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Partial automorphisms and injective partial endomorphisms of a finite undirected path

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Abstract

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids \(\mathrm {IEnd}(P_n)\) and \(\mathrm {PAut}(P_n)\) of all injective partial endomorphisms and of all partial automorphisms of the undirected path \(P_n\) with n vertices. We also describe Green’s relations of \(\mathrm {PAut}(P_n)\) and \(\mathrm {IEnd}(P_n)\) and calculate their cardinals.

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Acknowledgements

This work was produced, in part, during the visit of the first and third authors to CMA, FCT NOVA, Lisbon, in July 2019. The first author was supported by CMA through a visiting researcher fellowship.

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Author notes

  1. T. M. Quinteiro

    Present address: CMA, Nova School of Science and Technology, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica, Portugal

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    J. Koppitz

  2. Department of Mathematics, Faculty of Mathematics and Natural Science, South-West University “Neofit Rilski”, 2700, Blagoevgrad, Bulgaria

    I. Dimitrova

  3. CMA & Departamento de Matemática, Nova School of Science and Technology, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica, Portugal

    V. H. Fernandes

  4. Instituto Superior de Engenharia de Lisboa, 1950-062, Lisboa, Portugal

    T. M. Quinteiro

Authors
  1. I. Dimitrova
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  2. V. H. Fernandes
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  3. J. Koppitz
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  4. T. M. Quinteiro
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Correspondence to J. Koppitz.

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Communicated by László Márki.

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This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the project UIDB/00297/2020 (Center for Mathematics and Applications).

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Dimitrova, I., Fernandes, V.H., Koppitz, J. et al. Partial automorphisms and injective partial endomorphisms of a finite undirected path. Semigroup Forum 103, 87–105 (2021). https://doi.org/10.1007/s00233-021-10193-y

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  • DOI: https://doi.org/10.1007/s00233-021-10193-y

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