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 $\mathscr{ C}(m,A)$ is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $\mathscr{ C}(m,A)$ and to compute all the elements of $\mathscr{ C}(m,A)$ with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $\mathscr{ C}(m,A)$ when $A=\{ 1,\ldots,m-1\}^3$, $A=\{ (1,1),\ldots,(m-1,m-1)\}$, and $A=\{ 1,\ldots,m-1\}^2 \setminus \{ (1,1),\ldots,(m-1,m-1)\}$, respectively.
Categories: math.GR