On the set of molecules of numerical and Puiseux monoids

TL;DR

This paper explores the structure of molecules in Puiseux monoids, showing that for any natural number m ≥ 2, there exists a numerical monoid with exactly m molecules that are not atoms, thus addressing a recent conjecture.

Contribution

It proves that for every m ≥ 2, a numerical monoid can be constructed with exactly m molecules that are not atoms, confirming a recent realization conjecture.

Findings

Existence of numerical monoids with a prescribed number of non-atom molecules

Analysis of molecular sets in various subclasses of Puiseux monoids

Positive resolution of a recent conjecture on molecules in numerical monoids

Abstract

Additive submonoids of $\mathbb{Q}_{\ge 0}$, also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call such elements molecules. Molecules were first investigated by W. Narkiewicz in the context of algebraic number theory. More recently, F. Gotti and the first author studied molecules in the context of Puiseux monoids. Here we address some aspects related to the size of the sets of molecules of various subclasses of Puiseux monoids with different atomic behaviors. In particular, we positively answer the following recent realization conjecture: for each $m \in \mathbb{N}_{\ge 2}$ there exists a numerical monoid whose set of molecules that are not atoms has cardinality $m$.