arXiv 2202.009202022-02-02math.NT

数值半群的复杂性

The complexity of a numerical semigroup

J. I. García-García, M. A. Moreno-Frías, J. C. Rosales, A. Vigneron-Tenorio

摘要：

设$S$和$Δ$是数值半群.一个数值半群$S$是$\mathbf{I}(Δ)$ -{ \it半群}，如果$S\backslash \{0\}$是$Δ$的理想.我们将表示为$\mathcal{J}(Δ)=\{S \mid S \text{ is an $ \mathbf{I}（Δ）$-semigroup} \}.$我们将说$Δ$是{ \it是} $S$的理想扩张，如果$S\in \mathcal{J}(Δ).$在这项工作中，我们提出了一个算法，允许建立一个数值半群的所有理想扩张。对于所有的$k\in \mathbb{N}.$，我们可以递归地表示为$\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$和$\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$。数值半群$S$的复杂度是集合$\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$的最小值。

英文摘要：

Let $S$ and $Δ$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(Δ)$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $Δ$. We will denote by $\mathcal{J}(Δ)=\{S \mid S \text{ is an $\mathbf{I}(Δ)$-semigroup} \}.$ We will say that $Δ$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(Δ).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and $\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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