Arithmetic properties encoded in undermonoids

TL;DR

This paper investigates how certain algebraic properties of cancellative, commutative monoids can be inferred from their undermonoids, establishing conditions under which properties like atomicity and factoriality are preserved.

Contribution

It proves that for properties such as atomicity, bounded factorization, half-factoriality, and length-factoriality, checking undermonoids suffices to determine if all submonoids have these properties, and characterizes monoids with universally half- or length-factorial submonoids.

Findings

Atomicity of undermonoids implies the monoid satisfies ACCP.

Hereditarily atomic monoids satisfy ACCP.

Characterization of monoids with all submonoids half- or length-factorial.

Abstract

Let $M$ be a cancellative and commutative monoid. A submonoid $N$ of $M$ is called an undermonoid if the Grothendieck groups of $M$ and $N$ coincide. For a given property $\mathfrak{p}$, we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of $M$ satisfy $\mathfrak{p}$ to conclude that all submonoids of $M$ satisfy $\mathfrak{p}$? In this paper, we give a positive answer to this question for the property of being atomic, and then we prove that if $M$ is hereditarily atomic (i.e., every submonoid of $M$ is atomic), then $M$ must satisfy the ACCP, proving a recent conjecture posed by Vulakh and the first author. We also give positive answers to our main question for the following well-studied factorization properties: the bounded factorization property, half-factoriality, and length-factoriality. Finally, we determine all the monoids whose submonoids/undermonoids are half-factorial (or length-factorial).