Free local semigroup constructions
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 \(\widetilde {S} \to T\).
scientific article; zbMATH DE number 889762
scientific article