5 problems

Let $k_1,\ldots,k_n$ be elementary embeddings in $\mathcal{E}_{\lambda}$ such that $\mathcal{A}_{k_1,\ldots,k_n}$ is free, and let $\ell,p\in\mathcal{A}_{k_1,\ldots,k_n}$. The pull…

Freeness conjecture. The dendriform trialgebra ${\bf WQSym}$ is free.

Makar-Limanov's conjecture. If $F\otimes_K A$ contains a free $K$-algebra on at least two free generators, then $A$ also contains a free $K$-algebra on the same number of free gene…

Let $h$ be an associative presheaf, and let $\mathcal{A}(h)$ denote its pattern Hopf algebra. Freeness conjecture. The pattern Hopf algebra $\mathcal{A}(h)$ is free. This would est…

Makar-Limanov's involutional conjecture. If $D$ is infinite dimensional over $k$ and finitely generated as a division $k$-algebra, then there exist two symmetric elements in $D$ wh…