6 problems

Let $\mathbb R^n$ carry coordinate-wise tropical addition, and let an additive tropical set-variety mean an additive tropical set-variety in the sense of the source. A tropical set…

Let ${\mathbb T}=\mathbb R\cup\{-\infty\}$ be the tropical semifield, with tropical addition and multiplication given by $a\oplus b=\max\{a,b\}$ and $a\odot b=a+b$. A tropical poly…

Kubat–Okniński's conjecture. Every plactic monoid of finite rank satisfies a non-trivial semigroup identity.

Let $\mathbb{T}$ denote the tropical semifield, let $UT_2(\mathbb{T})$ be the monoid of all $2\times 2$ upper triangular matrices over $\mathbb{T}$, and let $\mathcal{B}$ be the bi…

Let $M_n(\mathbb{T})$ denote the monoid of $n\times n$ tropical matrices, where $n$ is a positive integer. Izhakian–Margolis conjecture. The monoid $M_n(\mathbb{T})$ satisfies an i…

Let the tropical semiring be the set $\mathbb{R}$ with operations $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. Matrix multiplication is defined using these tropical operations, an…