A characterization of finite factorization positive monoids

TL;DR

This paper characterizes positive monoids with the finite factorization property, showing that well-ordered generators guarantee it, while co-well-ordered generators do so if and only if they have bounded factorizations.

Contribution

It provides a complete characterization of finite factorization in positive monoids based on the order properties of their generating sets.

Findings

Positive monoids with well-ordered generators satisfy finite factorization.

Co-well-ordered generators satisfy finite factorization iff they have bounded factorization.

The characterization links order properties of generators to factorization properties.

Abstract

We provide a characterization of the positive monoids (i.e., additive submonoids of the nonnegative real numbers) that satisfy the finite factorization property. As a result, we establish that positive monoids with well-ordered generating sets satisfy the finite factorization property, while positive monoids with co-well-ordered generating sets satisfy this property if and only if they satisfy the bounded factorization property.