  The rank of a finite semigroup is the smallest number of elements required to
generate the semigroup. A formula is given for the rank of an arbitrary (non
necessarily regular) Rees matrix semigroup over a group. The formula is
expressed in terms of the dimensions of the structure matrix, and the relative
rank of a certain subset of the structure group obtained from subgroups
generated by entries in the structure matrix, which is assumed to be in Graham
normal form. This formula is then applied to answer questions about minimal
generating sets of certain natural families of transformation semigroups. In
particular, the problem of determining the maximum rank of a subsemigroup of
the full transformation monoid (and of the symmetric inverse semigroup) is
considered.
