The complexity of a numerical semigroup

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