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On semigroups of endomorphisms of a chain with restricted range

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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Correspondence to Vítor H. Fernandes.

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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.

Communicated by Jean-Eric Pin.

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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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Keywords

  • Transformations
  • Order-preserving
  • Restricted range
  • Rank