More on cancellative semigroups on manifolds
The paper deals with cancellative topological semigroups which, as topological spaces, are connected, paracompact, finite dimensional and locally Euclidean. \textit{D. R. Brown} and \textit{R. S. Houston} [Semigroup Forum 35, 279-302 (1987; Zbl 0626.22001)] proved that such a semigroup S can be locally embedded into a simply connected Lie group G(S). This embedding is achieved by constructing local groups of one sided quotients. In the same paper Brown and Houston use the explicit construction of this embedding to provide the semigroup with an analytic structure and give some functorial properties of the semigroups in question. The present paper shows how to obtain the analytic structure of the semigroup and various functorial properties using only the topological and algebraic properties of the Brown-Houston local embedding. The main step is to equip the direct product \(S\times G(S)\) with a topology for which both projections, \(S\times G(S)\to S\) and \(S\times G(S)\to G(S)\) are Hausdorff sheaves of topological spaces. From these sheaves various invariants are constructed which, in the future, may be helpful in deciding whether S can be embedded globally into G(S).
scientific article; zbMATH DE number 4034020
scientific article