Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T08:30:33.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct products of automatic semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

C. M. Campbell ,
E. F. Robertson ,
N. Ruškuc  and
R. M. Thomas
C. M. Campbell
Affiliation:
Mathematical Institute University of St AndrewsSt Andrews KY169SSScotland e-mail: [email protected]@[email protected]
E. F. Robertson
Affiliation:
Department of Mathematics and Computer Science University of LeicesterLeicester LE1 7RHEngland e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It 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.

MSC classification

Secondary: 20M05: Free semigroups, generators and relations, word problems 20M35: Semigroups in automata theory, linguistics, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 1 , August 2000 , pp. 19 - 24
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Baumslag, G., Gersten, S. M., Shapiro, M. and Short, H., ‘Automatics groups and amalgams’, J. Pure Appl. Algebra 76 (1991), 229–316.CrossRefGoogle Scholar
[2]Campbell, C. M., Robertson, E. F., Ruškuc, N. and Thomas, R. M., ‘Automatic semigroups’, Theoret Comput. Sci., to appear.Google Scholar
[3]Cannon, J. W., Epstein, D. B. A., Holt, D. F., Levy, S. V. F., Peterson, M. S. and Thurston, W. P., Word processing in groups (Jones and Barlett, Boston, 1992).Google Scholar
[4]Duncan, A. J., Roberston, E. F. and Ruškuc, N., ‘Automatic monoids and change of generators’, Math. Proc. Cambridge Philos. Soc. 127 (1999), 403–409.CrossRefGoogle Scholar
[5]Hoperoft, J. E. and Ullman, J. D., Introduction to automata theory, languages, and computation (Addison-Wesley, Reading, MA, 1979).Google Scholar
[6]Robertson, E. F., Ruškuc, N. and Wiegold, J., ‘Generators and relations of direct products of semigroups’, Trans. Amer. Math. Soc. 350 (1998), 2665–2685.CrossRefGoogle Scholar