Direct products of automatic semigroups

It is known that the direct product of two automatic groups is automatic. The notion of automaticity has been extended to semigroups, and this result for groups has been generalized to automatic monoids. However, the direct product of two automatic semigroups need not be finitely generated and hence not automatic.

Robertson, RuSkuc and Wiegold have determined necessary and sufficient conditions for the direct product of two finitely generated semigroups to be finitely generated. Building on this, we prove the following. Let S and T be automatic semigroups; if S and T are infinite, then S x T is automatic if and only if S-2 = S and T-2 = T; if S is finite and T is infinite, then S x T is automatic if and only if S-2 = S. As a consequence, we have that, if S and T are automatic semigroups, then S x T is automatic if and only if S x T is finitely generated.