Automatic semigroups

The area of automatic groups has been one in which significant advances hav e been made in recent years. While it is clear that the definition of an au tomatic group can easily be extended to that of an automatic semigroup, the re does not seem to have been a systematic investigation of such structures . It is the purpose of this paper to make such a study. We show that certain results from the group-theoretic situation hold in thi s wider context, such as the solvability of the word problem in quadratic t ime, although others do not, such as finite presentability. There are also situations which arise in the general theory of semigroups which do not occ ur when considering groups; for example, we show that a semigroup S is auto matic if and only if S with a zero adjoined is automatic, and also that S i s automatic if and only if S with an identity adjoined is automatic. We use this last result to show that any finitely generated subsemigroup of a fre e semigroup is automatic. (C) 2001 Elsevier Science B.V. All rights reserve d.