Generating countable sets of surjective functions
Volume 213 / 2011
Fundamenta Mathematicae 213 (2011), 67-93 MSC: Primary 20M20; Secondary 03E05. DOI: 10.4064/fm213-1-4
We prove that any countable set of surjective functions on an infinite set of cardinality $\aleph_n$ with $n\in\mathbb N$ can be generated by at most $n^2/2+9n/2+7$ surjective functions of the same set; and there exist $n^2/2+9n/2+7$ surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups such as the injective functions, the Baer–Levi semigroups, and the Schützenberger monoids.