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On semigroups of endomorphisms of a chain with restricted range

Semigroup Forum volume 89, pages 77–104 (2014)Cite this article

Abstract

Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\). In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\). Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.

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Acknowledgements

This research was mainly carried out during the visit of the second and fourth authors to Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa and Centro de Álgebra da Universidade de Lisboa between August and October 2011.

The authors would like to thank Cláudia and Francisco Coelho for their help in reviewing the text of this paper.

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Authors and Affiliations

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

    Vítor H. Fernandes

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

    Vítor H. Fernandes & Teresa M. Quinteiro

  3. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Preeyanuch Honyam

  4. Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, 1950-062, Lisboa, Portugal

    Teresa M. Quinteiro

  5. School of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University, Chiang Mai, 50300, Thailand

    Boorapa Singha

Authors
  1. Vítor H. Fernandes
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  2. Preeyanuch Honyam
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  3. Teresa M. Quinteiro
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  4. Boorapa Singha
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Corresponding author

Correspondence to Vítor H. Fernandes.

Additional information

Communicated by Jean-Eric Pin.

Dedicated to the memory of John M. Howie.

Vítor H. Fernandes work was developed within the research activities of Centro de Álgebra da Universidade de Lisboa, FCT´s project PEst-OE/MAT/UI0143/2013, and of Departamento de Matemática da Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa.

Teresa M. Quinteiro work was developed within the research activities of Centro de Álgebra da Universidade de Lisboa, FCT´s project PEst-OE/MAT/UI0143/2013, and of Instituto Superior de Engenharia de Lisboa.

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Fernandes, V.H., Honyam, P., Quinteiro, T.M. et al. On semigroups of endomorphisms of a chain with restricted range. Semigroup Forum 89, 77–104 (2014). https://doi.org/10.1007/s00233-013-9548-x

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  • DOI: https://doi.org/10.1007/s00233-013-9548-x

Keywords

  • Transformations
  • Order-preserving
  • Restricted range
  • Rank