35 problems

For every subsemiring $S\subseteq\mathbb{R}_{\ge 0}$ with $1\in S$, if the additive monoid $(S,+)$ and the nonzero multiplicative monoid $(S\setminus\{0\},\cdot)$ are both factoria…

Characterization conjecture. If $\mathsf D(G)\geq 4$, then $G$ and $G'$ are isomorphic.

Bi-UFS Positive Conjecture. A positive semidomain $S$ is a bi-UFS if and only if

Bi-UF Positive Conjecture. The prototypical semiring $\mathbb{N}_0$ is the only positive semiring that is a bi-UFS.

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…

Let $S$ be a numerical semigroup of multiplicity $m=2k$, where $k\in\mathbb{Z}$. A decomposition of $S$ is an irredundant intersection of irreducible numerical semigroups, and its…

Eventual upper-bound conjecture. If ${\sf D}(G)\ge 4$, then there exists $k_0\in\mathbb N$ such that

Let $G$ be a finite abelian group with $|G|>4$. Write $\rho(G)$ for its elasticity and let $\Delta{\rho}(G)$ be the set of all $dinmathbb N$ such that, for every $kinmathbb N$, ther…

Let $G$ be a finite abelian group, and let $\mathcal{F}(G)$ be the free abelian monoid on $G$. Define … and let … be the block monoid on $G$. The Betti elements of an affine monoid…

Finite catenary-degree conjecture. If $F_{\mathrm{red}}$ is finitely generated, then

Let $S$ be a positive semidomain. It is a bi-LFS if both $(S,+)$ and $S^\bullet$ are length-factorial monoids (LFMs), and a bi-UFS if both $(S,+)$ and $S^\bullet$ are unique-factor…

Let $S$ be a numerical monoid, and let $\mathcal P_{\mathrm{fin}}(S)$ denote its power monoid of finite nonempty subsets. For $A\in\mathcal P_{\mathrm{fin}}(S)$, let $\mathsf L(A)$…

Omega-primality finiteness conjecture. One has

Let $S$ be a positive semiring. Bi-UFS characterization conjecture. The semiring $S$ is a bi-UFS if and only if … The paper notes that $\mathbb{N}_0$ is a bi-UFS and presents no ot…

Molecularity conjecture for $\mathcal{C}_3$. The class $\mathcal{C}_3$ is molecular.

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

Let $G$ be a finite abelian group, and let $H$ be a transfer Krull monoid over $G$, meaning that there is a transfer homomorphism $H\to\mathcal B(G)$ to the monoid of zero-sum sequ…

Let $G$ be a finite abelian group, and call a sequence over $G$ a minimal zero-sum sequence if its sum is zero and no nonempty proper subsum is zero. Write $\mathsf D(G)$ for the D…

ACCP conjecture. If

Let $r\in\mathbb{Q}_{>1}\setminus\mathbb{N}$, and write $r=\mathsf{n}(r)/\mathsf{d}(r)$ in lowest terms. Let $S_r$ be the cyclic rational semiring generated by the geometric sequen…

Primary full-system conjecture. For every dimension $d\geq 3$, there exists a primary monoid in $\mathcal{C}_d$ having full system of sets of lengths.

Characterization conjecture. If

Let $H$ be a transfer Krull monoid over a finite abelian group $G$, and let $\mathcal B(G)$ be the monoid of zero-sum sequences over $G$. Write $\mathsf L(a)$ for the set of factor…

Let $H$ be a transfer Krull monoid over a finite abelian group $G$ with $|G|>4$. Let $\Delta_{\rho}(H)$ denote the set of all positive integers $d$ such that, for every…

Let Figure $1$ be the directed diagram of implications among unique factorization domains, atomic domains, semi-atomic domains, almost atomic domains, quasi-atomic domains, Fursten…