On the Atomic Structure of Puiseux Monoids

TL;DR

This paper explores the atomic properties of Puiseux monoids, generalizing numerical semigroups, and identifies conditions under which these monoids are atomic or non-atomic, including classifications of strongly bounded cases.

Contribution

It provides new characterizations of atomic Puiseux monoids and classifies atomic subfamilies over finite prime sets, expanding understanding of their structure.

Findings

Many Puiseux monoids are non-atomic or contain no atoms.

Characterization criteria for atomic Puiseux monoids are established.

Classification of atomic strongly bounded Puiseux monoids over finite primes.

Abstract

In this paper, we study the atomic structure of the family of Puiseux monoids. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-nineteenth century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criteria which force Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of strongly bounded Puiseux monoids over a finite set of primes.