5 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…

For each positive integer $n$, let $M_n(\mathbb T)$ be the monoid of $n\times n$ tropical matrices. The source notes that polynomial growth alone does not force a semigroup identit…

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…

Let $n$ be a positive integer. A group is said to admit a faithful representation by $n\times n$ tropical matrices if it embeds into the multiplicative semigroup of $n\times n$ tro…