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-Čtech compactification of the uniform monoid T. In itself ß _u T combines three different phenomena. First T acts uniformly on the compact space ß _u T, which is the orbit closure of j _u (0) in ß _u T. Furthermore using the maps Φ _x , we see that this action is the universal compact T action ambit [see (6.18)]. Next ß _u T is an Ellis semigroup mapping onto the enveloping semigroup, S _φ , by the homomorphism Φ^# [see (6.13)]. Finally we recall that ß _u T can be regarded as the space of maximal open filters on T as in Theorem 5.2. This connects ß _u T with all of the family constructions in Chapters 3 and 4.