Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1711.06961

Systems of sets of lengths of Puiseux monoids

Felix Gotti
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid $P = \langle 1/p \mid p \ \text{is prime} \rangle$ with the Goldbach's conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of Goldbach's numbers.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Felix Gotti. 2017-11-19. Systems of sets of lengths of Puiseux monoids. https://arxiv.org/abs/1711.06961

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

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

Nearly Gorenstein normal graded rings

We investigate nearly Gorenstein property for a normal graded ring $R = \bigoplus_{n\ge 0}R_n$ finitely generated over a field. For that purpose, we investigate ${K_R}^{-1}$, the inverse of $K_R$ (the canonical module of $R$) and introduce a new invariant $b(R)$ of $R$. We investigate nearly Gorenstein property of $R$ using $a(R)$ and $b(R)$ and $m(R)$, the initial degree of $R$. If $b(R)<0$, (and if $R$ is $\mathbb Q$-Gorenstein), then we believe that $R$ is log-terminal -- this is proved if $\dim R=2$ or $R$ is F-pure (or $F$-pure type). Then we determine the condition for a $2$-dimensional cone singularity over a smooth curve of genus $g\le 3$ to be nearly Gorenstein. We observe that ``almost Gorenstein" property and nearly Gorenstein property are drastically different for such rings.

math.AC↗
arXiv2026

Annhilators of local cohomology modules over modular invariant rings and Dickson polynomials

Let $\mathbb{F}_q$ be a finite field with $q = p^s$ elements. Let $V$ be a $d$ dimensional vector space over $\mathbb{F}_q$ and let $G$ be a subgroup of $GL(V)$. Let $R = \mathbb{F}_q[V] = \text{Sym}_{\mathbb{F}_q}(V^*)$ and let $G$ act naturally on $R$. Set $S = R^G$. Let $\mathbf{d}_{d,0}, \mathbf{d}_{d, 1}, \ldots, \mathbf{d}_{d, d-1} \in S$ be the Dickson polynomials with $°\mathbf{d}_{d,i} = q^d - q^i$. Let $I$ be a homogeneous ideal of $S$ and let $H^i_I(S)$ be the $i^{th}$-local cohomology module of $S$ with respect to $I$. Let $J_i = \sqrt{\text{ann} H^i_I(S)}$. Assume $J_i \neq 0$ and $\dim S/J_i = d - g$. Then we show that $\mathbf{d}_{d,0}, \ldots, \mathbf{d}_{d, d - g + 1} \in J_i$. We give several applications of our results. An application is a considerably simpler proof of Landweber-Stong conjecture.

math.AC↗