Representations and identities of involution Plactic-like monoids arising from the meet of the stalactic congruence and its dual

Let ${\mathsf{m}\mathsf{S}\mathsf{t}}_n$ be the plactic-like monoid obtained by factoring the free monoid over a finite alphabet ${\mathcal{A}}_n$ by the meet of the stalactic congruence and its dual. In this paper, we prove that ${\mathsf{m}\mathsf{S}\mathsf{t}}_n$ can be equipped with multiple involutions, and divide these involutions into $\lfloor \frac{n}{2}\rfloor +1$ types. A faithful representation of ${\mathsf{m}\mathsf{S}\mathsf{t}}_n$ under each of these involutions is obtained. We give transparent combinatorial characterizations of identities for ${\mathsf{m}\mathsf{S}\mathsf{t}}_n$ under each involution, and so the finite basis problem and identity checking problem for them are solved.