AbstractIf $$m \in {\mathbb {N}} \setminus \{0,1\}$$
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.
Modular Frobenius pseudo-varieties
