3 problems

Let $k$ be a field, let $A$ be a polytopal algebra over $k$, and let $B$ be a graded algebra retract of $A$. Bruns–Gubeladze conjecture. $B$ is again a polytopal algebra over $k$.…

Let $k$ be a field and let $\operatorname{Pol}(k)$ denote the category of polytopal algebras over $k$. A homomorphism is called tame if it is obtained from the identity mapping…

Tameness conjecture. Any homomorphism in $\operatorname{Pol}(k)$ is obtained by a sequence of these constructions, starting from the identity mapping $k\to k$. Moreover, there are…