Puiseux monoids and transfer homomorphisms

TL;DR

This paper explores the atomic structure of Puiseux monoids, showing that many arithmetical invariants do not transfer from other atomic monoids via homomorphisms, and classifies certain subclasses of Puiseux monoids.

Contribution

It demonstrates the non-existence of transfer homomorphisms from non-finitely generated Puiseux monoids to finitely generated monoids and classifies Puiseux monoids that are C-monoids.

Findings

Transfer homomorphisms from non-finitely generated Puiseux monoids to finitely generated monoids do not exist.

Many Puiseux monoids are not strongly primary.

The only Puiseux monoid with a transfer homomorphism to a Krull monoid is .

Abstract

There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades. The factorization theory of finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most systematically studied. Puiseux monoids, which are additive submonoids of $\mathbb{Q}_{\ge 0}$ consisting of nonnegative rational numbers, have only been studied recently. In this paper, we provide evidence that this family comprises plenty of monoids with a basically unexplored atomic structure. We do this by showing that the arithmetical invariants of the well-studied atomic monoids mentioned earlier cannot be transferred to most Puiseux monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms. Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux monoid to a finitely generated monoid do not exist. We also find a large family of Puiseux monoids that fail to be strongly primary. In addition, we prove that the only nontrivial Puiseux monoid that accepts a transfer homomorphism to a Krull monoid is $\mathbb{N}_0$. Finally, we classify the Puiseux monoids that happen to be C-monoids.