Representations and identities of hypoplactic monoids with involution

Abstract

Let $(\mathsf{\text{hyp}}{\mathsf{o}}_n{,}^{♯})$ be the hypoplactic monoid of finite rank n with Schützenberger’s involution ${}^{♯}$. In this paper, we exhibit a faithful representation of $(\mathsf{\text{hyp}}{\mathsf{o}}_n{,}^{♯})$ as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by $(\mathsf{\text{hyp}}{\mathsf{o}}_n{,}^{♯})$. Further, we prove that $(\mathsf{\text{hyp}}{\mathsf{o}}_n{,}^{♯})$ is non-finitely based if and only if n = 2, 3 and give a polynomial time algorithm to check whether a given word identity holds in $(\mathsf{\text{hyp}}{\mathsf{o}}_n{,}^{♯})$.

Communicated by Scott Chapman

KEYWORDS:

• Finite basis problem
• identity
• hypoplactic monoid
• representation
• Schützenberger’s involution

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M07
• 20M30
• 05E99
• 12K10
• 16Y60

Acknowledgments

The authors are very grateful to the anonymous referee for his/her careful reading and suggestions.

Additional information

Funding

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) and the Natural Science Foundation of Gansu Province (No. 23JRRA1055).