Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-29T03:34:56.762Z Has data issue: false hasContentIssue false
  • English
  • Français

An analogue for semigroups of a group problem of P. Erdös and B. H. Neumann

Published online by Cambridge University Press:  17 April 2009

Luise-Charlotte Kappe ,
John C. Lennox  and
James Wiegold
Luise-Charlotte Kappe
Affiliation:
Department of Math Sciences, Suny at Binghamton, Binghamton, NY 13902–6000, United States of America, e-mail: [email protected]
John C. Lennox
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, PO Box 926, Cardiff, CF24 4YH, Wales, United Kingdom, e-mail: [email protected]
James Wiegold
Affiliation:
Green College at the Radcliffe Observatory, Oxford OX2 6HG, United Kingdom, 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.

In response to a question posed by P. Erdös, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumann's result valid are cancellative semigroups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 1 , February 2001 , pp. 59 - 66
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bell, H., Klein, A. and Kappe, L. C., ‘An analogue for rings of a group problem of P. Erdös and B. H. Neumann’, Acta Math. Hunger. 77 (1997), 57–67.Google Scholar
[2]Brodie, M. A. and Kappe, L. C., ‘Finite coverings by subgroups with a given property’, Glasgow Math. J. 35 (1993), 179–188.CrossRefGoogle Scholar
[3]Clifford, A. H. and Preston, G. B, The algebraic theory of semigroups Vol. 2 (American Mathematical Society, Providence RI, 1967).Google Scholar
[4]Kim, P. S., Rhemtulla, A. and Smith, H., ‘A characterization of infinite metabelian groups’, Houston J. Math. 17 (1991), 429–437.Google Scholar
[5]Malcev, A., ‘On the immersion of an algebraic ring into a field’, Math. Ann. 113 (1937), 686–691.Google Scholar
[6]Neumann, B. H., ‘Some remarks on cancellative semigroups’, Math. Z. 117 (1970), 97–111.CrossRefGoogle Scholar
[7]Neumann, B. H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 21 (1976), 467–472.CrossRefGoogle Scholar
[8]Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Part I (Springer Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar