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Let $\Delta$ be a numerical semigroup. In this work we show that $𝒥\left(\Delta \right)=\{I\cup \{0\}:I\text{is}\text{an}\text{ideal}\text{of}\Delta \}$ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set ${𝒥}_a\left(\Delta \right)=\{S\in 𝒥\left(\Delta \right):max(\Delta \setminus S)=a\}$ for a given $a\in \Delta .$ As a consequence, we obtain another algorithm that computes all the elements of $𝒥\left(\Delta \right)$ with a fixed genus.

Let $S$ be a numerical semigroup. We say that $h\in \mathbb{N}\setminus S$ is an isolated gap of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $\mathrm{m}\left(S\right)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$, and $S\setminus \left\{\mathrm{m}\right(S\left)\right\}\in 𝒞$ for all $S\in 𝒞$ such that $S\ne min\left(𝒞\right).$ We prove that the set $𝒫\left(F\right)=\{S:S$ is a perfect numerical semigroup with...