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Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the operat …

[In a comment to the OP, I asked whether a statement along the lines of Corollary 1 below would count as an "interesting structure theorem", and this post expands on John Baez's yes to that question.] …

Let $H$ be a multiplicatively written commutative monoid. We use $\mathcal A(H)$ for the set of atoms of $H$ and $\pi_H$ for the canonical homomorphism $\mathscr F(\mathcal A(H)) \to H$, where $a \in …

In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. Bu …

Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$, and $H$ a submonoid of $\mathscr F(P)$. Given $x \in H \setminus \{1_H\}$, we let $\mathsf L_H(x)$ be the set of all $ …

The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it). Corollary. Let $R$ be a non-trivial Dedekin …

Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take $$XY := \{xy: x \in X,\, y \in Y\}.$$ We call a set $I \subseteq H$ an ideal of $H$ …

Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (wh …

Figured it out (sorry for answering my own question). I'll prove the following: Lemma. Let $H$ be a linearly orderable monoid and $R$ a domain whose group of units is trivial. Then $H$ embeds as a …

Edit (following the clarifications of the OP): This is not an answer, it is rather a long comment. Studying this kind of questions is part of the mission of factorization theory: The language of the t …

[I'm adding a new (non-)answer since what follows has almost nothing to do with my previous one.] Fix a real number $\alpha \ge 1$ and set $S_\alpha := \mathbb N \cup \mathbb R_{\ge \alpha}$. Of cours …

In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascen …

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is con …

Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is divisor-closed (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \mi …

I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite i …