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Subsemigroup, ideal and congruence growth of free semigroups

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Abstract

The growth of cofinite subsemigroups of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type n n for finite free rank greater than 1. Ideal growth is shown to be exponential with strict type 2n and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequence fits a polynomial of degree the index, whereas for congruences this fits an exponential equation of base the index. We use these results to describe an algorithm for computing values of these sequences and give a table of results for low rank and index.

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References

  1. http://alexanderbailey.github.io/software.

  2. The IRIDIS High Performance Computing Facility, (https://www.southampton.ac.uk/isolutions/staff/iridis.page).

  3. The On-Line Encyclopedia of Integer Sequences, published electronically at (http://oeis.org).

  4. J. Almeida, Profinite semigroups and applications, in Structural theory of automata, semigroups, and universal algebra, NATO Sci. Ser. II Math. Phys. Chem., Vol. 207, Springer, Dordrecht, 2005, Notes taken by Alfredo Costa, pp. 1–45.

    Chapter  Google Scholar 

  5. V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc. 125 (1997), x+78.

    MathSciNet  MATH  Google Scholar 

  6. V. Blanco, P. A. García-Sánchez and J. Puerto, Counting numerical semigroups with short generating functions, Internat. J. Algebra Comput. 21 (2011), 1217–1235.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Bras-Amorós, Fibonacci-like behavior of the number of numerical semigroups of a given genus, Semigroup Forum 76 (2008), 379–384.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bras-Amorós, Bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 213 (2009), 997–1001.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Bras-Amorós and S. Bulygin, Towards a better understanding of the semigroup tree, Semigroup Forum 79 (2009), 561–574.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. J. Cain and V. Maltcev, For a few elements more: A survey of finite rees index, arXiv:1307.8259.

  11. A. Distler and J. D. Mitchell., Smallsemi—a library of small semigroups, a gap 4 package [14], version 0.6.6, 2013, (http://tinyurl.com/jdmitchell/smallsemi/).

    Google Scholar 

  12. A. Distler and J. D. Mitchell, The number of nilpotent semigroups of degree 3, Electron. J. Combin. 19 (2012), Paper 51, 19.

    MathSciNet  MATH  Google Scholar 

  13. S. Elizalde, Improved bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 214 (2010), 1862–1873.

    Article  MathSciNet  MATH  Google Scholar 

  14. The GAP Group, Gap–groups, algorithms, and programming, version 4.6,5, 2013, (http://www.gap-system.org).

    Google Scholar 

  15. R. I. Grigorchuk, Semigroups with cancellations of degree growth, Mat. Zametki 43 (1988), 305–319, 428.

    MathSciNet  MATH  Google Scholar 

  16. R. Grigorchuk, Milnor’s problem on the growth of groups and its consequences, in Frontiers in complex dynamics, Princeton Math. Ser., Vol. 51, Princeton Univ. Press, Princeton, NJ, 2014, pp. 705–773.

    Google Scholar 

  17. M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), 53–73.

    Google Scholar 

  18. D. J. Kleitman, B. R. Rothschild and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227–232.

    MathSciNet  MATH  Google Scholar 

  19. A. Lubotzky, A. Mann and D. Segal, Finitely generated groups of polynomial subgroup growth, Israel J. Math. 82 (1993), 363–371.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Lubotzky and D. Segal, Subgroup growth, Progress in Mathematics, Vol. 212, Birkhäuser Verlag, Basel, 2003.

    Google Scholar 

  21. N. Ruškuc and R. M. Thomas, Syntactic and Rees indices of subsemigroups, J. Algebra 205 (1998), 435–450.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Voll, A newcomer’s guide to zeta functions of groups and rings, in Lectures on profinite topics in group theory, London Math. Soc. Stud. Texts, Vol. 77, Cambridge Univ. Press, Cambridge, 2011, pp. 99–144.

    Google Scholar 

  23. A. Zhai, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum 86 (2013), 634–662.

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Zhao, Constructing numerical semigroups of a given genus, Semigroup Forum 80 (2010), 242–254.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences, University of Southampton, Highfield, Southampton, UK, SO17 1BJ

    Alex Bailey & Robert Snocken

  2. Mathematische Institut, Georg-August-Universität Göttingen, Bunsenstrasse 3-5, 37073, Göttingen, Germany

    Martin Finn-Sell

Authors
  1. Alex Bailey
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  2. Martin Finn-Sell
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  3. Robert Snocken
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Correspondence to Alex Bailey.

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Bailey, A., Finn-Sell, M. & Snocken, R. Subsemigroup, ideal and congruence growth of free semigroups. Isr. J. Math. 215, 459–501 (2016). https://doi.org/10.1007/s11856-016-1384-8

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  • DOI: https://doi.org/10.1007/s11856-016-1384-8

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