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Centralizers in the Full Transformation Semigroup

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Abstract

For an arbitrary set X (finite or infinite), denote by T(X) the semigroup of full transformations on X. For α∈T(X), let C(α)={β∈T(X):αβ=βα} be the centralizer of α in T(X). The aim of this paper is to characterize the elements of C(α). The characterization is obtained by decomposing α as a join of connected partial transformations on X and analyzing the homomorphisms of the directed graphs representing the connected transformations. The paper closes with a number of open problems and suggestions of future investigations.

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Acknowledgements

The first author was partially supported by FCT through the following projects: PEst-OE/MAT/UI1043/2011, Strategic Project of Centro de Álgebra da Universidade de Lisboa; and PTDC/MAT/101993/2008, Project Computations in groups and semigroups.

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Authors and Affiliations

  1. Universidade Aberta, R. Escola Politécnica, 147, 1269-001, Lisboa, Portugal

    João Araújo

  2. Centro de Álgebra, Universidade de Lisboa, 1649-003, Lisboa, Portugal

    João Araújo

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

    Janusz Konieczny

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  1. João Araújo
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  2. Janusz Konieczny
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Correspondence to João Araújo.

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Communicated by Mikhail Volkov.

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Araújo, J., Konieczny, J. Centralizers in the Full Transformation Semigroup. Semigroup Forum 86, 1–31 (2013). https://doi.org/10.1007/s00233-012-9424-0

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