The minimal number of generators of a finite semigroup. (Q467521)

Known methods to compute the rank (minimal number of generators) of proper two-sided ideals of the full transformation monoid \(T_n\), rank of the symmetric inverse semigroup and rank of the partial transformation monoid are similar; here this similarity is used to introduce a general method to compute ranks of various types of finite semigroups: the rank of an arbitrary Rees matrix semigroup over a group, rank of a subsemigroup of \(T_n\) generated by mappings with prescribed kernels and images and the general problem: for a subset \(A\subseteq T_n\), what can be said about the rank of the semigroup generated by \(A\), e.g. (open problem 1) what is \(\max\{\text{rank}(S):S\leqslant T_n\}\)?