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Identities and bases in the sylvester and Baxter monoids

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Abstract

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.

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Acknowledgements

The authors thank the anonymous reviewers for their careful reading of the paper and helpful comments, in particular, for the observations on the varietal join and meet of the varieties generated by the sylvester and #-sylvester monoids, and the observation on the finite basis given for the variety generated by the Baxter monoid.

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  1. Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–513, Caparica, Portugal

    Alan J. Cain, António Malheiro & Duarte Ribeiro

  2. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-513, Caparica, Portugal

    António Malheiro & Duarte Ribeiro

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  1. Alan J. Cain
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  2. António Malheiro
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Correspondence to Duarte Ribeiro.

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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 projects UIDB/00297/2020 amd UIDP/00297/2020 (Center for Mathematics and Applications) and the project PTDC/MAT-PUR/31174/2017. The third author is funded by National Funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the studentship SFRH/BD/138949/2018.

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Cain, A.J., Malheiro, A. & Ribeiro, D. Identities and bases in the sylvester and Baxter monoids. J Algebr Comb 58, 933–966 (2023). https://doi.org/10.1007/s10801-022-01202-6

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