In this paper, we study the atomic structure of Puiseux monoids generated by monotone sequences. To understand this atomic structure, it is often useful to know whether the monoid has a bounded generating set. We provide necessary and sufficient conditions for the atomicity and boundedness to be transferred from a monotone Puiseux monoid to all its submonoids. Finally, we present two special subfamilies of monotone Puiseux monoids and fully classify their atomic structure.
A Puiseux monoid is a submonoid of $(\mathbb{Q},+)$ consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. In this paper, we use topological density to understand how much a Puiseux monoid, as well as its set of irreducibles, spread through $\mathbb{R}_{\ge 0}$. First, we separate Puiseux monoids according to their density in $\mathbb{R}_{\ge 0}$, and we characterize monoids in each of these classes in terms of generating sets and sets …
A Puiseux monoid is an additive submonoid of the nonnegative cone of rational numbers. Although Puiseux monoids are torsion-free rank-one monoids, their atomic structure is rich and highly complex. For this reason, they have been important objects to construct crucial examples in commutative algebra and factorization theory. In 1974 Anne Grams used a Puiseux monoid to construct the first example of an atomic domain not satisfying the ACCP, disproving Cohn's conjecture that every atomic domain satisfies the ACCP. Even recently, Jim Coykendall and Felix Gotti have used Puiseux monoids to constru…
A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.
In this paper, we study the atomic structure of the family of Puiseux monoids. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-nineteenth century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criter…
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…
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 t…
A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If $M$ is a Puiseux monoid, then the question of whether each non-invertible element of $M$ can be written as a sum of irreducible elements (that is, $M$ is atomic) is surprisingly difficult. Although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are atomic, no general characterization of such monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux monoids. We provide characterizations of when $M$ is finite…
A Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. We say that a Puiseux monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers, which contains $0$, such that $M$ is generated by the set $\{r^s \mid s \in S\}$. If $M$ is multiplicatively closed then we say that $M$ is an exponential Puiseux semiring. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization prop…
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…
Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux mono…
In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of $\mathbb{Q}_{\ge 0}$). We begin by presenting a BF-monoid $M$ with full system of sets of lengths, which means that for each subset $S$ of $\mathbb{Z}_{\ge 2}$ there exists an element $x \in M$ whose set of lengths $\mathsf{L}(x)$ is $S$. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent…
A Puiseux monoid is an additive submonoid of the real line consisting of rationals. We say that a Puiseux monoid is reciprocal if it can be generated by the reciprocals of the terms of a strictly increasing sequence of pairwise relatively primes positive integers. We say that a commutative and cancellative (additive) monoid is atomic if every non-invertible element $x$ can be written as a sum of irreducibles. The number of irreducibles in this sum is called a length of $x$. In this paper, we identify and investigate generalized classes of reciprocal Puiseux monoids that are atomic. Moreover, f…
Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The elasticity of $x$, denoted by $ρ(x)$, is the ratio of its largest factorization length to its shortest factorization length, and it measures how far is $x$ from having a unique factorization. The elasticity $ρ(M)$ of $M$ is the supremum of the elasticities of all non-unit elements of $M$. The monoid $M$ has accepted elasticity if $ρ(M) = ρ(m)$ for some $m \in M$. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of $\mathbb{Q}_{\ge 0}$). First, we characterize the Puiseux monoids $M$ ha…
An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures.
If $M$ is an atomic monoid and $x$ is a nonzero non-unit element of $M$, then the set of lengths $\mathsf{L}(x)$ of $x$ is the set of all possible lengths of factorizations of $x$, where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O'Neil studied the sets of elasticities $\mathcal{R}(P) := \{\sup \mathsf{L}(x)/\inf \mathsf{L}(x) : x \in P\}$ of Puiseux monoids $P$. Here we take this study a step forward and explore the local $k$-elasticities of the same class of monoids. We find conditions under which Puiseux mono…
We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study consists of all atomic monoids of the form $S_r := \langle r^n \mid n \in \mathbb{N}_0 \rangle,$ where $r$ is a positive rational. As the atomic monoids $S_r$ are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of $S_r$ is that all its sets of lengths are arithmetic …
Let $M$ be a commutative cancellative monoid, and let $R$ be an integral domain. The question of whether the monoid ring $R[x;M]$ is atomic provided that both $M$ and $R$ are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for $M = \mathbb{N}_0$: he constructed an atomic integral domain $R$ such that the polynomial ring $R[x]$ is not atomic. However, the question of whether a monoid algebra $F[x;M]$ over a field $F$ is atomic provided that $M$ is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any inf…
In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids $M$ that are hereditarily atomic (i.e., every submonoid of $M$ is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we…
Exponential Puiseux semirings are additive submonoids of $\qq_{\geq 0}$ generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmeti…
In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field $F$ and exponents in an additive submonoid $M$ of $\mathbb{Q}_{\ge 0}$ is called a Puiseux algebra and denoted by $F[M]$. Here we study the atomic structure of Puiseux algebras. To begin with, we answer the Isomorphism Problem for the class of Puiseux algebras, that is, we show that for a field $F$ if two Puiseux algebras $F[M_1]$ and $F[M_2]$ are isomorphic, then the monoids $M_1$ and $M_2$ must be isomorphic. Then we construct three classes of Puiseux algebras satisfying the following well-kn…
Let $S$ be a nonnegative semiring of the real line, called here a positive semiring. We study factorizations in both the additive monoid $(S,+)$ and the multiplicative monoid $(S\setminus\{0\}, \cdot)$. In particular, we investigate when, for a positive semiring $S$, both $(S,+)$ and $(S\setminus\{0\}, \cdot)$ have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP $\Righ…
Given an ambient ordered field $K$, a positive monoid is a countably generated additive submonoid of the nonnegative cone of $K$. In this paper, we first generalize several atomic features exhibited by Puiseux monoids of the field of rational numbers to the more general setting of positive monoids of Archimedean fields, accordingly arguing that such generalizations may fail if the ambient field is not Archimedean. In particular, we show that a positive monoid $P$ of an Archimedean field is a BF-monoid provided that $P \! \setminus \! \{0\}$ does not have $0$ as a limit point. Then, we prove ou…
A cancellative and commutative monoid $M$ is atomic if every non-invertible element of $M$ factors into irreducibles (also called atoms), and $M$ is hereditarily atomic if every submonoid of $M$ is atomic. In addition, $M$ is hereditary ACCP if every submonoid of $M$ satisfies the ascending chain condition on principal ideals (ACCP). Our primary purpose in this paper is to determine which abelian groups are hereditarily atomic. In doing so, we discover that in the class of abelian groups the properties of being hereditarily atomic and being hereditary ACCP are equivalent. Once we have determin…
