Free local semigroup constructions
type: Item
instance of: scholarly article
title: Free local semigroup constructions
zbMATH Open document ID: 0855.22002
DOI: 10.1007/BF01308723
published in: Monatshefte für Mathematik
publication date: 1997-01-27T00:00:00Z
full work available at URL: https://eudml.org/doc/178728
review text: Let \(G\) be a topological group and \(A\subseteq G\) a local subsemigroup such that the identity element \(e\) is contained in \(\overline {\text{int } A}\). A path \(\gamma :[0, T]\to G\) with \(\gamma (0) =e\) is called causal (with respect to \(A\)) if for any \(t\in [0, T]\) there exists a neighborhood \(V\) of \(t\) such that \(t_1> t_2\), \(t_1, t_2\in V\), implies that \(\gamma (t_2)\in \gamma (t_1) A\). The set of causal paths forms a monoid with respect to the natural concatenation product. One also has a corresponding concept of a causal homotopy defining a congruence relation on the set of causal paths and hence a monoid structure on the set \(\Gamma (A)\) of causal homotopy classes. One of the main results of the paper is that continuous local homomorphisms \(A\to T\), where \(T\) is a topological semigroup, extend to semigroup homomorphisms \(\Gamma (A)\to T\). Furthermore it is shown that for the important class of Ol'shanskij subsemigroups in Lie groups the semigroup \(\Gamma (S)\) can be identified with the universal covering semigroup \(\widetilde {S}\). As an important consequence one obtains that in this case a continuous local homomorphism of an identity neighborhood \(U\) in \(S\) into a topological semigroup \(T\) always lifts to global homomorphism of the universal covering semigroup \(\widetilde {S} \to T\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 55T99
zbMATH DE Number: 889762
zbMATH Keywords: Lie semigroup
zbMATH Keywords: Ol'shanskij semigroup
zbMATH Keywords: topological group
zbMATH Keywords: causal paths
zbMATH Keywords: monoid
zbMATH Keywords: causal homotopy
zbMATH Keywords: topological semigroup
zbMATH Keywords: Ol'shanskij subsemigroups
zbMATH Keywords: universal covering semigroup
author: Jimmie D. Lawson
reviewed by: Karl-Hermann Neeb
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 889762