On the atomic structure of torsion-free 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.

16 pages

References in corpus (4)

• On strongly primary monoids, with a focus on Puiseux monoids
• Factorizations in upper triangular matrices over information semialgebras
• Factorization under Local Finiteness Conditions
• Length-Factoriality and Pure Irreducibility

Cited by in corpus (1)

• Semigroup Graded Stillman's Conjecture