Alfred Geroldinger, Qinghai Zhong
Let $\mathcal L^*$ be a family of finite subsets of $\mathbb N_0$ having the following properties. (a). $\{0\}, \{1\} \in \mathcal L^*$ and all other sets of $\mathcal L^*$ lie in $\mathbb N_{\ge 2}$. (b). If $L_1, L_2 \in \mathcal L^*$, then the sumset $L_1 + L_2 \in \mathcal L^*$. We show that there is a Dedekind domain $D$ whose system of sets of lengths equals $\mathcal L^*$.
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We show that for every finite nonempty set L of integers greater than or equal to 2 there are a numerical monoid H and a squarefree element a $\in$ H whose set of lengths L(a) is equal to L.
We explicitly determine those sets of nonnegative integers which occur as sets of lengths in all numerical monoids.
Let $H$ be an atomic monoid. The set of distances $Δ(H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. It is well-known (and easy to show) that, if $Δ(H)$ is nonempty, then $\min Δ(H) = \gcd Δ(H)$. In this paper we show conversely that for every finite nonempty set $Δ\su…
The delta set of a numerical semigroup $S$, denoted $Δ(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min Δ(S) = \gcd Δ(S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$.
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 positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x^n in R[x], where (R, m) is an Artinian local ring with m^2 = 0.
On the structure of length sets with maximal elasticity
研究具有最大弹性的元素的长度集结构，证明其通常为区间。
The structure theorem for sets of length for numerical semigroups
本文针对任意数值半群证明了特化版长度集结构定理，利用由原半群生成元导出的两个数值半群的Apéry集及其间隙刻画大元素长度集的缺失长度，并给出满足定理的元素的明确界值。
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…
Let $H$ be an atomic monoid. For $x \in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H) = \{\mathsf{L}(x) \mid x \in H\}$. On the other hand, the elasticity of $x$, denoted by $ρ(x)$, is the quotient $\sup \mathsf{L}(x)/\inf \mathsf{L}(x)$ and the elasticity of $H$ is the supremum of the set $\{ρ(x) \mid x \in H\}$. The system of sets of lengths and the elasticity of $H$ both measure how far is $H$ from being half-factorial, i.e., $|\mathsf{L}(x)| = 1$ for each $x \in H$. …
Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multipl…
Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains
研究整数值多项式环Int(M,D)中长度集的构造与非转移克鲁尔域性质，探讨其在不同环结构下的应用。
Let $S=\langle a_1,\ldots,a_p\rangle$ be a numerical semigroup, $s\in S$ and ${\sf z}(s)$ its set of factorizations. The set of length is denoted by ${\mathcal L}(s)=\{{\tt L}(x_1,\dots,x_p)\mid (x_1,\dots,x_p)\in{\sf Z}(s)\}$ where ${\tt L}(x_1,\dots,x_p)=x_1+\ldots+x_p$. From these definitions, the following sets can be defined ${\textsf W}(n)=\{s\in S\mid \exists x\in{\sf z}(s) \textrm{ such that } {\tt{L}}(x)=n\}$, $ν(n)=\cup_{s\in {\textsf W}(n)} {\mathcal L}(s)=\{l_1<l_2<\ldots< l_r\}$ and $Δν(n)=\{l_2-l_1,\ldots,l_r-l_{r-1}\}$. In this paper, we prove that the set $Δν(S)=\cup_{n\in{\mat…
Some of the fundamental notions related to sets of lengths of Krull monoids with finite class group are discussed, and a survey of recent results is given. These include the elasticity and related notions, the set of distances, and the structure theorem for sets of lengths. Several open problems are mentioned.
Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\mathcal L (H)$ is additively closed under set addition.
Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, s…
Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. Then sets of lengths in H have a well-defined structure which just depends on the class group G. With methods from additive combinatorics we establish a characterization of those class groups G guaranteeing that all sets of lengths are (almost) arithmetical progressions.
We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $Δ_ρ (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $ρ(H)$. We study $Δ_ρ (H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal t…
For an element $a$ of a monoid $H$, its set of lengths $\mathsf L (a) \subset \mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \cdot \ldots \cdot u_k$ into $k$ atoms. We study the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ with a focus on the unions $\mathcal U_k (H) \subset \mathbb N$ which are the unions of all sets of lengths containing a given $k \in \mathbb N$. The Structure Theorem for Unions -- stating that for all sufficiently large $k$, the sets $\mathcal U_k (H)$ are almost arithmetical progressions with the same difference a…
On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences
本文研究加法阿贝尔群上 plus-minus 加权零和序列幺半群中的长度集合，证明了当群有限时长度集合高度结构化，而当群无限时 $\mathbb N_{\ge 2}$ 的任意有限非空子集均可作为某序列的长度集合。
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal L (H)$ of all sets of lengths of $H$ and study when $\mathcal L (H)$ contains or is contained in a system $\mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $\mathsf D (G')=\mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m \ge 7$ or an elementary $2$-group of rank $m-1 \ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $\ma…
Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \ldots u_k$, and the number of factors $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the group…
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