arXiv subjects

Felix Gotti

Publications and source records attributed to Felix Gotti.

At least 19 recordsLinked to original sources

arXiv2026

Additive factorizations in finitely generated semidomains

For finite $S \subseteq \mathbb{C}$, we study additive factorizations in $\mathbb{N}_0[S]$ and their behavior under adjoining generators. We construct rational and quadratic examples with infinitely many atoms but without atomicity, including a quadratic example in which every pure power of each generator is an atom. Adjoining generators algebraically independent over $\mathbb{Q}(S)$ preserves atomicity and the standard factorization properties, while unique factorization ascends from the individual monoids under joint linear disjointness for positive algebraic generators. Using the theorem of Evertse, Schlickewei, and Schmidt, we prove that positive real generators give finitely many factorizations of each prescribed length. Thus, bounded and finite factorization are equivalent, and half-factoriality implies finite factorization. We exhibit a set of two quadratic generators that separates half-factoriality from unique factorization and give an augmentation criterion for half-factoriality when every monomial is an atom. Lastly, we characterize complete integral closure for positive rational generators.

math.AC↗

arXiv2026

The Bi-LF Positive Conjecture Holds

We prove that a semidomain has length-factorial additive and multiplicative monoids if and only if it is isomorphic to the semiring of nonnegative integers or is a unique factorization domain. Consequently, bi-length-factoriality and bi-factoriality coincide for all semidomains, settling the Bi-LF Positive Conjecture. The main step shows that additive length-factoriality and multiplicative factoriality force additive factoriality, without an additive reducedness assumption. Multiplicative factoriality forces every semidomain with a nonzero additive unit to be a unique factorization domain. In the additively reduced case, a uniform multiplier for lattice points in a finitely generated cone yields cancellation of divisibility across sums, which leads to an obstruction to additive atoms dividing positive integers and completes the structural argument. We also consider related questions on quasi-factoriality and classify the rational and transcendental monogenic positive semirings whose multiplicative monoids are quasi-$n$-factorial for a prescribed integer $n \ge 2$.

math.AC↗

arXiv2026

Maximal common divisors in monoid algebras

A commutative cancellative monoid has the MCD property if every nonempty finite subset has a maximal common divisor, and it has the MCD-finite property if every such subset has only finitely many maximal common divisors up to associates. We study these properties in monoid algebras. We first identify atomic and antimatter families of rank-$1$ torsion-free MCD monoids. We then prove that, for a field $F$ and a torsion-free monoid $M$, the MCD property ascends from $M$ to $F[M]$ whenever the group algebra $F[\mathcal{G}(M)]$ satisfies the ascending chain condition on principal ideals ($\mathcal{G}(M)$ denotes the Grothendieck group of $M$). We also record the pre-Schreier case as a consequence of classical GCD ascent. In contrast, for every prime $p$, we construct a reduced rank-$3$ MCD monoid $M$ such that $\mathbb{F}_p[M]$ is not an MCD domain, and an atomic rank-$1$ MCD monoid whose algebra over $\mathbb{F}_p$ is not atomic. We further prove that the prime reciprocal monoids considered here are MCD-finite and that MCD-finiteness ascends to polynomial extensions. Finally, for every prime $p$, we construct a rank-$1$ torsion-free q-GCD monoid whose algebra over $\mathbb{F}_p$ is not q-GCD, where the q-GCD property requires at most one maximal common divisor up to associates for every nonempty finite subset.

math.AC↗

arXiv2026

Factorizations in algebraic monogenic semidomains

We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_α$. We first relate their structure to the conjugates of $α$, characterize when $S_α$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_α^\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $α$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[α]$ is a finite factorization domain for every algebraic number $α$, establish criteria ensuring that $S_α$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[β]$ for arbitrary algebraic $β$, and prove that, for algebraic integers $α$, unique factorization ascends from $\mathbb{Z}[α]$ to $\mathbb{Z}[α/n]$ for every $n\in\mathbb{N}$.

math.NT↗

arXiv2026

An unrestricted notion of the finite factorization property

A nonzero element of an integral domain (or an element of a commutative cancellative monoid) is called atomic if it is a unit or can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted version of the finite factorization property, extending the work on unrestricted UFDs carried out by Coykendall and Zafrullah in 2004. An integral domain is said to have the unrestricted finite factorization (U-FF) property if every atomic element has only finitely many factorizations, or equivalently, if its atomic submonoid has the finite factorization (FF) property. We position the U-FF property within the hierarchy of classical finiteness conditions, showing that every IDF domain is a U-FFD but not conversely. Then we analyze the behavior of the U-FF property under standard constructions, including localization, polynomial extensions, and the $D+M$ construction: in each case, we find sufficient conditions under which the U-FF property is inherited by the construction. We also prove that, as it is the case with the IDF property, the U-FF property does not ascend to polynomial extensions. Then we extend the known characterization of FFDs as atomic IDF domains by arguing that the FF property and the U-FF property are equivalent over the class of nearly atomic domains (i.e., integral domains having a nonzero principal ideal whose nonzero elements are atomic). These results demonstrate that the U-FF property behaves in many ways analogously to the IDF property. We conclude taking a look at the class of all unrestricted UFDs.

math.AC↗

arXiv2026

Factorizations in rational monogenic semidomains

For $α\in \mathbb{C}$, the monogenic semidomain generated by $α$ is the smallest subsemiring $S_α$ of the complex field $\mathbb{C}$ containing $α$. We initiate a systematic study of the arithmetic and factorizations of the monogenic semidomains $S_q$ generated by rational parameters $q$. After some preliminaries, we introduce and investigate the monoid of technical fractions $T_q$, which is a divisor-closed submonoid of the multiplicative monoid of $S_q$ that encodes a significant amount of arithmetic information about $S_q$. We then study several fundamental factorization properties of $S_q$: the bounded factorization (BF) and finite factorization (FF) properties, the unique factorization (UF) property, and the half-factorial (HF) property. First, we prove that $S_q$ satisfies the UF property if and only if it satisfies the HF property, which happens when $q \in \mathbb{N} \cup \mathbb{N}^{-1}$. We determine all the positive rational values of the parameter $q$ for which $S_q$ satisfies the FF property. Then we show that, over the class of rational monogenic semidomains, the BF property is equivalent to the ascending chain condition on principal ideals. Finally, we prove that $S_q$ is a Krull semidomain if and only if it is root-closed, which happens precisely when $S_q$ satisfies the UF property.

math.AC↗

arXiv2026

The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.

math.GM↗

arXiv2026

On the additive structure of algebraic valuations of polynomial semirings II

For $α\in \mathbb{C}$, let $\mathbb{N}_0[α]$ be the subsemiring of~$\mathbb{C}$ obtained as a homomorphic image of the $α$-evaluation map $\mathbb{N}_0[x] \to \mathbb{C}$ defined as $p(x) \mapsto p(α)$ for each polynomial $p(x) \in \mathbb{N}_0[x]$. Fundamental arithmetic and atomic aspects of the additive structure of $\mathbb{N}_0[α]$ were first studied by the second author and Correa-Morris (2022). In this paper, we continue the investigation, now from the valuation-theoretic perspective. We show that for any algebraic number $α$, the additive monoid of $\mathbb{N}_0[α]$ contains no additive irreducibles if and only if it is isomorphic to the direct product of finitely many isomorphic valuation monoids (monoids whose principal ideals form a chain under inclusion). For any algebraic number $α\in (0,1)$, these valuation monoids are precisely those where $α^{-1}$ is a Perron number having no positive conjugates other than itself. In addition, we offer a description of the algebraic parameters $α$ for which the additive structure of $\mathbb{N}_0[α]$ is a valuation monoid. Finally, we argue that the subset of $(0,1)$ consisting of all algebraic parameters $α$ such that the additive structure of $\mathbb{N}_0[α]$ is a valuation monoid is dense in $(0,1)$.

math.AC↗

arXiv2026

On near atomicity and a characterization of the FF property

A commutative cancellative monoid is atomic if every nonunit factors into atoms, and an integral domain is atomic if its multiplicative monoid of nonzero elements is atomic. Several weakenings of atomicity have been introduced and studied during the past decade, including near atomicity, almost atomicity, and quasi-atomicity. Although nearly atomic monoids that are not atomic were already known, whether there exist nearly atomic integral domains that are not atomic had remained open. We answer this question affirmatively by constructing an explicit nearly atomic integral domain that is not atomic. We also strengthen the classical Anderson--Anderson--Zafrullah characterization of the finite factorization property by proving that an integral domain is an FFD if and only if it is both nearly atomic and IDF. We conclude by showing that near atomicity cannot be weakened to almost atomicity in this characterization, even within the class of IDF domains.

math.AC↗

arXiv2025

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↗

arXiv2025

On the ascent of almost and quasi-atomicity to monoid semidomains

A commutative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral (semi)domain is atomic if its multiplicative monoid is atomic. Notions weaker than atomicity have been introduced and studied during the past decade, including almost atomicity and quasi-atomicity, which were coined and first investigated by Boynton and Coykendall in their study of graphs of divisibility of integral domains. The ascent of atomicity to polynomial extensions was settled by Roitman back in 1993 while the ascent of atomicity to monoid domains was settled by Coykendall and the second author in 2019 (in both cases the answer was negative). The main purpose of this paper is to study the ascent of almost atomicity and quasi-atomicity to polynomial extensions and monoid domains. Under certain reasonable conditions, we establish the ascent of both properties to polynomial extensions (over semidomains). Then we construct an explicit example illustrating that, with no extra conditions, quasi-atomicity does not ascend to polynomial extensions. Finally, we show that, in general, neither almost atomicity nor quasi-atomicity ascend to monoid domains, improving upon a construction first provided by Coykendall and the second author for the non-ascent of atomicity.

math.AC↗

arXiv2025

On finitary power monoids of linearly orderable monoids

A commutative monoid $M$ is called a linearly orderable monoid if there exists a total order on $M$ that is compatible with the monoid operation. The finitary power monoid of a commutative monoid $M$ is the monoid consisting of all nonempty finite subsets of $M$ under the so-called sumset. In this paper, we investigate whether certain atomic and divisibility properties ascend from linearly orderable monoids to their corresponding finitary power monoids.

math.AC↗

arXiv2024

Arithmetic properties encoded in undermonoids

Let $M$ be a cancellative and commutative monoid. A submonoid $N$ of $M$ is called an undermonoid if the Grothendieck groups of $M$ and $N$ coincide. For a given property $\mathfrak{p}$, we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of $M$ satisfy $\mathfrak{p}$ to conclude that all submonoids of $M$ satisfy $\mathfrak{p}$? In this paper, we give a positive answer to this question for the property of being atomic, and then we prove that if $M$ is hereditarily atomic (i.e., every submonoid of $M$ is atomic), then $M$ must satisfy the ACCP, proving a recent conjecture posed by Vulakh and the first author. We also give positive answers to our main question for the following well-studied factorization properties: the bounded factorization property, half-factoriality, and length-factoriality. Finally, we determine all the monoids whose submonoids/undermonoids are half-factorial (or length-factorial).

math.AC↗

arXiv2024

On primality and atomicity of numerical power monoids

In the first part of this paper, we establish a variation of a recent result by Bienvenu and Geroldinger on the (almost) non-existence of absolute irreducibles in (restricted) power monoids of numerical monoids: we argue the (almost) non-existence of primal elements in the same class of power monoids. The second part of this paper, devoted to the study of the atomic density of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$, is motivated by work of Shitov, a recent paper by Bienvenu and Geroldinger, and some questions pointed out by Geroldinger and Tringali. In the same, we study atomic density through the lens of the natural partition $\{ \mathcal{A}_{n,k} : k \in \mathbb{N}_0\}$ of $\mathcal{A}_n$, the set of atoms of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$ with maximum at most $n$: \[ \mathcal{A}_{n,k} = \{A \in \mathcal{A} : \max A \le n \text{ and } |A| = k\} \] for all $n,k \in \mathbb{N}$, where $\mathcal{A}$ is the set of atoms of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$. We pay special attention to the sequence $(α_{n,k})_{n,k \ge 1}$, where $α_{n,k}$ denote the size of the block $\mathcal{A}_{n,k}$. First, we establish some bounds and provide some asymptotic results for $(α_{n,k})_{n,k \ge 1}$. Then, we take some probabilistic approach to argue that, for each $n \in \mathbb{N}$, the sequence $(α_{n,k})_{k \ge 1}$ is almost unimodal. Finally, for each $n \in \mathbb{N}$, we consider the random variable $X_n : \mathcal{A}_n \to \mathbb{N}_0$ defined by the assignments $X_n : A \mapsto |A|$, whose probability mass function is $\mathbb{P}(X_n=k) = α_{n,k}/| \mathcal{A}_n|$. We conclude proving that, for each $m \in \mathbb{N}$, the sequence of moments $(\mathbb{E}(X_n^m))_{n \ge 1}$ behaves asymptotically as that of a sequence $(\mathbb{E}(Y_n^m))_{n \ge 1}$, where $Y_n$ is a binomially distributed random variable with parameters $n$ and $\frac12$.

math.CO↗

arXiv2024

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid $M$ and an integral domain $R$ are both atomic implies that the monoid algebra $R[M]$ of $M$ over $R$ is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

math.AC↗

arXiv2024

One-dimensional monoid algebras and ascending chains of principal ideals

An integral domain $R$ is called atomic if every nonzero nonunit of $R$ factors into irreducibles, while $R$ satisfies the ascending chain condition on principal ideals if every ascending chain of principal ideals of $R$ stabilizes. It is well known and not hard to verify that if an integral domain satisfies the ACCP, then it must be atomic. The converse does not hold in general, but examples are hard to come by and most of them are the result of crafty and technical constructions. Sporadic constructions of such atomic domains have appeared in the literature in the last five decades, including the first example of a finite-dimensional atomic monoid algebra not satisfying the ACCP recently constructed by the second and third authors. Here we construct the first known one-dimensional monoid algebras satisfying the almost ACCP but not the ACCP (the almost ACCP is a notion weaker than the ACCP but still stronger than atomicity). Although the two constructions we provide here are rather technical, the corresponding monoid algebras are perhaps the most elementary known examples of atomic domains not satisfying the ACCP.

math.AC↗

arXiv2024

Atomicity in integral domains

In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and, therefore, divisibility by and factorizations into atoms are not possible. In the other (more interesting) side of the spectrum, we find the atomic algebraic structures, where essentially every element factors into atoms (the study of such objects is known as factorization theory). In this paper, we survey some of the most fundamental results on the atomicity of cancellative commutative monoids and integral domains, putting our emphasis on the latter. We mostly consider the realm of atomic domains. For integral domains, the distinction between being atomic and satisfying the ascending chain condition on principal ideals, or ACCP for short (which is a stronger and better-behaved algebraic condition) is subtle, so atomicity has been often studied in connection with the ACCP: we consider this connection at many parts of this survey. We discuss atomicity under various classical algebraic constructions, including localization, polynomial extensions, $D+M$ constructions, and monoid algebras. Integral domains having all their subrings atomic are also discussed. In the last section, we explore the middle ground of the spectrum of atomicity: some integral domains where some of but not all the elements factor into atoms, which are called quasi-atomic and almost atomic. We conclude providing techniques from homological algebra to measure how far quasi-atomic and almost atomic domains are from being atomic.

math.AC↗

arXiv2024

Riemann zeta functions for Krull monoids

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.

math.NT↗