Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid

Abstract

If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M\hookrightarrow \text{gp}(M)\hookrightarrow \mathbb{F}{\otimes }_{\mathbb{Z}}\text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid $M$ in $\mathcal{C}$ are connected to the combinatorics and geometry of its conic hull $\text{cone}(M)\subseteq \mathbb{F}{\otimes }_{\mathbb{Z}}\text{gp}(M)$. First, we show that the submonoids of $M$ determined by the faces of $\text{cone}(M)$ account for all divisor-closed submonoids of $M$. Then we appeal to the geometry of $\text{cone}(M)$ to characterize whether $M$ is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in $\mathcal{C}$. Along the way, we determine the cones that can be realized by monoids in $\mathcal{C}$ and by finitary monoids in $\mathcal{C}$.

40 pages, 4 figures; to appear in Linear Algebra and Its Applications

References in corpus (1)

• On strongly primary monoids, with a focus on Puiseux monoids

Cited by in corpus (3)

• On the additive structure of algebraic valuations of polynomial semirings
• On half-factoriality of transfer Krull monoids
• Atomicity and Factorization of Puiseux Monoids