2 problems

Let $\mathbb{P}_n$ denote the plactic monoid on rank $n$, and let a tropical representation of $\mathbb{P}_n$ be faithful when it distinguishes all elements of $\mathbb{P}_n$. The…

Strict identity-variety growth conjecture. For every positive integer $n$, there is a semigroup identity satisfied by $UT_n(b$mathbb{T})$but not by$UT{n+1}(b$\mathbb{T})$.