12 problems

Let $H$ be a numerical monoid, meaning a cofinite additive submonoid of the non-negative integers $\mathbb N$. Let $\mathcal P_{\mathrm{fin},0}(H)$ be the monoid of finite subsets…

Let $H$ and $K$ be numerical monoids, and let $\mathcal{P}_{\operatorname{fin},1}(H)$ and $\mathcal{P}_{\operatorname{fin},1}(K)$ denote their reduced finitary power monoids, consi…

Fan–Tringali conjecture. Every nonempty finite subset of $\mathbb{N}_{\geq 2}$ occurs as a length set in each of the power monoids $\mathcal{P}_{{\rm fin}}(\mathbb{N}_0)$ and…

Let $\mathcal P_{\mathrm{fin},0}(\mathbb Z)$ be the monoid of finite subsets of the additive group $\mathbb Z$ containing $0$, with setwise addition. Tringali–Wen's conjecture. The…

Let the reduced finitary power monoid of the additive group $(\mathbb{Z},+)$ be the quotient of the finitary power monoid by translation equivalence, and let $X\mapsto -X$ denote t…

Let $H$ be a numerical monoid, and for $k,l\in\mathbb N$ let $\alpha_{k,l}(H)$ be the number of $k$-element atoms of $\mathcal P_{\mathrm{fin},0}(H)$ whose maximum is at most $l$.…

Let $\mathcal O$ be a class of finite monoids, and for a monoid $H$ write the system of length sets of $\mathcal P_{\mathrm{fin},1}(H)$ for the collection of all its length sets. F…

Let $H$ be a monoid that is not torsion, and let $\mathcal P_{\mathrm{fin},1}(H)$ be its finitary power monoid containing the identity. A length set is the set of factorization len…

A numerical monoid is a submonoid $S$ of $(\mathbb N,+)$ such that $\mathbb N\smallsetminus S$ is finite. Let the reduced power monoid of $S$ be the reduced power monoid associated…

Let $S$ and $S'$ be numerical monoids. For a numerical monoid $T$, let $\mathcal P_{\mathrm{fin},0}(T)$ denote the power monoid of finite subsets of $T$ containing $0$. Isomorphism…

Let $\mathcal P_{\mathrm{fin}}(\mathbb N_0)$ be the monoid of finite nonempty subsets of $\mathbb N_0$ under sumset addition, where … For an element $A$ of this monoid, let…

Power-monoid sets-of-lengths conjecture. The systems of sets of lengths of $\mathcal P_{\rm fin}(H)$ and $\mathcal P_{{\rm fin},\times}(H)$ contain every non-empty finite subset of…