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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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.
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
- Inner translations
- Cayley functions
- Functional digraphs