J. Aust. Math. Soc. 83 (2007), no. 3, pp. 357–367.

On finite generation and presentability of Schützenberger products

Peter Gallagher Nik Ruškuc
School of Mathematics and Statistics
University of St Andrews
St Andrews
Scotland, UK
Peterg@mcs.st-and.ac.uk
School of Mathematics and Statistics
University of St Andrews
St Andrews
Scotland, UK
nik@mcs.st-and.ac.uk
Received 25 July 2005; revised 5 May 2006
Communicated by D. Easdown

Abstract

The finite generation and presentation of Schützenberger products of semigroups are investigated. A general necessary and sufficient condition is established for finite generation. The Schützenberger product of two groups is finitely presented as an inverse semigroup if and only if the groups are finitely presented, but is not finitely presented as a semigroup unless both groups are finite.

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2000 Mathematics Subject Classification: primary 20M05; secondary 20M18
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References

  1. G. M. S. Gomes, J-E. Pin and H. Sezinando, ‘Presentations of the Schützenberger product of n groups’, Comm. Algebra to appear. MR2220809
  2. R. Gray and N. Ruškuc, ‘Generators and relations for subsemigroups via boundaries in Cayley graphs’, submitted.
  3. J. M. Howie, Automata and languages (Clarendon Press, Oxford, 1991). MR1254435
  4. J. M. Howie, Fundamentals of Semigroup Theory, volume 12 of London Math. Soc. Monogr. Ser. (Clarendon Press, New York, 1995). MR1455373
  5. J. M. Howie and N. Ruškuc, ‘Constructions and presentations for monoids’, Comm. Algebra 22 (1994), 6209–6224. MR1302999
  6. S. W. Margolis and J-E. Pin, ‘Expansions, free inverse semigroups and Schützenberger product’, J. Algebra 110 (1987), 298–305. MR910385
  7. M. Petrich, Inverse semigroups (John Wiley and Sons Publication, 1984). MR752899
  8. E. F. Robertson, N. Ruškuc and J. Wiegold, ‘Generators and relations of direct products of semigroups’, Trans. Amer. Math. Soc. 350 (1998), 2665–2685. MR1451614
  9. N. Ruškuc, Semigroup presentations (Ph.D. Thesis, University of St Andrews, 1995).
  10. B. M. Schein, ‘Free inverse semigroups are not finitely presented’, Acta Math. Acad. Scient. Hung. 26 (1975), 41–52. MR360878
  11. M. P. Schützenberger, ‘On finite monoids having only trivial subgroups’, Information and Control 8 (1965), 190–194. MR176883
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