Representations and identities of Baxter monoids with involution

TL;DR

This paper studies the Baxter monoid with involution, providing a faithful matrix representation, characterizing its identities, and analyzing its finite basis and computational complexity.

Contribution

It introduces a faithful involution monoid representation of the Baxter monoid and characterizes its identities and computational properties.

Findings

Faithful matrix representation over semirings including tropical semiring.

Transparent combinatorial characterization of word identities.

Finite basis property depends on the rank n, with polynomial-time identity checking.

Abstract

Let $(\mathsf{baxt}_n,~^\sharp)$ be the Baxter monoid of finite rank $n$ with Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, it is shown that $(\mathsf{baxt}_n,~^\sharp)$ 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 $(\mathsf{baxt}_n,~^\sharp)$ is given. Further, it is proved that $(\mathsf{baxt}_n,~^\sharp)$ is finitely based if and only if $n\neq 3$, and shown that the identity checking problem for $(\mathsf{baxt}_n,~^\sharp)$ can be done in polynomial time.