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Presentations for monoids of finite partial isometries

Abstract

In this paper we give presentations for the monoid \({\mathcal {DP}}_n\) of all partial isometries on \(\{1,\ldots ,n\}\) and for its submonoid \({\mathcal {ODP}}_n\) of all order-preserving partial isometries.

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References

  1. 1.

    Aĭzenštat, A.Ya.: The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sibirsk. Mat. 3, 161–169 (1962). (Russian)

    MathSciNet  Google Scholar 

  2. 2.

    Al-Kharousi, F., Kehinde, R., Umar, A.: Combinatorial results for certain semigroups of partial isometries of a finite chain. Aust. J. Combin. 58, 365–375 (2014)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Al-Kharousi, F., Kehinde, R., Umar, A.: On the semigroup of partial isometries of a finite chain, Communications in Algebra. To appear

  4. 4.

    Cowan, D.F., Reilly, N.R.: Partial cross-sections of symmetric inverse semigroups. Int. J. Algebra Comput. 5, 259–287 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Derech, V.D.: On quasi-orders over certain inverse semigroups, Sov. Math. 35 (1991), 74–76; translation from. Izv. Vyssh. Uchebn. Zaved. Mat. 3(346), 76–78 (1991) (Russian)

  6. 6.

    Delgado, M., Fernandes, V.H.: Abelian kernels of some monoids of injective partial transformations and an application. Semigroup Forum 61, 435–452 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Delgado, M., Fernandes, V.H.: Abelian kernels of monoids of order-preserving maps and of some of its extensions. Semigroup Forum 68, 335–356 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: a new class of divisors. Semigroup Forum 54, 230–236 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fernandes, V.H.: Normally ordered inverse semigoups. Semigroup Forum 56, 418–433 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Fernandes, V.H.: Presentations for some monoids of partial transformations on a finite chain: a survey. In: Gomes, G.M.S., Pin, J.-E., Silva, P.V. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 363–378. World Scientific, Singapore (2002)

    Chapter  Google Scholar 

  12. 12.

    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: another class of divisors. Izv. Vyssh. Uchebn. Zaved. Mat. 3(478), 51–59 (2002). (Russian)

    Google Scholar 

  13. 13.

    Fernandes, V.H.: Normally ordered semigroups. Glasg. Math. J. 50, 325–333 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Fernandes, V.H., Gomes, G.M.S., Jesus, M.M.: Presentations for some monoids of injective partial transformations on a finite chain. Southeast Asian Bull. Math. 28, 903–918 (2004)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ganyushkin, O., Mazorchuk, V.: On the structure of \(IO_n\). Semigroup Forum 66, 455–483 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    GAP Group: GAP—Groups, Algorithms, and Programming, v. 4.7.5. http://www.gap-system.org (2014)

  17. 17.

    Gould, H.W.: Combinatorial Identities. Morgantown, West-Virginia (1972)

    MATH  Google Scholar 

  18. 18.

    Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45, 272–282 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Higgins, P.M.: Divisors of semigroups of order-preserving mappings on a finite chain. Int. J. Algebra Comput. 5, 725–742 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  21. 21.

    Howie, J.M.: Product of idempotents in certain semigroups of transformations. Proc. Edinb. Math. Soc. 17, 223–236 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Lallement, G.: Semigroups and Combinatorial Applications. Wiley, New York (1979)

    MATH  Google Scholar 

  23. 23.

    Laradji, A., Umar, A.: Combinatorial results for semigroups of order-preserving partial transformations. J. Algebra 278, 342–359 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Laradji, A., Umar, A.: Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72, 51–62 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    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). (Russian)

    MathSciNet  Google Scholar 

  26. 26.

    Ruškuc, N.: Semigroup Presentations, Ph.D. Thesis, University of St Andrews, St Andrews (1995)

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Acknowledgments

V. H. Fernandes: This work was developed within the FCT Project PEst-OE/MAT/UI0143/2014 of CAUL, FCUL, and of Departamento de Matemática da Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa. T. M. Quinteiro: This work was developed within the FCT Project PEst-OE/MAT/UI0143/2014 of CAUL, FCUL, and of Instituto Superior de Engenharia de Lisboa.

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Correspondence to Vítor H. Fernandes.

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Communicated by Jorge Almeida.

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Fernandes, V.H., Quinteiro, T.M. Presentations for monoids of finite partial isometries. Semigroup Forum 93, 97–110 (2016). https://doi.org/10.1007/s00233-015-9759-4

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Keywords

  • Presentations
  • Transformations
  • Order-preserving
  • Partial isometries