Automatic semigroups

• Published in: Theoretical computer science Vol. 250; no. 1; pp. 365 - 391
• Main Authors: Campbell, Colin M., Robertson, Edmund F., Ruškuc, Nikola, Thomas, Richard M.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2001; Elsevier
• Subjects: Automatic; Exact sciences and technology; Group; Group theory; Group theory and generalizations; Mathematics; Regular; Sciences and techniques of general use; Semigroup; Group; Semigroup; Regular; Automatic; Automatic group; Regularity; Group theory
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/S0304-3975(99)00151-6

The area of automatic groups has been one in which significant advances have been made in recent years. While it is clear that the definition of an automatic group can easily be extended to that of an automatic semigroup, there 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 this wider context, such as the solvability of the word problem in quadratic time, although others do not, such as finite presentability. There are also situations which arise in the general theory of semigroups which do not occur when considering groups; for example, we show that a semigroup S is automatic if and only if S with a zero adjoined is automatic, and also that S is 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 free semigroup is automatic.