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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: cmc@st-andrews.ac.ukedmund@dcs.st-andrews.ac.uknik@dcs.st-andrews.ac.uk
E. F. Robertson
Affiliation:
Department of Mathematics and Computer Science University of LeicesterLeicester LE1 7RHEngland e-mail: rmt@mcs.le.ac.uk
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Abstract

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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

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