Direct products of automatic semigroups

Abstract

AbstractIt is known that the direct product of two automatic groups is automatic. The notion of automaticity bas been extended to semigroups, and this 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, Ruškuc 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 × T is automatic if and only if S2 = S and T2 = T; if S is finite and T is infinite, then S × T is automatic if and only if S2 = S. As a consequence, we have that, if S and T are automatic semigroups, then S × T is automatic if and only if S × T is finitely generated.