4 problems

Shortest-identity conjecture for $T_5$. This identity is the shortest identity of $T_5$. The preceding theorem establishes that this identity holds, while the conjecture concerns i…

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$\mathbb{T})$.

Finite-rank plactic identity conjecture. For each $n \geq 4$, there is a non-trivial identity satisfied by ${\mathsf{plac}}_n$.

For each positive integer $n$, let $M_n(\mathbb T)$ be the monoid of $n\times n$ tropical matrices. A nontrivial semigroup identity is an identity between distinct semigroup words…