Generating self-map monoids of infinite sets

Abstract

Let I be a countably infinite set, S = Sym(I) the group of permutations of I, and E = End(I) the monoid of self-maps of I. Given two subgroups G, G' of S, let us write G \approx_S G' if there exists a finite subset U of S such that the groups generated by G \cup U and G' \cup U are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to \approx_S. Letting \approx denote the obvious analog of \approx_S for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups G, G' of S which are closed in the function topology on S, we have G \approx_S G' if and only if G \approx G' (as submonoids of E), and that cl_S (G) \approx cl_E (G) for every subgroup G of S (where cl_S (G) denotes the closure of G in the function topology in S and cl_E (G) its closure in the function topology in E).