left translations are continuous; the present author would call these semigroup right topological, because the continuous argument is on the right side. Let us accede to the author and stick with his terminology throughout the review. Topological compact semigroups occur naturally as compact subsemigroups of the semigroup of all real matrices of degree n and as a great variety of compact semilattices such as they may arise from continuous lattices, and they occur less naturally as constructs in compact semigroup theory as a branch of compact topological algebra. While it would be exaggerated to call their theory classical, after its liftetime of over thirty years it is very well developed, and a number of books and monographs is devoted to it [\textit{A. B. Paalman-de Miranda}, Topological semigroups (1964; Zbl 0136.269); the reviewer and \textit{P. S. Mostert}, Elements of compact semigroups (1966; Zbl 0161.019); the reviewer, The duality of compact semigroups and \(C^*\)-bigebras (Lect. Notes Math. 129) (1970; Zbl 0211.433); \textit{J. H. Carruth, J. A. Hildebrant}, and \textit{R. J. Koch}, The theory of topological semigroups, Volumes I and II (1983; Zbl 0515.22003) and (1986; Zbl 0581.22001)]. This does not even count books in which a substantial portion is devoted to special compact topological semigroups such as compact semilattices [\textit{G. Gierz}, the reviewer, \textit{K. Keimel, J. D. Lawson, M. Mislove}, and \tex…
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 57S99
Mathematics Subject Classification ID: 43A60
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 3979388
zbMATH Keywords: compact manifold semigroups
zbMATH Keywords: topological lattices
zbMATH Keywords: continuous lattices
zbMATH Keywords: compact topological semigroups
zbMATH Keywords: semitopological semigroups
zbMATH Keywords: almost periodic functions
zbMATH Keywords: harmonic analysis
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: right topological semigroups
zbMATH Keywords: cohomology
zbMATH Keywords: homotopy
zbMATH Keywords: transformation groups
zbMATH Keywords: history
zbMATH Keywords: bibliographical references
zbMATH Keywords: joint continuity
zbMATH Keywords: minimal ideals
zbMATH Keywords: separate continuity
zbMATH Keywords: isotropy groups
zbMATH Keywords: weak almost periodic compactification
zbMATH Keywords: semitopological semigroups on compact manifolds
zbMATH Keywords: peripherality methods
zbMATH Keywords: circle group
zbMATH Keywords: unsolved problems
author: Wolfgang A. F. Ruppert
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0073675
OpenAlex ID: W1540942463
DOI: 10.1007/BFB0073675
description: scientific article; zbMATH DE number 3979388