We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions.(i) Is every countable subset F of S also a subset of a finitely generated subsemigroup of S? If so, what is the least number n such that for every countablesubset F of S there exist n elements of S that generate a subsemigroup of Scontaining F as a subset.(ii) Given a subset U of S, what is the least cardinality of a subset A of S suchthat the union of A and U is a generating set for S?(iii) Define a preorder relation ≤ on the subsets of S as follows. For subsets V andW of S write V ≤ W if there exists a countable subset C of S such that Vis contained in the semigroup generated by the union of W and C. Given asubset U of S, where does U lie in the preorder ≤ on subsets of S?Semigroups S for which we answer question (i) include: the semigroups of the injec-tive functions and the surjective functions on a countably infinite set; the semigroupsof the increasing functions, the Lebesgue measurable functions, and the differentiablefunctions on the closed unit interval [0, 1]; and the endomorphism semigroup of therandom graph. We investigate questions (ii) and (iii) in the case where S is the semigroup Ω[superscript Ω] of all functions on a countably infinite set Ω. Subsets U of Ω[superscript Ω] under considerationare semigroups of Lipschitz functions on Ω with respect to discrete metrics on Ω andsemigroups of endomorphisms of binary relations on Ω such as graphs or preorders.
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