Modular Frobenius pseudo-varieties

If $$m \in {\mathbb {N}} \setminus \{0,1\}$$ m ∈ N \ { 0 , 1 } and A is a finite subset of $$\bigcup _{k \in {\mathbb {N}} \setminus \{0,1\}} \{1,\ldots ,m-1\}^k$$ ⋃ k ∈ N \ { 0 , 1 } { 1 , … , m - 1 } k , then we denote by $$\begin{aligned} {\mathscr {C}}(m,A) =&\{ S\in {\mathscr {S}}_m \mid s_1+\cdots +s_k-m \in S \text { if } (s_1,\ldots ,s_k)\in S^k \text { and } \\ {}&\qquad (s_1 \bmod m, \ldots , s_k \bmod m)\in A \}. \end{aligned}$$ C ( m , A ) = { S ∈ S m ∣ s 1 + ⋯ + s k - m ∈ S if ( s 1 , … , s k ) ∈ S k and ( s 1 mod m , … , s k mod m ) ∈ A } . In this work we prove that $${\mathscr {C}}(m,A)$$ C ( m , A ) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $${\mathscr {C}}(m,A)$$ C ( m , A ) and to compute all the elements of $${\mathscr {C}}(m,A)$$ 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 $${\mathscr {C}}(m,A)$$ C ( m , A ) when $$A=\{1,\ldots ,m-1\}^3$$ A = { 1 , … , m - 1 } 3 , $$A=\{(1,1),\ldots ,(m-1,m-1)\}$$ A = { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , and $$A=\{1,\ldots ,m-1\}^2 \setminus \{(1,1),\ldots ,(m-1,m-1)\}$$ A = { 1 , … , m - 1 } 2 \ { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , respectively.