12 problems

Let the single-theater game have escalation set $S=\{1,\dots,m\}$. Its misère quotient is the commutative monoid obtained by identifying positions that are interchangeable in every…

Let … be the multivariable exponential generating function for submonoids of products of finite commutative monoids. Rationality conjecture. The generating function $G(x,y,z)$ is a…

He–et al.'s conjecture. There is some $n\in\mathbb{N}$ such that $V^{\otimes n}$ contains a direct summand of the form $\operatorname{Ind}(W)$ for a $kG$-module $W$.

Henckell–Rhodes conjecture. Every finite inverse monoid $M$ has a finite $F$-inverse cover: there is an idempotent-separating homomorphism from a finite $F$-inverse monoid onto $M$…

Let $M$ be a finite monoid, and let $K_{\rm G}(M)$ denote its $\rm G$-kernel, the intersection of $\mu^{-1}(1)$ over all relational morphisms from $M$ to finite groups, where…

Let $K$ be a simply connected finite simplicial complex, and let $M$ be a finite monoid. Fiedorowicz's conjecture. Every homotopy type of a simply connected finite simplicial compl…

Let $M$ be a finite monoid, let $G$ be its group of units, and let $V$ be a $kM$-module. Write $a(n)$ for the asymptotic growth quantity associated with $V^{\otimes n}$, and let…

Cartan nonsingularity conjecture for controlled stabilizers. If $C(\mathbb C M)$ is nonsingular, then $C(kM)$ is nonsingular for every field $k$ of characteristic $p$.

Characteristic-zero Cartan nonsingularity conjecture. If $C(\mathbb C M)$ is nonsingular, then $C(KM)$ is nonsingular for every field $K$.

Type II conjecture. The monoid $K_{\mathsf{G}}(M)$ is the smallest submonoid of $M$ that contains the idempotents and is closed under weak conjugation.

The Trotter-Weil hierarchy consists of varieties of finite monoids, including its join levels and intersection levels; the separation problem asks whether two specified languages c…

Let $\mathcal{PODI}_n$ be the monoid of partial order-preserving or order-reversing permutations, let $\mathcal{POI}_n$ be the monoid of partial order-preserving permutations, and…