arXiv subjects

Felix Gotti

Publications and source records attributed to Felix Gotti.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the subatomicity of polynomial semidomains

A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although atomicity and divisibility in integral domains have been systematically investigated for more than thirty years, the same aspects in the more general context of semidomains have been considered just recently. Here we study subatomicity in the context of semidomains, focusing on whether certain subatomic properties ascend from a semidomain to its polynomial extension and its Laurent polynomial extension. We investigate factorization and divisibility notions generalizing that of atomicity. First, we consider the Furstenberg property, which is due to P. Clark and motivated by the work of H. Furstenberg on the infinitude of primes. Then we consider the almost atomic and quasi-atomic properties, both introduced by J. G. Boynton and J. Coykendall in their study of divisibility in integral domains.

math.AC↗

arXiv2023

Factoriality inside Boolean lattices

Given a join semilattice $S$ with a minimum $\hat{0}$, the quarks (also called atoms in order theory) are the elements that cover $\hat{0}$, and for each $x \in S \setminus \{\hat{0}\}$ a factorization (into quarks) of $x$ is a minimal set of quarks whose join is $x$. If every element $x \in S \setminus \{\hat{0}\}$ has a factorization, then $S$ is called factorizable. If for each $x \in S \setminus \{\hat{0}\}$, any two factorizations of $x$ have equal (resp., distinct) size, then we say that $S$ is half-factorial (resp., length-factorial). Let $B_\mathbb{N}$ be the Boolean lattice consisting of all finite subsets of $\mathbb{N}$ under intersections and unions. Here we study factorizations into quarks in join subsemilattices of $B_\mathbb{N}$, focused on the notions of half-factoriality and length-factoriality. We also consider the unique factorization property, which is the most special and relevant type of half-factoriality, and the elasticity, which is an arithmetic statistic that measures the deviation from half-factoriality.

math.CO↗

arXiv2023

Hereditary atomicity and ACCP in abelian groups

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 determined the abelian groups that are hereditarily atomic, we will use this knowledge to determine the commutative group algebras that are hereditarily atomic, that is, the commutative group algebras satisfying that all their subrings are atomic. The interplay between atomicity and the ACCP is a subject of current active investigation. Throughout our journey, we will discuss several examples connecting (hereditary) atomicity and the ACCP, including, for each integer $d$ with $d \ge 2$, a construction of a rank-$d$ additive submonoid of $\mathbb{Z}^d$ that is atomic but does not satisfy the ACCP.

math.AC↗

arXiv2023

On the additive structure of algebraic valuations of polynomial semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[α]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $\mathbb{N}_0[α]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $\mathbb{N}_0[α]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $\mathbb{N}_0[α]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $α$. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.

math.NT↗

arXiv2022

On the atomic structure of torsion-free monoids

Let $M$ be a cancellative and commutative (additive) monoid. The monoid $M$ is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, $M$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

math.AC↗

arXiv2022

Divisibility and a weak ascending chain condition on principal ideals

An integral domain $R$ is atomic if each nonzero nonunit of $R$ factors into irreducibles. In addition, an integral domain $R$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. Although it is not hard to verify that every integral domain satisfying ACCP is atomic, examples of atomic domains that do not satisfy ACCP are notoriously hard to construct. The first of such examples was constructed by A. Grams back in 1974. In this paper we delve into the class of atomic domains that do not satisfy ACCP. To better understand this class, we introduce the notion of weak-ACCP domains, which generalizes that of integral domains satisfying ACCP. Strongly atomic domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. It turns out that every weak-ACCP domain is strongly atomic, and so we introduce a taxonomic classification on our class of interest: ACCP implies weak-ACCP, which implies strong atomicity, which implies atomicity. We study this chain of implications, putting special emphasis on the weak-ACCP property. This allows us to provide new examples of atomic domains that do not satisfy ACCP.

math.AC↗

arXiv2022

Integral domains and the IDF property

An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P.~Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that the IDF property ascends in the class of GCD-domains. We put special emphasis on IDF-domains where every nonunit is divisible by an irreducible, which we call TIDF-domains, and we also consider PIDF-domains, which form a special class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate both the TIDF and the PIDF properties under taking polynomial rings and localizations. We also delve into their behavior under monoid domain and $D+M$ constructions

math.AC↗

arXiv2021

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.

math.AC↗

arXiv2021

Atomic semigroup rings and the ascending chain condition on principal ideals

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals terminates. It was asserted by Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by Grams in the seventies with an explicit construction of a neat example. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams' first construction. In the first part of this paper, we generalize Grams' construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct what seems to be the first atomic semigroup ring without the ACCP in the existing literature.

math.AC↗

arXiv2021

Divisibility in rings of integer-valued polynomials

In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.

math.AC↗

arXiv2021

On semigroup algebras with rational exponents

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-known atomic properties: the ACCP property, the bounded factorization property, and the finite factorization property. We show that there are bounded factorization Puiseux algebras with extremal systems of sets of lengths, which allows us to prove that Puiseux algebras cannot be determined up to isomorphism by their arithmetic of lengths. Finally, we give a full description of the seminormal closure, root closure, and complete integral closure of a Puiseux algebra, and use such description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles).

math.AC↗

arXiv2021

Bi-atomic classes of positive semirings

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 $\Rightarrow$ BFP and FFP $\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$ atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.

math.AC↗

arXiv2021

Length-factoriality in commutative monoids and integral domains

An atomic monoid $M$ is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element $x \in M$ no two distinct factorizations of $x$ have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is in general weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid $M$, and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of $M$. Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely short and a purely long irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

math.AC↗

arXiv2020

Bounded and finite factorization domains

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive integer $N$ such that for any factorization $x = a_1 \cdots a_n$ of $x$ into irreducibles $a_1, \dots, a_n$ in $R$, the inequality $n \le N$ holds. In addition, we say that $R$ is a finite factorization domain if it is atomic and every nonzero nonunit in $R$ factors into irreducibles in only finitely many ways (up to order and associates). The notions of bounded and finite factorization domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in their systematic study of factorization in atomic integral domains. Here we provide a survey of some of the most relevant results on bounded and finite factorization domains.

math.AC↗

arXiv2020

On strongly primary monoids, with a focus on Puiseux monoids

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 monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

math.AC↗

arXiv2020

Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid

If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid $M$ in $\mathcal{C}$ are connected to the combinatorics and geometry of its conic hull $\text{cone}(M) \subseteq \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$. First, we show that the submonoids of $M$ determined by the faces of $\text{cone}(M)$ account for all divisor-closed submonoids of $M$. Then we appeal to the geometry of $\text{cone}(M)$ to characterize whether $M$ is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in $\mathcal{C}$. Along the way, we determine the cones that can be realized by monoids in $\mathcal{C}$ and by finitary monoids in $\mathcal{C}$.

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

Increasing positive monoids of ordered fields are FF-monoids

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 our main result: every increasing positive monoid of an ordered field is an FF-monoid. Finally, we deduce that every increasing positive monoid is hereditarily atomic.

math.AC↗