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Infinite dual symmetric inverse monoids

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Abstract

Let \({\mathcal {I}}^*_X\) be the dual symmetric inverse monoid on an infinite set X. We show that \({\mathcal {I}}^*_X\) may be generated by the symmetric group \({\mathcal {S}}_X\) together with two (but no fewer) additional block bijections, and we classify the pairs \(\alpha ,\beta \in {\mathcal {I}}^*_X\) for which \({\mathcal {I}}^*_X\) is generated by \({\mathcal {S}}_X\,\cup \,\{\alpha ,\beta \}\). Among other results, we show that any countable subset of \({\mathcal {I}}^*_X\) is contained in a 4-generated subsemigroup of \({\mathcal {I}}^*_X\), and that the length function on \({\mathcal {I}}^*_X\) (and its finitary power semigroup) is bounded with respect to any generating set.

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References

  1. J.M. André, V.H. Fernandes, J.D. Mitchell, Largest 2-generated subsemigroups of the symmetric inverse semigroup. Proc. Edinb. Math. Soc. (2) 50(3), 551–561 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Banach, Sur un Thèoréme de M. Sierpiński. Fund. Math. 25, 5–6 (1935)

    Article  MATH  Google Scholar 

  3. G.M. Bergman, Generating infinite symmetric groups. Bull. Lond. Math. Soc. 38, 429–440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Easdown, J. East, D.G. FitzGerald, A presentation of the dual symmetric inverse monoid. Int. J. Algebra Comput. 18(2), 357–374 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. East, The factorizable braid monoid. Proc. Edinb. Math. Soc. (2) 49(3), 609–636 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. East, A presentation of the singular part of the symmetric inverse monoid. Commun. Algebra 34, 1671–1689 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. East, Braids and partial permutations. Adv. Math. 213(1), 440–461 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. East, Generators and relations for partition monoids and algebras. J. Algebra 339, 1–26 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. East, Generation of infinite factorizable inverse monoids. Semigroup Forum 84(2), 267–283 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. East, Infinite partition monoids. Int. J. Algebra Comput. 24(4), 429–460 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. East, D.G. FitzGerald, The semigroup generated by the idempotents of a partition monoid. J. Algebra 372, 108–133 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. East, J. Mitchell, Y. Péresse, Maximal subsemigroups of the semigroup of all mappings on an infinite set. Trans. Am. Math. Soc. 367(3), 1911–1944 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. V.H. Fernandes, The monoid of all injective order preserving partial transformations on a finite chain. Semigroup Forum 62(2), 178–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. D.G. FitzGerald, A presentation for the monoid of uniform block permutations. Bull. Aust. Math. Soc. 68, 317–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. D.G. FitzGerald, J. Leech, Dual symmetric inverse semigroups and representation theory. J. Aust. Math. Soc. 64, 146–182 (1998)

    Article  MATH  Google Scholar 

  16. P. Gallagher, On the finite and non-finite generation of finitary power semigroups. Semigroup Forum 71(3), 481–494 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Gallagher, N. Ruškuc, Finitary power semigroups of infinite groups are not finitely generated. Bull. Lond. Math. Soc. 37(3), 386–390 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Galvin, Generating countable sets of permutations. J. Lond. Math. Soc. (2) 51(2), 230–242 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Higgins, J. Howie, J. Mitchell, N. Ruskuc, Countable versus uncountable rank in infinite semigroups of transformations and relations. Proc. Edinb. Math. Soc. 46(3), 531–544 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Higgins, J. Howie, N. Ruškuc, Generators and factorisations of transformation semigroups. Proc. R. Soc. Edinb. Sect. A 128, 1355–1369 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Howie, N. Ruškuc, P. Higgins, On relative ranks of full transformation semigroups. Commun. Algebra 26(3), 733–748 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Hyde, Personal Communication (2013)

  23. J. Hyde, Y. Péresse, Sierpiński Rank of the Symmetric Inverse Semigroup, Preprint (2012)

  24. T. Jech, Set Theory, The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics (Springer, Berlin, 2003)

  25. M.V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries (World Scientific Publishing Co., Inc, River Edge, 1998)

    Book  MATH  Google Scholar 

  26. S. Lipscombe, Symmetric Inverse Semigroups (American Mathematical Society, Providence, 1996)

    Google Scholar 

  27. V. Maltcev, On a new approach to the dual symmetric inverse monoid \(\cal{I}_X^*\). Int. J. Algebra Comput. 17(3), 567–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. V. Maltcev, J. Mitchell, N. Ruškuc, The Bergman property for semigroups. J. Lond. Math. Soc. (2) 80(1), 212–232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Mitchell, Y. Pèresse, Generating countable sets of surjective functions. Fund. Math. 213(1), 67–93 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Petrich, Inverse Semigroups. Pure and Applied Mathematics (Wiley, New York, 1984)

  31. L.M. Popova, Defining relations in some semigroups of partial transformations of a finite set. Uchenye Zap. Leningrad Gos. Ped. Inst. 218, 191–212 (1961). (in Russian)

    Google Scholar 

  32. W. Sierpiński, Sur les Suites Infinies de Fonctions Définies dans les Ensembles Quelconques. Fund. Math. 24, 209–212 (1935)

    Article  MATH  Google Scholar 

  33. T. Tamura, J. Shafer, Power semigroups. Math. Jpn. 12, 25–32 (1967)

    MathSciNet  MATH  Google Scholar 

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  1. Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith, NSW, 2751, Australia

    James East

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East, J. Infinite dual symmetric inverse monoids. Period Math Hung 75, 273–285 (2017). https://doi.org/10.1007/s10998-017-0194-z

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