Generating self-map monoids of infinite sets
Zachary Mesyan · 2012
Let Ω be a countably infinite set, S = Sym(Ω) the group of permutations of Ω, and E = End(Ω) the monoid of self-maps of Ω. Given two subgroups G1,G2 ⊆ S, let us write G1 ≈S G2 if there exists a finite subset U ⊆ S such that the groups generated by G1 ∪ U and G2 ∪ 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 ≈S. Letting ≈ denote the obvious analog of ≈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 S ≈ E, that given two subgroups G1,G2 ⊆ S which are closed in the function topology on S, we have G1 ≈S G2 if and only if G1 ≈ G2 (as submonoids of E), and that clS(G) ≈ clE(G) for every subgroup G ⊆ S (where clS(G) denotes the closure of G in the function topology in S and clE(G) its closure in the function topology in E). 1