Representations and identities of Baxter monoids with involution

Let $({\mathsf{b}\mathsf{a}\mathsf{x}\mathsf{t}}_n,{}^{\mathrm{♯}})$ be the Baxter monoid of finite rank $n$ with Schützenberger’s involution ${}^{\mathrm{♯}}$. In this paper, it is shown that $({\mathsf{b}\mathsf{a}\mathsf{x}\mathsf{t}}_n,{}^{\mathrm{♯}})$ 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 $({\mathsf{b}\mathsf{a}\mathsf{x}\mathsf{t}}_n,{}^{\mathrm{♯}})$ is given. Further, it is proved that $({\mathsf{b}\mathsf{a}\mathsf{x}\mathsf{t}}_n,{}^{\mathrm{♯}})$ is finitely based if and only if $n\mathrm{\ne }3$, and shown that the identity checking problem for $({\mathsf{b}\mathsf{a}\mathsf{x}\mathsf{t}}_n,{}^{\mathrm{♯}})$ can be done in polynomial time.