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Partial transformation monoids preserving a uniform partition

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Abstract

The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.

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References

  1. Aĭzenštat, AJa: Defining relations of finite symmetric semigroups. Mat. Sb. N.S. 45(87), 261–280 (1958)

    MathSciNet  Google Scholar 

  2. Araújo, I.M.: Presentations for semigroup constructions and related computational methods. PhD thesis, University of St Andrews (2001)

  3. Araújo, J., Schneider, Cs: The rank of the endomorphism monoid of a uniform partition. Semigroup Forum 78(3), 498–510 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. East, J.: A presentation of the singular part of the symmetric inverse monoid. Commun. Algebra 34(5), 1671–1689 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. East, J.: A presentation for the singular part of the full transformation semigroup. Semigroup Forum 81(2), 357–379 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. East, J.: Presentations for singular subsemigroups of the partial transformation semigroup. Int. J. Algebra Comput. 20(1), 1–25 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. East, J.: On the singular part of the partition monoid. Int. J. Algebra Comput. 21(1–2), 147–178 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Fernandes, V.H.: Presentations for some monoids of partial transformations on a finite chain: a survey. In: Gomes, G.M.S., Pin, J.-É., Silva, P. V. (eds.). Semigroups. Algorithms, Automata and Languages (Coimbra, 2001), pp. 363–378. World Sci. Publ, River Edge, NJ (2002)

  11. Fernandes, V.H., Quinteiro, T.M.: On the monoids of transformations that preserve the order and a uniform partition. Commun. Algebra 39(8), 2798–2815 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fernandes, V.H., Quinteiro, T.M.: The cardinal of various monoids of transformations that preserve a uniform partition. Bull. Malays. Math. Sci. Soc. (2) 35(4), 885–896 (2012)

    MATH  MathSciNet  Google Scholar 

  13. Fernandes, V.H., Quinteiro, T.M.: On the ranks of certain monoids of transformations that preserve a uniform partition. Commun. Algebra 42(2), 615–636 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups, An introduction, volume 9 of Algebra and Applications. Springer London Ltd, London (2009)

    Google Scholar 

  15. Huisheng, P., Dingyu, Z.: Green’s equivalences on semigroups of transformations preserving order and an equivalence relation. Semigroup Forum 71(2), 241–251 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Huisheng, P.: On the rank of the semigroup \(T_E(X)\). Semigroup Forum 70(1), 107–117 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lavers, T.G.: Presentations of general products of monoids. J. Algebra 204(2), 733–741 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Moore, E.H.: Concerning the abstract groups of order \(k!\) and \(\frac{1}{2} k!\) holohedrically isomorphic with the symmetric and the alternating substitution-groups on \(k\) letters. Proc. Lond. Math. Soc. 28(1), 357–367 (1896)

    Article  Google Scholar 

  19. Popova, L.M.: Defining relations of a semigroup of partial endomorphisms of a finite linearly ordered set. Leningrad. Gos. Ped. Inst. Učen. Zap. 238, 78–88 (1962)

    MathSciNet  Google Scholar 

  20. Rhodes, J., Steinberg, B.: The \(q\)-theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, New York (2009)

    Google Scholar 

  21. Ruškuc, N.: Semigroup presentations. PhD thesis, University of St Andrews (1995)

  22. Sun, L., Pei, H., Cheng, Z.: Regularity and Green’s relations for semigroups of transformations preserving orientation and an equivalence. Semigroup Forum 74(3), 473–486 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

Cicalò gratefully acknowledges support of FCT and PIDDAC, within the Project PTDC/MAT/69514/2006 of CAUL, and support of the Regione Sardegna, within the Project Master & Back, PR-MAB-A2009-837. She also thanks the Department of Mathematics of the University of Trento for its hospitality from 2011. Fernandes gratefully acknowledges support of FCT and PIDDAC, within the Projects ISFL-1-143 and PTDC/MAT/69514/2006 of CAUL. Schneider was supported by the research funds PTDC/MAT/101993/2008 (FCT), 302660/2013-5 (CNPq, Produtividade em Pesquisa), 475399/2013-7 (CNPq, Universal), and by APQ-00452-13 (Fapemig, Universal).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124, Cagliari, Italy

    Serena Cicalò

  2. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica, Portugal

    Vítor H. Fernandes

  3. Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal

    Vítor H. Fernandes

  4. Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG, 30123-970, Brazil

    Csaba Schneider

Authors
  1. Serena Cicalò
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  2. Vítor H. Fernandes
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  3. Csaba Schneider
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Corresponding author

Correspondence to Csaba Schneider.

Additional information

Communicated by Mikhail Volkov.

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Cicalò, S., Fernandes, V.H. & Schneider, C. Partial transformation monoids preserving a uniform partition. Semigroup Forum 90, 532–544 (2015). https://doi.org/10.1007/s00233-014-9629-5

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  • DOI: https://doi.org/10.1007/s00233-014-9629-5

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