82 problems

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…

Bi-UF Positive Conjecture. The only such positive semiring is the prototypical semidomain $\mathbb{N}_0$.

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…

Atomicity conjecture. If $M_r$ is atomic, then $\mathbb{Q}[M_r]$ is atomic.

The catenary-degree equality conjecture. For every group $G$ with

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…

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

Equal-elasticity conjecture. If

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…

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…

A numerical semigroup is an additive subsemigroup of the nonnegative integers containing $0$ and having finite complement. Every numerical semigroup can be expressed as an irredund…

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 $G$ and $G'$ be finite abelian groups. Write $\mathcal{B}(G)$ for the monoid of zero-sum sequences over $G$, $\mathcal{L}(\mathcal{B}(G))$ for its system of sets of lengths, an…

Atomic integral-extension boundary conjecture. If $S$ is atomic, then $S$ cannot contain a nonunit of boundary $0$.

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

Let a hereditarily atomic commutative monoid be a commutative monoid whose every submonoid is atomic, and let ACCP denote the ascending chain condition on principal ideals. The ACC…

Let an hereditarily atomic integral domain be an integral domain whose every subring is atomic, and let ACCP denote the ascending chain condition on principal ideals. The ACCP conj…

Let $K$ be a real quadratic field and, for each positive integer $f$, let $\mathcal{O}_f=\mathbb{Z}+f\mathcal{O}_K$ be the order of conductor $f$. A domain is half-factorial if eve…

Let $K$ be a real quadratic field with discriminant $ath?]$ and let $\mathcal{O}_f=\mathbb{Z}+f\mathcal{O}_K$ be the order of conductor $f$. Call $f$ split-free if it is not divisi…

Let $G$ be a finite group. Write $\mathcal L(G)$ for the collection of sets of lengths of product-one sequences over $G$; explicitly,…

Half-factorial generating set conjecture. Every abelian group has a half-factorial generating set.

Characterization conjecture. If

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…