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Representations and identities of Baxter monoids with involution

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Abstract

Let \((\textsf{baxt}_n,^\sharp )\) be the Baxter monoid of finite rank n with Schützenberger’s involution \(^{\sharp }\). In this paper, it is shown that \((\textsf{baxt}_n,^\sharp )\) admits a faithful representation by an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. Then a transparent combinatorial characterization of the word identities satisfied by \((\textsf{baxt}_n,^\sharp )\) is given. Further, it is proved that \((\textsf{baxt}_n,^\sharp )\) is finitely based if and only if \(n\ne 3\), and shown that the identity checking problem for \((\textsf{baxt}_n,^\sharp )\) is decidable in polynomial time.

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Acknowledgements

The authors are very grateful to the anonymous referee for his/her careful reading and detailed suggestions which significantly improve the quality and readability of this paper. In particular, the authors thank the anonymous referee for pointing out the missing conditions in Theorems 4.2 and 4.3 and suggesting us considering the axiomatic rank of \((\textsf{baxt}_2,^{\sharp })\) and \((\textsf{baxt}_n,^{\sharp })\) with \(n\ge 4\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Bin Bin Han, Wen Ting Zhang & Yan Feng Luo

  2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, Inner Mongolia, People’s Republic of China

    Jin Xing Zhao

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  1. Bin Bin Han
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Correspondence to Wen Ting Zhang.

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Communicated by Mikhail Volkov.

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This research was partially supported by the National Natural Science Foundation of China (Nos. 12271224, 12171213, 12161062), the Fundamental Research Funds for the Central University (No. lzujbky-2023-ey06), the Natural Science Foundation of Inner Mongolia (No. 2021MS01005) and the Natural Science Foundation of Gansu Province (No. 23JRRA1055)

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Han, B.B., Zhang, W.T., Luo, Y.F. et al. Representations and identities of Baxter monoids with involution. Semigroup Forum 107, 424–458 (2023). https://doi.org/10.1007/s00233-023-10388-5

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