8 problems

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…

Interval-closure conjecture. For every ideal extension $S$ of $\mathbb{N}^{(I)}$, the set $2\mathcal{A}(S)$ is closed under intervals.

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…

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

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.

Rational-or-infinite elasticity conjecture. The elasticity of any monoid in the class of submonoids of finite-rank free commutative monoids is either rational or infinite.

Strong commutative monoid axiom conjecture. Under these hypotheses, the strong commutative monoid axiom holds in $\mathcal{M}$. This conjecture is motivated by the corresponding re…