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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generators and relations
of direct products of semigroups


Authors: E. F. Robertson, N. Ruskuc and J. Wiegold
Journal: Trans. Amer. Math. Soc. 350 (1998), 2665-2685
MSC (1991): Primary 20M05
MathSciNet review: 1451614
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Abstract: The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup $S$ of order 11 such that $S\times T$ is finitely generated but not finitely presented for every finitely generated infinite semigroup $T$. By way of contrast we show that, if $S$ and $T$ belong to a wide class of semigroups, then $S\times T$ is finitely presented if and only if both $S$ and $T$ are finitely presented, exactly as in the case of groups and monoids.


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Additional Information

E. F. Robertson
Affiliation: Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland
Email: efr@st-and.ac.uk

N. Ruskuc
Affiliation: Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland
Email: nr1@st-and.ac.uk

J. Wiegold
Affiliation: School of Mathematics, University of Wales, College of Cardiff, Senghenydd Road, Cardiff, CF2 4AG, Wales
Email: SMAJW@cardiff.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02074-1
PII: S 0002-9947(98)02074-1
Received by editor(s): September 24, 1996
Article copyright: © Copyright 1998 American Mathematical Society