# On the molecules of numerical semigroups, Puiseux monoids, and Puiseux algebras

## TL;DR

This paper investigates the structure of molecules in Puiseux monoids and their algebras, revealing their properties, classifications, and the existence of infinitely many non-isomorphic atomic monoids with specific molecule characteristics.

## Contribution

It provides new characterizations of molecules in Puiseux monoids and algebras, including classifications and constructions of monoids with particular molecular properties.

## Key findings

- Infinite non-isomorphic atomic Puiseux monoids with only atoms as molecules.
- Characterization of molecules in Puiseux monoids generated by rationals with prime denominators.
- Existence of infinitely many Puiseux algebras with infinitely many reducible molecules.

## Abstract

A molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules of their corresponding semigroup algebras, which we call Puiseux algebras. We begin by presenting, in the context of numerical semigroups, some results on the possible cardinalities of the sets of molecules and the sets of reducible molecules (i.e., molecules that are not irreducibles/atoms). Then we study the molecules in the more general context of Puiseux monoids. We construct infinitely many non-isomorphic atomic Puiseux monoids all whose molecules are atoms. In addition, we characterize the molecules of Puiseux monoids generated by rationals with prime denominators. Finally, we turn to investigate the molecules of Puiseux algebras. We provide a characterization of the molecules of the Puiseux algebras corresponding to root-closed Puiseux monoids. Then we use such a characterization to find an infinite class of Puiseux algebras with infinitely many non-associated reducible molecules.