Let be the semigroup of all mappings on the natural numbers , and let U and V be subsets of . We write U≼V if there exists a countable subset C of such that U is contained in the subsemigroup generated by V and C. We give several results about the structure of the preorder ≼. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis.
The preorder ≼ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on . The results in this paper suggest that the preorder on subsemigroups of is much more complicated than that on subgroups of the symmetric group.