Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T17:06:21.701Z Has data issue: false hasContentIssue false
  • English
  • Français

CANCELLATIVE AND MALCEV PRESENTATIONS FOR FINITE REES INDEX SUBSEMIGROUPS AND EXTENSIONS

Part of: Semigroups

Published online by Cambridge University Press:  01 February 2008

ALAN J. CAIN ,
EDMUND F. ROBERTSON  and
NIK RUŠKUC
ALAN J. CAIN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (email: alanc@mcs.st-and.ac.uk)
EDMUND F. ROBERTSON
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (email: edmund@mcs.st-and.ac.uk)
NIK RUŠKUC*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (email: nik@mcs.st-and.ac.uk)
*
✉For correspondence; e-mail: nik@mcs.st-and.ac.uk
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, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.)

Keywords

Maclev presentationcancellativesubsemigroupfinite indexrewriting

MSC classification

Secondary: 20M05: Free semigroups, generators and relations, word problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 39 - 61
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Adjan, S. I., ‘Defining relations and algorithmic problems for groups and semigroups’, Tr. Mat. Inst. Steklova 85 (1966) (in Russian. See [2] for a translation).Google Scholar
[2]Adjan, S. I., ‘Defining relations and algorithmic problems for groups and semigroups’, P. Steklov Inst. Math. 85 (1966) (translated from the Russian by M. Greendlinger).Google Scholar
[3]Araújo, I. M., Silva, P. V. and Ruškuc, N., ‘Presentations for inverse subsemigroups with finite complement’, submitted.Google Scholar
[4]Cain, A. J., ‘Presentations for subsemigroups of groups’, PhD Thesis, University of St Andrews, 2005. http://www-groups.mcs.st-and.ac.uk/∼alanc/publications/c_phdthesis/c_phdthesis.pdf.Google Scholar
[5]Cain, A. J., ‘Malcev presentations for subsemigroups of direct products of coherent groups’, J. Pure Appl. Algebra, to appear.Google Scholar
[6]Cain, A. J., ‘Automatism of subsemigroups of Baumslag–Solitar semigroups’, submitted.Google Scholar
[7]Cain, A. J., ‘Malcev presentations for subsemigroups of groups—a survey’, in: Groups St Andrews 2005, Vol. 1, London Mathematical Society Lecture Note Series, 339 (eds. C. M. Campbell, M. Quick, E. F. Robertson and G. C. Smith) (Cambridge University Press, Cambridge, 2007),pp. 256–268.CrossRefGoogle Scholar
[8]Cain, A. J., Robertson, E. F. and Ruškuc, N., ‘Subsemigroups of groups: presentations, Malcev presentations, and automatic structures’, J. Group Theory 9 (2006), 397–426.CrossRefGoogle Scholar
[9]Cain, A. J., Robertson, E. F. and Ruškuc, N., ‘Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness’, Math. Proc. Cambridge Philos. Soc. 141 (2006), 57–66.CrossRefGoogle Scholar
[10]Campbell, C. M., Robertson, E. F., Ruškuc, N. and Thomas, R. M., ‘Reidemeister–Schreier type rewriting for semigroups’, Semigroup Forum 51 (1995), 47–62.CrossRefGoogle Scholar
[11]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1961).Google Scholar
[12]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. II, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1967).Google Scholar
[13]Croisot, R., ‘Automorphismes intérieurs d’un semi-groupe’, Bull. Soc. Math. France 82 (1954), 161–194 (in French).CrossRefGoogle Scholar
[14]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P., Word processing in groups (Jones & Bartlett, Boston, MA, 1992).CrossRefGoogle Scholar
[15]Higgins, P. M., Techniques of semigroup theory (Clarendon Press, Oxford University Press, New York, 1992).CrossRefGoogle Scholar
[16]Howie, J. M., Fundamentals of semigroup theory, London Mathematical Society Monographs, 12 (New Series) (Clarendon Press, Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[17]Jura, A., ‘Determining ideals of a given finite index in a finitely presented semigroup’, Demonstratio Math. 11 (1978), 813–827.Google Scholar
[18]Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89 (Springer, Berlin, 1977).Google Scholar
[19]Malcev, A. I., ‘On the immersion of associative systems in groups’, Mat. Sbornik 6 (1939), 331–336 (in Russian).Google Scholar
[20]Ruškuc, N., ‘Semigroup presentations’, PhD Thesis, University of St Andrews, 1995. http://turnbull.mcs.st-and.ac.uk/∼nik/Papers/thesis.ps.Google Scholar
[21]Ruškuc, N., ‘On large subsemigroups and finiteness conditions of semigroups’, Proc. London Math. Soc. (3) 76 (1998), 383–405.CrossRefGoogle Scholar
[22]Spehner, J.-C., ‘Présentations et présentations simplifiables d’un monoïde simplifiable’, Semigroup Forum 14 (1977), 295–329 (in French).CrossRefGoogle Scholar
[23]Spehner, J.-C., ‘Every finitely generated submonoid of a free monoid has a finite Malcev’s presentation’, J. Pure Appl. Algebra 58 (1989), 279–287.CrossRefGoogle Scholar