On the atomic structure of torsion-free monoids

TL;DR

This paper characterizes torsion-free monoids satisfying the ACCP as those with all submonoids atomic, and explores the atomic structure of specific classes of positive monoids, advancing understanding of their algebraic properties.

Contribution

It provides a characterization of torsion-free monoids satisfying the ACCP and investigates the atomic structure of particular positive monoids, offering new insights into their algebraic behavior.

Findings

Torsion-free monoids satisfying ACCP have all submonoids atomic.

Positive monoids are torsion-free and their atomic structure is analyzed.

Characterization of atomicity in classes of positive monoids.

Abstract

Let $M$ be a cancellative and commutative (additive) monoid. The monoid $M$ is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, $M$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.