幺半群的纯性与半群环中的无特征分裂

Apéry sets and the ideal class monoid of a numerical semigroup

The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$ whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the Apéry sets and the Kunz coordinates of $S$. We study some combinatorial and algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to…

数值半群与算术同余幺半群分解的$p$-长度的若干渐近结果

Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids

本文引入$p$-长度（分解长度的广义概念，基于分解指数序列的$\ell_p$-范数），给出数值半群与算术同余幺半群中大型元素分解极值$p$-长度的渐近结果，揭示其组合特性与一般幺半群的微妙性。

On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if $G$ is abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper w…

Factorizations of the same length in abelian monoids

Let $\mathcal S \subseteq \mathbb Z^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in $\mathcal S$ having at least two factorizations of the same length, namely the ideal $\mathcal L_{\mathcal S}$. To this end, we work with a certain (lattice) ideal associated to the monoid $\mathcal S$. Our study can be seen as a new approach generalizing \cite{chapman:2011}, which only studies the case of numerical semigroups. When $\mathcal S$ is a numerical semigroup we give three main results: (1) we compute explicitly a set…

同构意义下的广义数值半群

Generalized Numerical semigroups up to isomorphism

本文研究了广义数值半群的同构问题，通过坐标置换刻画同构关系，提出了生成给定亏格下所有同构广义数值半群的程序，并展示了相关计算数据与性质。

枚举与数值半群相关的数值集合

Enumerating numerical sets associated to a numerical semigroup

该研究引入数值半群的空隙偏序集，建立以给定半群为原子幺半群的数值集合与该偏序集序理想的双射关系，以此解答小类型半群下的反原子问题。

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

We give an algorithm to compute the $ω$-primality of finitely generated atomic monoids. Asymptotic $\w$-primality is also studied and a formula to obtain it in finitely generated quasi-Archimedean monoids is proven. The formulation is applied to numerical semigroups, obtaining an expression of this invariant in terms of its system of generators.

On the Atomic Structure of Puiseux Monoids

In this paper, we study the atomic structure of the family of Puiseux monoids. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-nineteenth century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criter…