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The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

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Abstract

In this work, we focus on the set-theoretical solutions of the Yang–Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions \(r_S\) and \(r_T\) is of finite order if and only if \(r_S\) and \(r_T\) are. Furthermore, we show that with sufficient information on \(r_S\) and \(r_T\), we can precisely establish the order of the matched product. Finally, we prove that if B is a finite semi-brace, then the associated solution r satisfies \(r^n=r\), for an integer n closely linked with B.

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Acknowledgements

We thank the referee for the thorough and detailed review.

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Authors and Affiliations

  1. Department of Mathematics and Physics “Ennio De Giorgi”, University of Salento, Via per Arnesano, 73100, Lecce, LE, Italy

    Francesco Catino, Ilaria Colazzo & Paola Stefanelli

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  1. Francesco Catino
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  2. Ilaria Colazzo
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  3. Paola Stefanelli
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Correspondence to Francesco Catino.

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This work was partially supported by the Department of Mathematics and Physics “Ennio De Giorgi”—University of Salento. The authors are members of GNSAGA (INdAM).

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Catino, F., Colazzo, I. & Stefanelli, P. The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order. Mediterr. J. Math. 17, 58 (2020). https://doi.org/10.1007/s00009-020-1483-y

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  • DOI: https://doi.org/10.1007/s00009-020-1483-y

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