On transfer Krull monoids

TL;DR

This paper systematically studies transfer Krull monoids, exploring how their root closures influence their properties and providing characterizations under various conditions, with implications for understanding factorization in algebraic structures.

Contribution

It offers the first in-depth analysis of transfer Krull monoids, characterizing their properties based on root closure types and establishing new criteria for when they are half-factorial or Krull.

Findings

Transfer Krull monoids with DVM root closures are characterized by inertness.

Factorial root closures imply transfer Krull monoids are characterized by atom sets.

Half-factorial root closures lead to specific atom inclusion conditions.

Abstract

Let $H$ be a cancellative commutative monoid, let $\mathcal{A}(H)$ be the set of atoms of $H$ and let $\widetilde{H}$ be the root closure of $H$. Then $H$ is called transfer Krull if there exists a transfer homomorphism from $H$ into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counter examples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if $\widetilde{H}$ is a DVM, then $H$ is transfer Krull if and only if $H\subseteq\widetilde{H}$ is inert. Moreover, we prove that if $\widetilde{H}$ is factorial, then $H$ is transfer Krull if and only if $\mathcal{A}(\widetilde{H})=\{u\varepsilon\mid u\in\mathcal{A}(H),\varepsilon\in\widetilde{H}^{\times}\}$. We also show that if $\widetilde{H}$ is half-factorial, then $H$ is transfer Krull if and only if $\mathcal{A}(H)\subseteq\mathcal{A}(\widetilde{H})$. Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.