论文标题
数值半群的复杂性
The complexity of a numerical semigroup
论文作者
论文摘要
令$ s $和$δ$为数值半群。如果$ s \ s \ backslash \ {0 \} $是$Δ$的理想，则数值半群$ s $是$ \ mathbf {i}（δ）$ - {\ it semigroup}。我们将以$ \ MATHCAL {J}（δ）= \ {s \ mid s \ text {是$ \ mathbf {i}（δ）$ - semigroup} \}。允许构建数值半群的所有理想扩展的算法。我们可以用$ \ Mathcal {J}^0（\ Mathbb {n}）= \ Mathbb {n}，$ $ $ \ Mathcal {J}^1（\ Mathbb {N}）= \ Mathcal = \ Mathcal {J}（J Mathbb {n n}）$递归表示$ \ MATHCAL {J}^{K+1}（\ MathBb {n}）= \ Mathcal {J}（\ Mathcal {\ Mathcal {J}^{K}^{K}（\ Mathbb {n}）$ in \ Mathbb {n}。 $ \ {k \ in \ mathbb {n} \ mid s \ in \ mathcal {j}^k（\ mathbb {n}）\} \ $ $。此外，我们还将给出一种算法，使我们能够计算具有固定多重性和复杂性的所有数字半元素。
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.
