Atomicity of Positive Monoids

TL;DR

This paper surveys recent advances in the study of positive monoids, focusing on their atomic and arithmetic structures, with particular attention to rational and geometric sequence-generated monoids.

Contribution

It provides a comprehensive overview of recent research on positive monoids, highlighting their complex atomic and arithmetic properties through numerous examples.

Findings

Positive monoids exhibit complex atomic structures.

Rational and geometric sequence-generated monoids have unique properties.

Recent research has advanced understanding of positive monoid arithmetic.

Abstract

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures.