1. Introduction
Let be the set of integer numbers, and . A submonoid of is a subset of a closed under addition and contains 0. A numerical semigroup S is a submonoid of , such that is finite.
If S is a numerical semigroup, then , , and , the cardinal of the set , are three important invariants of S called the multiplicity, the Frobenius number, and the genus of S, respectively.
If
A is a non-empty subset of
, then we denote by
the submonoid of
generated by
A, i.e.,
. In ([
1], Lemma 2.1), it is shown that
is a numerical semigroup if and only if
.
If
M is a submonoid of
and
, then we say that
A is a
system of generators of M. Furthermore, if
for all
, then we say that
A is a
minimal system of generators of M. In ([
1], Corollary 2.8), it is shown that for any
M submonoid of
, there exists a unique minimal system of generators, and it is finite. We denote this minimal system of generators as
. Its cardinality is called the
embedding dimension ofM; we will denote it as
.
The Frobenius problem for numerical semigroups (see [
2]) is to find formulas to compute the Frobenius number and the genus of a numerical semigroup using its minimal system of generators. This problem was solved in [
3] for numerical semigroups with an embedding dimension equal to 2. Today, the problem is still open for numerical semigroups with a higher embedding dimension.
An unfamiliar reader with the numerical semigroup can find these names a bit strange; however, there exists a large number of publications devoted to the study of analytically irreducible unidimensional local domains using the semigroups of values, which is a numerical semigroup and where the reader can find the connection and the interpretation of these invariants (see, for example, [
4,
5,
6,
7,
8]). Following this question, there also appear interesting classes of numerical semigroups as numerical semigroups having a maximal embedding dimension, called
MED-semigroups (see [
4,
9,
10]).
In ([
1], Proposition 2.10), it is shown that for any numerical semigroup
S, we have
. We say that
S is a maximal embedding dimension semigroup if
. In ([
1], Proposition 3.12), it is shown that a numerical semigroup has a maximal embedding dimension if and only if
for every
.
Following the terminology used in [
11], a numerical semigroup is
second-level (or a
second-level semigroup) if
, for every
. Note that every numerical semigroup with a maximal embedding dimension is a second-level numerical semigroup.
Let
m be an integer greater than or equal to 2. We denote
In
Section 2, we show that
is a Frobenius pseudo-variety. Using several results from [
12], we deduce an algorithm to obtain all the elements in
for a fixed genus. Let
f be a positive integer. We denote
. In
Section 3, we show that
is a ratio-covariety. The results in [
13] allow us to present an algorithm to compute all the elements in
.
A
-
monoid is a submonoid of
, which can be written as the intersection of elements in
. For
, we denote by
the intersection of all elements in
containing
A. It is clear that
is the smallest
-monoid containing
A. In
Section 4, we give an algorithm to compute
. We also show that
belongs to
if and only if
.
If
, then we say that
A is a
-
system of generators for
M. Moreover, if
for every
, then we say that
A is a
minimal-
system of generators for
M. In
Section 4, we show that every
-monoid
M has a unique minimal
-system of generators, which will be denoted as
-
. We define the
-
rank of
M as the cardinality of
-
.
In
Section 5, we focus on the elements of
with the
-rank being equal to 1. These semigroups will be
for some
, such that
and
. We provide the minimal system of generators for
and solve the Frobenius problem for these semigroups.
Given S, a numerical semigroup, and x, an integer number, we say that x is a pseudo-Frobenius number of S if and for every . The set of all pseudo-Frobenius numbers will be denoted as . We define the type of S as the cardinality of , and we will denote it as .
In
Section 5, we show that
for any
with the
-rank being equal to 1.
For a numerical semigroup
S, we denote the set of
small numbers by
, and the cardinality of N(S) is denoted as
. H.S. Wilf conjectured in [
14] that
. Today, this question is still open, and it has become one of the most important challenges in the theory of numerical semigroups. This conjecture has been proved for a large number of families of numerical semigroups. Readers interested in this problem may refer to [
15,
16,
17,
18,
19,
20].
In
Section 5, we will show that the elements in
with the
-rank being equal to 1 verify Wilf’s conjecture.
2. Elements in for a Fixed Genus
In this section, m will be an integer greater than or equal to 2.
Let g be a positive integer. We denote . The aim of this section is to obtain an algorithm to compute .
The following result is easy to check:
Lemma 1. is the maximum of with respect to the inclusion order, where means that every integer bigger than belongs to the set.
For such that , we have .
If , then . For with , we denote .
Proposition 1. Take . Then, if and only if .
Proof. Necessity. If , then and . Applying Lemma 1, we have so .
Sufficiency. If , it is clear that . Applying Lemma 1, we have . On the other hand, the number of positive elements in smaller than is , so , and . □
A Frobenius pseudo-variety is a non-empty family of numerical semigroups satisfying the following properties:
The family has a maximum element (with respect to the inclusion order);
If , then ;
If and , then .
The following result can be deduced from [
11], Proposition 2.6 and Corollary 3.4.
Proposition 2. is a Frobenius pseudo-variety and is its maximum.
A graph G is a pair , where V is a non-empty set and . The elements in V are called the vertices of G, and the elements in E are called the edges of G. A path of length n connecting the vertices x and y of G is a sequence of different elements in E, , where and .
A graph G is called a tree if there exists a vertex r named the root of G, such that for every x of G there exists a unique path joining x and r. Finally, if is an edge of a tree, we say that x is a child of y.
We define the graph as follows: The vertices of this graph are the elements of , and is an edge if and only if .
As a consequence of Proposition 2, ([
12], Lemmas 11 and 12, and Theorem 3), we can obtain the following result:
Theorem 1. is a tree whose root is . Moreover, the set of children of a vertex S of is .
The following result is easy to check:
Lemma 2. Let S be a numerical semigroup, and let . Then, is a numerical semigroup if and only if . Moreover, .
Let S be a numerical semigroup. We consider . If , then we denote .
Lemma 3 ([
11], Proposition 3.5)
. Let , and let . Then, we have if and only if . A tree can be recurrently built starting from the root and by connecting each vertex with its children using an edge. Also, note that elements in
with genus
g are children of elements in
with genus
. Following this idea and the above results, we present Algorithm 1 to compute
.
Algorithm 1 |
Input: Integers m and g, such that . Output: . , ; If , then return A, and the algorithm is finished; For every compute ; ; and go back to (2).
|
Example 1. Using Algorithm 1, we are going to compute .
, ;
;
, ;
; ; ;
, and the algorithm is finished.
Algorithm 1 returns .
3. Elements in for a Fixed Frobenius Number
In this section, m will be an integer greater than or equal to 2.
Let f be a positive integer. We consider . The objective of this section is to obtain an algorithm to compute .
Proposition 3. Take . Then, if and only if and .
Proof. Necessity. If , then , so . Moreover, since , then . Hence, .
Sufficiency. Observe that . □
If and , then . In the rest of this section, we assume , such that and .
Let S be a numerical semigroup, such that . The ratio of S is . Note that .
A ratio-covariety is a family of numerical semigroups satisfying the following properties:
The family has a minimum element (with respect to the inclusion order);
If , then ;
If and , then .
The following result is easy to prove:
Lemma 4. Let be two numerical semigroups. Then, is also a numerical semigroup, and .
We denote .
Proposition 4. is a ratio-covariety and is its minimum.
Proof. It is clear that is the minimum of . If , then using Lemma 4, we deduce . To finish the proof, we see that if such that , then . As , then , and we obtain that is a numerical semigroup with multiplicity m and a Frobenius number f using Lemma 2. On the other hand, if , then . Furthermore, , so . From this, we have . □
Let S be a numerical semigroup, and . We say that x is a special gap of S if and is a numerical semigroup. We denote by the set of special gaps of S.
We define the graph as follows: The set of vertices is , and is an edge if and only if .
Using Proposition 4, [
13], and Propositions 3 and 4, we obtain the following result:
Theorem 2. is a tree whose root is . Moreover, the set of children of a vertex S is .
The following result is Proposition 3.1 of [
11].
Proposition 5. Let S be a numerical semigroup with multiplicity m. Then, if and only if for every .
Corollary 1. Consider , and , such that . Then, if and only if for every .
Proof. To apply Proposition 5, we need to observe that . □
Let
S be a numerical semigroup, and
, we define (in honor of [
21]) the Apéry set of
n in
S as
. The following result appears in [
1], Lemma 2.4:
Lemma 5. Let S be a numerical semigroup and . Then, , where is the smallest element in S congruent with i modulo n.
Remark 1. Let S be a numerical semigroup and assume that , is known; then, we have the following:
- 1.
We can solve the problem of belonging to S as follows: Any integer x belongs to S if and only if .
- 2.
We can obtain the special gaps of S. From ([22], Remark 1), we have an easy way to compute them as [22], Lemma 3.4, where [23], Lemma 10. Here, is the partial order defined on as , if and only if . - 3.
From ([22], Lemma 3.5), we have , for every . - 4.
Finally, as , we can obtain from (3).
Using the above ideas, we present Algorithm 2. It allow to compute
being
f and
m integers under certain conditions.
Algorithm 2 |
Input: Integers f and m such that and . Output: . and . Compute . For every , compute . If , then return , and the algorithm is finished. . and . For every , compute and go back to (3).
|
Note that the computation of in line 7 is used to compute the set in a new loop, as we point out in Remark 1.
Example 2. Using Algorithm 2, we compute .
First loop of the algorithm is:
- (1)
and .
- (2)
.
Therefore, . .
Moreover, .
Meanwhile, and .
Then, we have
- (3)
, as .
- (5)
.
- (6)
and .
- (7)
and .
Therefore,
- (a)
. .
Moreover, .
Meanwhile, and .
- (b)
. .
Moreover, .
Meanwhile, and .
Then, second loop is
- (3)
, as
and , as .
- (5)
.
- (6)
and .
- (7)
.
Therefore, . .
Moreover, .
Meanwhile, and .
Then, the third loop is
- (3)
, as .
- (4)
and the algorithm is finished.
Algorithm 2 returns .
4. -Systems of Generators
It is clear that and . In this section, m will be an integer greater than or equal to 3.
From Proposition 2, we know that the intersection of a finite number of elements in is again an element in . However, when we consider an infinite number of them, the intersection is generally not an element in , as shown in the following example:
Example 3. For any , we consider . Clearly, . However, .
The finite or infinite intersection of elements in is always a submonoid of . This fact allows us to give the following definition. A -monoid is a submonoid of that can be written as an intersection of elements in .
The following result is easy to check:
Lemma 6. The intersection of -monoids is again a -monoid.
Consider . We denote by the intersection of all the -monoids containing A. As a consequence of Lemma 6, we find that is the smallest (with respect to the inclusion) -monoid containing A.
We can easily reach the following result:
Lemma 7. Let A be a subset of . Then, is equal to the intersection of all the elements in containing A.
If , then we will say that A is a -system of generators of M. Furthermore, if for any , then A is called a minimal-system of generators of M.
The following result is easy to deduce from Proposition 2 and [
12], Corollary 1.
Proposition 6. Every -monoid has a unique minimal -system of generators. Furthermore, such a minimal -system of generators is finite.
Let M be a -monoid. We denote - as the minimal -system of generators of M. We define the -rank of M as the cardinality of this minimal -system of generators.
Proposition 7. Let M be a submonoid of . Then, the following statements are equivalent.
M is a -monoid.
and for every .
and for every .
Proof. . If M is a -monoid, then there exists a family of elements in , such that . Hence, . On the other hand, if , then for every , so for every , and then .
. Trivial.
. For every , we consider . From Proposition 5, we easily deduce that for every . It is clear that , so M is a -monoid. □
Proposition 7 allows us to present Algorithm 3 which compute
for any
.
Algorithm 3 |
Input: A finite subset A of . Output: . . If , then return M, and the algorithm is finished. and go back to (2).
|
Now, we present two different examples to show how Algorithm 3 works. In the first example, we calculate , and we will see that belongs to . In the second example, we calculate , and we will show that does not belong to . This raises the following question: What is the condition for to belong to ?
Example 4. Let us calculate using Algorithm 3.
Algorithm 3 returns .
Example 5. Let us calculate using Algorithm 3.
Algorithm 3 returns . Since , then is not a numerical semigroup; therefore, .
Proposition 8. Consider . Then, if and only if .
Proof. Necessity. If , then using Proposition 7, we deduce is a -monoid. It is clear that and . From this, we have that is not a numerical semigroup. Consequently, .
Sufficiency. We have that is a submonoid of and . As , we deduce , and then is a numerical semigroup. Consequently, belongs to . □
5. Elements in with -Rank Equal to 1
In this section, m denotes an integer greater than or equal to 3.
Proposition 9. Consider . Then, the following statements are equivalent:
- 1.
The -rank of S is equal to 1.
- 2.
for some integer r, such that and .
Proof. . If -rank , then there exists such that -msg . Hence, and . Using Proposition 8, we have , which implies .
. It is clear that -msg . Then, -rank . □
In the rest of this section, r will be an integer greater than m such that . We also denote .
Proposition 10. .
Proof. First of all, we will show, by induction on k, that . For , the result is clear because . We assume the statement to be true for k; then, we prove it for . As and , then and so .
Therefore, we have that and consequently .
Now, we will see that . For this, we only need to prove .
Consider . Then, as . Then, Proposition 5 ensures that .
Consequently, we have . □
As a consequence of Proposition 10, we have . We also have . Then, for every . Then, we have the following result.
Lemma 8. If , then we have one of the following asserts:
- 1.
for some .
- 2.
for some .
As
, we have
Theorem 3. If , thenfor every . Proof. If k is odd, then there exists such that , so . However, if k is even, then for some , and we have because .
Now, we will first see that in the case in which k is odd, we have . Assume the contrary. By Lemma 8, we consider two cases:
There exist and such that . Then, . However, since , we have . Hence, there exists such that . Consequently, and then . Therefore, . Hence, , and , which is a contradiction, since . Observe that . Otherwise, we have , which is impossible.
There exist and , such that . Then, . Hence, for some . Therefore, . Then, following the same reasoning as in (1), we have . Hence, . Consequently, , which is again a contradiction. Observe that . Otherwise, we have , which is impossible.
We then focus on the case where k is even and try to prove that . As above, we assume the contrary, and using Lemma 8 we consider two cases:
There exist and such that . Then, , since we have , and so there exists such that . Then, and then . Therefore, , which implies that and , and we obtain the same contradiction. As we have commented in the above paragraph, l must be different from 0. Otherwise, .
There exist and such that . Then, , and we can write for some . Therefore, . Consequently, . Then, , allowing us to obtain , which is a contradiction. Again, we can ensure since .
□
We give the following example to illustrate Theorem 3:
Example 6. Using Theorem 3, we calculate .
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The following result is easy to prove:
Lemma 9. Let S be a numerical semigroup, and . Then, .
Lemma 10. .
Proof. Applying Lemma 9 and Theorem 3, we deduce that . However, when j is even, we have , so . Therefore, . □
Proposition 11. The embedding dimension of is . Moreover, .
Proof. Suppose, on the contrary, that for some . Then, there exists some such that . Theorem 3 ensures that for our k, which ensures that . Then, using Lemma 10, we can assume for some . Therefore, we have the following: and we also have . Furthermore, we have , since . So, by Theorem 3, we can take . Thus, using Theorem 3, we have . However, this implies , which is absurd, and the proof is finished. □
Example 7. Using Proposition 11, we can compute .
Now, we are going to solve the Frobenius problem for elements in
with the
-rank being equal to 1. We will use Theorem 3 and the following result from [
24]:
Lemma 11. Let S be a numerical semigroup and . Then, we have the following:
- 1.
.
- 2.
.
Proof. Using Theorem 3, we deduce the following:
Moreover, with Lemma 11, we finish the proof. □
Example 8. - 1.
.
- 2.
.
Lemma 12. If for all , then we have Proof. By Theorem 3, we can write the elements in as for all . Moreover, from Lemmas 11 and 12, we have
If m is odd: .
If m is even: .
□
Example 9. - 1.
.
- 2.
.
As we pointed out in Remark 1, the next result appears in [
23], Lemma 10.
Lemma 13. Let S be a numerical semigroup, and . Then, Proposition 14. . For .
Proof. If is odd, then , so is not a maximal element in .
On the other hand, if is even, from Proposition 11, we have that every element in can be written as for any i odd number, such that . Therefore, using Theorem 3, we have , which is also stated in Theorem 3. Furthermore, for every , such that j is even. Therefore, is maximal in when j is even. Clearly, is also maximal in . □
Corollary 2. .
Example 10. Using Corollary 2, we can compute .
We have obtained in Examples 6, 7 and 10 the Apéry set of the multiplicity, the minimal system of generators, and the pseudo-Frobenius numbers of . Now, we are going to change the parity of m to show an example where m is odd.
Example 11. Using Theorem 3, Proposition 11, and Corollary 2, we are going to compute the Apéry set of the multiplicity, the minimal system of generators, and the pseudo-Frobenius numbers of .
The Apéry set is | |
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| |
| |
| |
From the above, we have , and .
Corollary 3. if m is even or if m is odd. Summarizing .
The following result comes from Proposition 11 and Corollary 3:
Corollary 4. We have for every with the -rank being equal to 1.
On page 15 of [
1], it is shown that
for any numerical semigroup
S (it was earlier obtained in [
7]). Then, using Corollary 4, we achieve the following result:
Proposition 15. Every element in with the -rank being equal to 1 satisfies Wilf’s conjecture.