Generating countable sets of surjective functions

We prove that any countable set of surjective functions on an infinite set of cardinality ℵn with n∈N can be generated by at most n2/2+9n/2+7 surjective functions of the same set; and there exist n2/2+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.