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Let $(\mathsf{hypo}_n,~^\sharp)$ be the hypoplactic monoid of finite rank $n$ with Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, we exhibit a faithful representation of $(\mathsf{hypo}_n,~^\sharp)$ 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 $(\mathsf{hypo}_n,~^\sharp)$. Further, we prove that $(\mathsf{hypo}_n,~^\sharp)$ 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 $(\mathsf{hypo}_n,~^\sharp)$.

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., Comment: arXiv admin note: substantial text overlap with arXiv:2301.12449