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

Let $\mathsf{mSt}_n$ 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 $\mathsf{mSt}_n$ can be equipped with multiple involutions, and divide these involutions into $\lfloor\frac{n}{2}\rfloor+1$ types. A faithful representation of $\mathsf{mSt}_n$ under each of these involutions is obtained. We give transparent combinatorial characterizations of identities for $\mathsf{mSt}_n$ under each involution, and so the finite basis problem and identity checking problem for them are solved.