18 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…

Let $\Gamma$ be a numerical monoid with least generator $n_1$, and let $\omega(n)$ denote its omega-function. Exact-period conjecture. For all sufficiently large $n$, the function…

Let $\b X$ be a formal curve with associated numerical monoid … For a numerical monoid, let the associated numerical monoid of a blowup be the numerical monoid corresponding to its…

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 $\mathscr{T}(e,m)$ and $\mathscr{D}(e,m)$ denote the functions defined in the paper for the type and the relevant upper bound of numerical monoids. Asymptotic strictness conjec…

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 $n\geq 2$, let $M=\mathbf P(n,\mathbb N)$, and let $S$ be a unipotent numerical monoid in $M$. Let $\varphi(S)$ be its image in $\mathbb N^{n-1}$ under the monoid isomorphism…

Let $G$ be a unipotent linear algebraic group, and let $M=G_{\mathbb N}$. Let $S$ be a unipotent numerical monoid in $M$, with relative conductor $\mathtt{c}_M(S)$, relative sporad…

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…

Realizability conjecture. For every $n \in \mathbb{N}_{\ge 2}$ there exists a numerical monoid $N$ such that

Let $M$ be the generalized arithmetic monoid under consideration, let $m\in M$, and let $\mathcal Z(m)$ be its set of factorizations. For factorizations $z,z'\in\mathcal Z(m)$, wri…

Let $M$ be the generalized arithmetic monoid under consideration, with parameters $h$ and $d$, and let $c(M)$, $c_{mon}(M)$, and $c_{eq}(M)$ denote its regular, monotone, and equiv…

Realization conjecture for sets of distances. Every finite set $\Delta \subset \mathbb N$ satisfying

Set-of-length-sets conjecture. If

Length-set invariance conjecture. If $|G|,|G'|>3$ and

Let $\Gamma = \langle n_1,n_2,n_3\rangle$ be a numerical monoid minimally generated by $n_1<n_2<n_3$, and let $\omega(n_i)$ denote the omega-value of $n_i$. The forbidden ordering…

Let $S$ be a commutative cancellative monoid with finite abelian class group $G$, and let $\rho_k(S)$ denote the $k$th higher elasticity and $\rho(S)$ the elasticity. Define … In t…