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Directed graphs of inner translations of semigroups

Semigroup Forum volume 94, pages 650–673 (2017)Cite this article

Abstract

A mapping \(\alpha :S\rightarrow S\) is called a Cayley function if there exist an associative operation \(\mu :S\times S\rightarrow S\) and an element \(a\in S\) such that \(\alpha (x)=\mu (a,x)\) for every \(x\in S\). The aim of the paper is to give a characterization of Cayley functions in terms of their directed graphs. This characterization is used to determine which elements of the centralizer of a permutation on a finite set are Cayley functions. The paper ends with a number of problems.

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Acknowledgments

We thank the referee for an excellent report on the paper. The first author was supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013, and through project “Hilbert’s 24th problem” PTDC/MHC-FIL/2583/2014. The second author has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PCOFUND-GA-2009-246542 and from the Foundation for Science and Technology of Portugal under PCOFUND-GA-2009-246542 and SFRH/BCC/52684/2014, and acknowledges that this work was developed within FCT projects CAUL (PEst-OE/MAT/UI0143/2014) and CEMAT-CIÊNCIAS (UID/Multi/04621/2013). The third author was supported by a 2013–14 University of Mary Washington Faculty Research Grant.

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Authors and Affiliations

  1. Departamento de Ciências Exatas e Tecnológicas, Universidade Aberta, R. Escola Politécnica, 147, 1269-001, Lisboa, Portugal

    João Araújo

  2. CEMAT-Ciências, Universidade de Lisboa, Campo Grande 4, 1649-003, Lisboa, Portugal

    João Araújo

  3. School of Mathematics and Physical Sciences, University of Hull, Kingston upon Hull, HU6 7RX, UK

    Wolfram Bentz

  4. Department of Mathematics, University of Mary Washington, Fredericksburg, VA, 22401, USA

    Janusz Konieczny

Authors
  1. João Araújo
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  2. Wolfram Bentz
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  3. Janusz Konieczny
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Corresponding author

Correspondence to João Araújo.

Additional information

Communicated by Mikhail Volkov.

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Araújo, J., Bentz, W. & Konieczny, J. Directed graphs of inner translations of semigroups. Semigroup Forum 94, 650–673 (2017). https://doi.org/10.1007/s00233-016-9821-x

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  • DOI: https://doi.org/10.1007/s00233-016-9821-x

Keywords

  • Inner translations
  • Cayley functions
  • Functional digraphs