Modular Frobenius pseudo-varieties

If $m\in \mathbb{N}$ and $A$ is a finite subset of $\bigcup_{k \in $\mathbb{N}$ \setminus {0,1}} {1,\ldots,m-1}^k$, then we denote by \begin{align*} \mathscr{C}(m,A) = \left{S\in \mathscr{S}_m \mid s_1+\cdots+s_k-m \in S \mbox{ if } (s_1,\ldots,s_k)\in S^k \mbox{ and }\right. \ \left.(s_1 \bmod m, \ldots, s_k \bmod m)\in A \right}. \end{align*} In this work we prove that $\mathcal{C}(m,A)$ is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $\mathcal{C}(m,A)$ and to compute all the elements of $\mathcal{C}(m,A)$ with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $\mathcal{C}(m,A)$ when $A=\{1,\dots ,m-1{\}}^3$, $A=\{(1,1),\dots ,(m-1,m-1)\}$, and $A={1,\ldots,m-1}^2 \setminus {(1,1),\ldots,(m-1,m-1)}$, respectively.