论文标题
Abelian Monoid中相同长度的因素化
Factorizations of the same length in abelian monoids
论文作者
论文摘要
令$ \ MATHCAL S \ subseteq \ Mathbb z^m \ oplus t $成为有限生成和减少的单体。在本文中，我们制定了一种一般策略来研究$ \ Mathcal s $中至少两个相同长度的因素的元素集，即理想的$ \ MATHCAL L _ {\ MATHCAL S} $。为此，我们与与Monoid $ \ Mathcal S $相关的某种理想（晶格）合作。我们的研究可以看作是一种概括\ cite {Chapman：2011}的新方法，它仅研究数值半群的情况。当$ \ mathcal s $是数值半群时，我们给出三个主要结果：（1）我们明确计算一组理想的$ \ Mathcal l _ {\ Mathcal s} $的发电机时，当$ \ Mathcal s $几乎是由几乎算术序列生成的$ \ MATHCAL S $时； （2）我们提供了一个无限的数字半群家族，例如$ \ MATHCAL L _ {\ MATHCAL S} $是主要的理想； （3）我们将不在$ \ MATHCAL L _ {\ MATHCAL S} $中的最大整数分类为$ \ MATHCAL {NP} $ - 硬问题。
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 of generators of the ideal $\mathcal L_{\mathcal S}$ when $\mathcal S$ is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that $\mathcal L_{\mathcal S}$ is a principal ideal; (3) we classify the computational problem of determining the largest integer not in $\mathcal L_{\mathcal S}$ as an $\mathcal{NP}$-hard problem.
