The translational hull of a topological semigroup

Abstract

This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull Ω ( S ) \Omega (S) is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact-open topology in this case), then Ω ( S ) \Omega (S) is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section.