The Bergman-Shelah preorder on transformation semigroups



Let equation image be the semigroup of all mappings on the natural numbers equation image, and let U and V be subsets of equation image. We write UV if there exists a countable subset C of equation image 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 equation image. The results in this paper suggest that the preorder on subsemigroups of equation image is much more complicated than that on subgroups of the symmetric group.