4 problems

Let $M_4 = \operatorname{Mon}\langle a,b\mid aba^2=ba\rangle$ and let $M_7$ denote its anti-isomorphic one-relation monoid. A faithful tropical representation of a monoid is an inj…

Let $M_5 = \operatorname{Mon}\langle a,b\mid aba=ba\rangle$ be the one-relation monoid with defining relation $aba=ba$. Its UT-rank is the least $n$ such that $M_5$ admits a faithf…

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$\mathbb{T})$.

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…