唯一表现的n-生成交换幺半群

On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element $n$ of a cancellative commutative monoid $S$ is a nonnegative integer measuring the distance between the irreducible factorizations of $n$. The catenary degree of the monoid $S$, defined as the supremum over all catenary degrees occurring in $S$, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set $\mathsf C(S)$ of catenary degrees achieved by elements of $S$ as a factorization invariant, focusing on the case where $S$ in finitely generated (where $\mathsf C(S)$ is known to be finite). Answering an open qu…

Systems of sets of lengths of Puiseux monoids

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…

W-素性和渐近W-素性的计算及其在数值半群中的应用

Computation of the w-primality and asymptotic w-primality with applications to numerical semigroups

我们给出了一个算法来计算随机生成的原子幺半群的$ω$ -素性。渐近$\w$素性也进行了研究，并证明了一个公式，以获得它在生成的拟阿基米德幺半群。该配方适用于数值半群，获得表达的不变量在其系统的发电机。

Factorization Theory in Commutative Monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and th…

交换幺半群上的预序空间

The space of preorders on a commutative monoid

研究有限生成交换幺半群上与幺半群结构兼容的全预序，用分治技术给出构造性描述，证明其通用理论可判定，还得到多项式环上自由模单项式序新描述。

Noetherianity for infinite-dimensional toric varieties

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a …

从单式词到词再到图

From Monomials to Words to graphs

给定一个有限的字母表X和字母的排序，映射σ将X上的每个单项式发送到单项式中字母幂的有序乘积的单词。受Groebner基问题的启发，我们用生成集的形式刻画了自由交换幺半群中的理想I，使得自由交换幺半群中由σ（I）生成的理想<σ（I）>是σ-生成的.是否存在一个序使得<σ（I）>是非线性生成的，证明是NP-完全的。后者的问题是密切相关的识别问题的可比性图。

分裂F-正则幺半群代数的有限生成

Finite generation of split F-regular monoid algebras

本文证明素特征域上分裂F-正则幺半群代数是有限生成的，支持相关诺特性猜想，关键工具为凸几何中的丢番图逼近。

Idempotence of finitely generated commutative semifields

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

Factorization in monoids and rings

Let $H^\times$ be the group of units of a multiplicatively written monoid $H$. We say $H$ is acyclic if $xyz \ne y$ for all $x, y, z \in H$ with $x \notin H^\times$ or $z \notin H^\times$; unit-cancellative if $yx \ne x \ne xy$ for all $x, y \in H$ with $y \notin H^\times$; f.g.u. if there is a finite set $A \subseteq H$ such that every non-unit of $H$ is a finite product of elements of the form $uav$ with $u, v \in H^\times$ and $a \in A$; l.f.g.u. if, for each $x \in H$, the smallest divisor-closed submonoid of $H$ containing $x$ is f.g.u; and atomic if every non-unit can be written as a fin…

自由交换幺半群的理想扩张

Ideal extensions of free commutative monoids

本文引入间隙吸收幺半群，证明其链次度至多四且长度集为区间，并给出理想扩张的ω-素性界及有限间隙的等价条件。