Resumen de Modular Frobenius pseudo-varieties

Aureliano Matías Robles Pérez, José Carlos Rosales González

• If m \in {\mathbb {N}} \setminus \{0,1\} 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{aligned} {\mathscr {C}}(m,A) =&\{ S\in {\mathscr {S}}_m \mid s_1+\cdots +s_k-m \in S \text { if } (s_1,\ldots ,s_k)\in S^k \text { and } \\ {}&\qquad (s_1 \bmod m, \ldots , s_k \bmod m)\in A \}. \end{aligned} 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.