On automatic Rees matrix semigroups

We consider a Rees matrix semigroup S = M [ U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that p U-1 = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.