Atomicity and Factorization of Puiseux Monoids

TL;DR

This paper investigates the atomic structure and factorization properties of Puiseux monoids, providing conditions for atomicity and constructing classes with diverse atomic behaviors, highlighting their complexity and importance in algebra.

Contribution

It offers new sufficient conditions for atomicity and related properties in Puiseux monoids, and constructs classes with varied atomic structures, advancing understanding of their factorization theory.

Findings

Established conditions for atomicity and ACCP in Puiseux monoids

Constructed classes with diverse atomic and factorization properties

Provided evidence that Puiseux monoids exhibit rich atomic behavior

Abstract

A Puiseux monoid is an additive submonoid of the nonnegative cone of rational numbers. Although Puiseux monoids are torsion-free rank-one monoids, their atomic structure is rich and highly complex. For this reason, they have been important objects to construct crucial examples in commutative algebra and factorization theory. In 1974 Anne Grams used a Puiseux monoid to construct the first example of an atomic domain not satisfying the ACCP, disproving Cohn's conjecture that every atomic domain satisfies the ACCP. Even recently, Jim Coykendall and Felix Gotti have used Puiseux monoids to construct the first atomic monoids with monoid algebras (over a field) that are not atomic, answering a question posed by Robert Gilmer back in the 1980s. This dissertation is focused on the investigation of the atomic structure and factorization theory of Puiseux monoids. Here we established various sufficient conditions for a Puiseux monoid to be atomic (or satisfy the ACCP). We do the same for two of the most important atomic properties: the finite-factorization property and the bounded-factorization property. Then we compare these four atomic properties in the context of Puiseux monoids. This leads us to construct and study several classes of Puiseux monoids with distinct atomic structure. Our investigation provides sufficient evidence to believe that the class of Puiseux monoids is the simplest class with enough complexity to find monoids satisfying almost every fundamental atomic behavior.