arXiv subjects

Marly Gotti

Publications and source records attributed to Marly Gotti.

15 recordsLinked to original sources

arXiv2026

The Bi-UFS Positive Conjecture for algebraic semidomains

A semidomain is called bi-UFS if both its additive monoid and its nonzero multiplicative monoid are unique factorization monoids. The Bi-UFS Positive Conjecture predicts that the only positive semidomain with this property is the nonnegative integers. We prove this conjecture for finitely generated algebraic positive semidomains. In the cyclic case, we show that for every positive algebraic number $\alpha$, the semidomain $\mathbb{N}_0[\alpha]$ is bi-UFS if and only if $\alpha \in \mathbb{N}$, equivalently $\mathbb{N}_0[\alpha]=\mathbb{N}_0$. The proof separates the quadratic case, where an analysis of the least additive atom larger than $1$ leaves only the examples $\mathbb{N}_0[\sqrt 2]$ and $\mathbb{N}_0[(1+\sqrt 5)/2]$ to exclude, from the higher-degree case, where explicit multiplicative identities force the minimal polynomial into impossible forms. We then give a Perron-Frobenius argument showing that if $\alpha_1,\ldots,\alpha_n$ are positive algebraic numbers and $\mathbb{N}_0[\alpha_1,\ldots,\alpha_n]$ is bi-UFS then this semidomain is $\mathbb{N}_0$. Finally, we prove a reduction theorem for complex semidomains: every bi-UFS subsemidomain of $\mathbb{C}$ with finitely many additive atoms admits an isomorphic realization as a positive semidomain. Consequently, every finitely generated algebraic bi-UFS semidomain over $\mathbb{C}$ is isomorphic to $\mathbb{N}_0$.

math.AC↗

arXiv2025

On the set of atoms and strong atoms in additive monoids of cyclic semidomains

Let $M$ be a cancellative and commutative monoid. A non-invertible element of $M$ is called an atom (or irreducible element) if it cannot be factored into two non-invertible elements, while an atom $a$ of $M$ is called strong if $a^n$ has a unique factorization in $M$ for every $n \in \mathbb{N}$. The monoid $M$ is atomic if every non-invertible element factors into finitely many atoms (repetitions allowed). For an algebraic number $\alpha$, we let $M_\alpha$ denote the additive monoid of the subsemiring $\mathbb{N}_0[\alpha]$ of $\mathbb{C}$. The atomic structure of $M_\alpha$ reflects intricate interactions between algebraic number theory and additive semigroup theory. For $m, n \in \mathbb{N}_0 \cup \{ \infty \}$ (with $m \le n$), the pair $(m,n)$ is called realizable if there exists an algebraic number $\alpha \in \mathbb{C}$ such that $M_\alpha$ has $m$ strong atoms and $n$ atoms. Our primary goal is to identify classes of realizable pairs with the long-term goal of providing a complete description of the full set of realizable pairs.

math.AC↗

arXiv2021

Factorizations in reciprocal Puiseux monoids

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, for the atomic monoids in the identified classes, we study the ascending chain condition on principal ideals and also the sets of lengths of their elements.

math.AC↗

arXiv2021

Atomicity of Positive Monoids

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.

math.AC↗

arXiv2021

On the set of molecules of numerical and Puiseux monoids

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 $m$.

math.AC↗

arXiv2020

Atomicity and Factorization of Puiseux Monoids

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 construct the first atomic monoids with monoid algebras (over a field) that are not atomic, answering a question posed by Robert Gilmer back in the 1980s. This dissertation is focused on the investigation of the atomic structure and factorization theory of Puiseux monoids. Here we established various sufficient conditions for a Puiseux monoid to be atomic (or satisfy the ACCP). We do the same for two of the most important atomic properties: the finite-factorization property and the bounded-factorization property. Then we compare these four atomic properties in the context of Puiseux monoids. This leads us to construct and study several classes of Puiseux monoids with distinct atomic structure. Our investigation provides sufficient evidence to believe that the class of Puiseux monoids is the simplest class with enough complexity to find monoids satisfying almost every fundamental atomic behavior.

math.AC↗

arXiv2020

Atomicity and boundedness of monotone Puiseux monoids

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.

math.AC↗

arXiv2020

When is a Puiseux monoid atomic?

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 finitely generated, factorial, half-factorial, other-half-factorial, Prüfer, seminormal, root-closed, and completely integrally closed. In addition to the atomicity, characterizations are also not known for when $M$ satisfies the ACCP, the bounded factorization property, or the finite factorization property. In each of these cases, we construct an infinite class of Puiseux monoids satisfying the corresponding property.

math.AC↗

arXiv2020

Atomicity and Density of Puiseux Monoids

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 of irreducibles. Then we study the density of the difference group, the root closure, and the conductor semigroup of a Puiseux monoid. Finally, we prove that every Puiseux monoid generated by a strictly increasing sequence of rationals is nowhere dense in $\mathbb{R}_{\ge 0}$ and has empty conductor.

math.AC↗

arXiv2020

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

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.

math.AC↗

arXiv2019

Factorization invariants of Puiseux monoids generated by geometric sequences

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 sequences of the same distance, namely $|a-b|$, where $a,b \in \mathbb{N}$ are such that $r = a/b$ and $\text{gcd}(a,b) = 1$. We prove this, and then use it to study the elasticity and tameness of $S_r$.

math.AC↗

arXiv2019

How do elements really factor in $\mathbb{Z}[\sqrt{-5}]$?

Most undergraduate level abstract algebra texts use $\mathbb{Z}[\sqrt{-5}]$ as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of how elements actually factor in this ring of algebraic integers. We offer here an interactive framework which shows that while $\mathbb{Z}[\sqrt{-5}]$ is not a UFD, it does satisfy a slightly weaker factorization condition, known as half-factoriality. The arguments involved revolve around the Fundamental Theorem of Ideal Theory in algebraic number fields.

math.HO↗

arXiv2018

On the local k-elasticities of Puiseux monoids

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 monoids have all their local elasticities finite as well as conditions under which they have infinite local $k$-elasticities for sufficiently large $k$. Finally, we focus our study of the $k$-elasticities on the class of primary Puiseux monoids, proving that they have finite local $k$-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain $0$ as a limit point.

math.AC↗

arXiv2017

The Chain Group of a Forest

For every labeled forest $\mathsf{F}$ with set of vertices $[n]$ we can consider the subgroup $G$ of the symmetric group $S_n$ that is generated by all the cycles determined by all maximal paths of $\mathsf{F}$. We say that $G$ is the chain group of the forest $\mathsf{F}$. In this paper we study the relation between a forest and its chain group. In particular, we find the chain groups of the members of several families of forests. Finally, we prove that no copy of the dihedral group of cardinality $2n$ inside $S_n$ can be achieved as the chain group of any forest.

math.CO↗

arXiv2016

On three families of dense Puiseux monoids

A positive monoid is a submonoid of the nonnegative cone of a linearly ordered abelian group. The positive monoids of rank $1$ are called Puiseux monoids, and their atomicity, arithmetic of length, and factorization have been systematically investigated for about ten years. Each Puiseux monoid can be realized as an additive submonoid of the nonnegative cone of $\mathbb{Q}$. We say that a Puiseux monoid is dense if it is isomorphic to an additive submonoid of $\mathbb{Q}_{\ge 0}$ that is dense in $\mathbb{R}_{\ge 0}$ with respect to the Euclidean topology. Every non-dense Puiseux monoid is known to be a bounded factorization monoid. However, the atomic structure as well as the arithmetic and factorization properties of dense Puiseux monoids turn out to be quite interesting. In this paper, we study the atomic structure and some arithmetic and factorization aspects of three families of dense Puiseux monoids.

math.AC↗