Semigroups and Families

Abstract

In this chapter we consider a uniform action φ: T × X → X with X compact, and compare the semigroup and family viewpoints. Recall that by Lemma 1.2 any topological action of a uniform monoid on a compact space is a uniform action. The focus of our comparison is the uniform Stone-Ctech compactification of the uniform monoid T. In itself s u T combines three different phenomena. First T acts uniformly on the compact space s u T, which is the orbit closure of j u (0) in s u T. Furthermore using the maps Φ x , we see that this action is the universal compact T action ambit [see (6.18)]. Next s u T is an Ellis semigroup mapping onto the enveloping semigroup, S φ , by the homomorphism Φ# [see (6.13)]. Finally we recall that s u T can be regarded as the space of maximal open filters on T as in Theorem 5.2. This connects s u T with all of the family constructions in Chapters 3 and 4.