Semigroups with commuting threads
type: Item
instance of: scholarly article
title: Semigroups with commuting threads
zbMATH Open document ID: 0672.22007
DOI: 10.1007/BF02573298
author: S. H. Smith
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134987
review text: Let \(\hat S\) denote a compact topological semigroup satisfying the following two conditions: (1) (\(\forall x\in \hat S\), \(n\in {\mathbb{N}})\) (\(\exists !y\in \hat S)\) \(y^ n=x\) (unique divisibility), and (2) if A and B denote one-parameter semigroups in \(\hat S\) then \(AB=BA\) (commuting thread property). Then there is a zero 0 and \(S=\hat S\setminus \{0\}\) is (isomorphic to) a closed subsemigroup of a Lie group G. Moreover, G is almost abelian, and S is a Lie semialgebra. Specifically, a Lie algebra \({\mathfrak g}\) is almost abelian if every proper vector subspace is a subalgebra iff \({\mathfrak g}\) is either abelian or the split extension of some \({\mathbb{R}}^ n\) by R acting via scalar multiplication. A connected Lie group is almost abelian if its Lie algebra is almost abelian. These Lie groups are exponential. A Lie semialgebra in an almost abelian Lie group G is the diffeomorphic image, under exp, of a closed convex cone in \({\mathfrak g}\). (For background on the Lie theory of semigroups, see \textit{J. Hilgert}, \textit{K. H. Hofmann}, and \textit{J. D. Lawson}, Lie groups, convex cones, and semigroups [Oxford Univ. Press, 1989, \(xxxiii+646 pp.]\). Preprints of the paper under review circulated in 1981.)
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 4099573
zbMATH Keywords: compact topological semigroup
zbMATH Keywords: unique divisibility
zbMATH Keywords: one-parameter semigroups
zbMATH Keywords: commuting thread
zbMATH Keywords: Lie semialgebra
zbMATH Keywords: almost abelian
zbMATH Keywords: connected Lie group
zbMATH Keywords: almost abelian Lie group
MaRDI profile type: Publication
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OpenAlex ID: W2055793163
description: scientific article; zbMATH DE number 4099573