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Article

Second-Level Numerical Semigroups

by
David Llena
1,*,† and
José Carlos Rosales
2,†
1
Departamento de Matemáticas, Universidad de Almería, E-04120 Almería, Spain
2
Departamento de Álgebra, E-18071 Granada, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2025, 13(4), 593; https://doi.org/10.3390/math13040593
Submission received: 14 January 2025 / Revised: 4 February 2025 / Accepted: 9 February 2025 / Published: 11 February 2025

Abstract

:
Let S be a numerical semigroup with multiplicity m ( S ) . Then, S is called a second-level numerical semigroup if x + y + z m ( S ) S for every { x , y , z } S { 0 } . In this paper, we present some algorithms to compute all the second-level numerical semigroups with multiplicity, genus, and a Frobenius fixed number. For m and r, which are positive integers, such that m < r and gcd ( m , r ) = 1 , we show that there exists the minimal second-level numerical semigroup with multiplicity m containing r. We solve the Frobenius problem for these semigroups and show that they satisfy Wilf’s conjecture.

1. Introduction

Let Z be the set of integer numbers, and  N = { x Z x 0 } . A submonoid of ( N , + ) is a subset of a closed N under addition and contains 0. A numerical semigroup S is a submonoid of ( N , + ) , such that N S = { n N n S } is finite.
If S is a numerical semigroup, then m ( S ) = min ( S { 0 } ) , F ( S ) = max ( Z S ) , and  g ( S ) , the cardinal of the set N S , 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 N , then we denote by A the submonoid of ( N , + ) generated by A, i.e., A = { λ 1 a 1 + + λ n a n n N { 0 } , { a 1 , , a n } A , a n d { λ 1 , , λ n } N } . In ([1], Lemma 2.1), it is shown that A is a numerical semigroup if and only if gcd ( A ) = 1 .
If M is a submonoid of ( N , + ) and M = A , then we say that A is a system of generators of M. Furthermore, if  M B for all B A , 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 ( N , + ) , there exists a unique minimal system of generators, and it is finite. We denote this minimal system of generators as msg ( M ) . Its cardinality is called the embedding dimension ofM; we will denote it as e ( M ) .
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 e ( S ) m ( S ) . We say that S is a maximal embedding dimension semigroup if e ( S ) = m ( S ) . In ([1], Proposition 3.12), it is shown that a numerical semigroup has a maximal embedding dimension if and only if x + y m ( S ) S for every { x , y } S { 0 } .
Following the terminology used in [11], a numerical semigroup is second-level (or a second-level semigroup) if x + y + z m ( S ) S , for every { x , y , z } S { 0 } . 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
L 2 ( m ) = { S S is a   second-level semigroup with m ( S ) = m } .
In Section 2, we show that L 2 ( m ) is a Frobenius pseudo-variety. Using several results from [12], we deduce an algorithm to obtain all the elements in L 2 ( m ) for a fixed genus. Let f be a positive integer. We denote L 2 ( f , m ) = { S L 2 ( m ) F ( S ) = f } . In Section 3, we show that L 2 ( f , m ) is a ratio-covariety. The results in [13] allow us to present an algorithm to compute all the elements in L 2 ( f , m ) .
A L 2 ( m ) -monoid is a submonoid of ( N , + ) , which can be written as the intersection of elements in L 2 ( m ) . For  A { m , } , we denote by L 2 ( m ) [ A ] the intersection of all elements in L 2 ( m ) containing A. It is clear that L 2 ( m ) [ A ] is the smallest L 2 ( m ) -monoid containing A. In Section 4, we give an algorithm to compute L 2 ( m ) [ A ] . We also show that L 2 ( m ) [ A ] belongs to L 2 ( m ) if and only if gcd ( A { m } ) = 1 .
If M = L 2 ( m ) [ A ] , then we say that A is a L 2 ( m ) -system of generators for M. Moreover, if  M L 2 ( m ) [ B ] for every B A , then we say that A is a minimal L 2 ( m ) -system of generators for M. In Section 4, we show that every L 2 ( m ) -monoid M has a unique minimal L 2 ( m ) -system of generators, which will be denoted as L 2 ( m ) - msg ( M ) . We define the L 2 ( m ) -rank of M as the cardinality of L 2 ( m ) - msg ( M ) .
In Section 5, we focus on the elements of L 2 ( m ) with the L 2 ( m ) -rank being equal to 1. These semigroups will be L 2 ( m ) [ { r } ] for some r N , such that m < r and gcd ( m , r ) = 1 . We provide the minimal system of generators for L 2 ( m ) [ { r } ] 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 x S and x + s S for every s S { 0 } . The set of all pseudo-Frobenius numbers will be denoted as PF ( S ) . We define the type of S as the cardinality of PF ( S ) , and we will denote it as t ( S ) .
In Section 5, we show that t ( S ) + 1 = e ( S ) for any S L 2 ( m ) with the L 2 ( m ) -rank being equal to 1.
For a numerical semigroup S, we denote the set of small numbers by N ( S ) = { s S s < F ( S ) } , and the cardinality of N(S) is denoted as n ( S ) . H.S. Wilf conjectured in [14] that F ( S ) + 1 e ( S ) n ( S ) . 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 L 2 ( m ) with the L 2 ( m ) -rank being equal to 1 verify Wilf’s conjecture.

2. Elements in L 2 ( m ) 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 L 2 ( m , g ) = { S L 2 ( m ) g ( S ) = g } . The aim of this section is to obtain an algorithm to compute L 2 ( m , g ) .
The following result is easy to check:
Lemma 1. 
  • Δ ( m ) = { 0 , m , } is the maximum of L 2 ( m ) with respect to the inclusion order, where { a 1 , , a n , } means that every integer bigger than a n belongs to the set.
  • For k N such that k m , we have m { k , } L 2 ( m ) .
If q Q , then q = max { z Z z q } . For  { a , b } Z with b 0 , we denote a mod b = a a b b .
Proposition 1. 
Take g N . Then, L 2 ( m , g ) if and only if g m 1 .
Proof. 
Necessity. If  S L 2 ( m , g ) , then S L 2 ( m ) and g ( S ) = g . Applying Lemma 1, we have S Δ ( m ) so g = g ( S ) g ( Δ ( m ) ) = m 1 .
Sufficiency. If  g m 1 , it is clear that g m 1 + g + 1 m . Applying Lemma 1, we have S = m { g m 1 + g + 1 , } L 2 ( m ) . On the other hand, the number of positive elements in m smaller than g m 1 + g is g m 1 + g m = g m 1 + g m = g + ( m 1 ) g ( m 1 ) m = m g m ( m 1 ) = g m 1 , so g ( S ) = g , and  S L 2 ( m , g ) .    □
A Frobenius pseudo-variety  P is a non-empty family of numerical semigroups satisfying the following properties:
  • The family P has a maximum element (with respect to the inclusion order);
  • If { S , T } P , then S T P ;
  • If S P and S max ( P ) , then S { F ( S ) } P .
The following result can be deduced from [11], Proposition 2.6 and Corollary 3.4.
Proposition 2. 
L 2 ( m ) is a Frobenius pseudo-variety and Δ ( m ) is its maximum.
A graph G is a pair ( V , E ) , where V is a non-empty set and E { ( u , v ) V × V u v } . 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, ( v 0 , v 1 ) , ( v 1 , v 2 ) , , ( v n 1 , v n ) , where v 0 = x and v n = y .
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  ( x , y ) is an edge of a tree, we say that x is a child of y.
We define the graph G ( L 2 ( m ) ) as follows: The vertices of this graph are the elements of L 2 ( m ) , and  ( S , T ) L 2 ( m ) × L 2 ( m ) is an edge if and only if T = S { F ( S ) } .
As a consequence of Proposition 2, ([12], Lemmas 11 and 12, and Theorem 3), we can obtain the following result:
Theorem 1. 
G ( L 2 ( m ) ) is a tree whose root is Δ ( m ) . Moreover, the set of children of a vertex S of G ( L 2 ( m ) ) is { S { x } x msg ( S ) w i t h x > F ( S ) a n d S { x } L 2 ( m ) } .
The following result is easy to check:
Lemma 2. 
Let S be a numerical semigroup, and let  x S . Then, S { x } is a numerical semigroup if and only if x msg ( S ) . Moreover, g ( S { x } ) = g ( S ) + 1 .
Let S be a numerical semigroup. We consider msg ( S ) = { n 1 , , n e } . If  s S , then we denote L S ( s ) = max { a 1 + + a e ( a 1 , , a e ) N e   and   a 1 n 1 + + a e n e = s } .
Lemma 3 
([11], Proposition 3.5). Let S L 2 ( m ) , and let  x msg ( S ) { m } . Then, we have S { x } L 2 ( m ) if and only if L S { x } ( x + m ) 2 .
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 L 2 ( m ) with genus g are children of elements in L 2 ( m ) with genus g 1 . Following this idea and the above results, we present Algorithm 1 to compute L 2 ( m , g ) .
Algorithm 1
Input: Integers m and g, such that 1 m 1 g .
Output: L 2 ( m , g ) .
  • A = { Δ ( m ) } , i = m 1 ;
  •  If i = g , then return A, and the algorithm is finished;
  •  For every S A compute B S = { x msg ( S ) { m } x > F ( S ) a n d L S { x } ( x + m ) 2 } ;
  • A : = S A { S { x } x B S } ; i : = i + 1 and go back to (2).
Example 1. 
Using Algorithm 1, we are going to compute L 2 ( 4 , 5 ) .
  • A = { 4 , 5 , 6 , 7 } , i = 3 ;
  • B 4 , 5 , 6 , 7 = { 5 , 6 , 7 } ;
  • A = { 4 , 6 , 7 , 9 , 4 , 5 , 7 , 4 , 5 , 6 } , i = 4 ;
  • B 4 , 6 , 7 , 9 = { 6 , 7 , 9 } ; B 4 , 5 , 7 = { 7 } ; B 4 , 5 , 6 = ;
  • A = { 4 , 7 , 9 , 10 , 4 , 6 , 9 , 11 , 4 , 6 , 7 , 4 , 5 , 11 } , i = 5 = g and the algorithm is finished.
Algorithm 1 returns L 2 ( 4 , 5 ) = { 4 , 7 , 9 , 10 , 4 , 6 , 9 , 11 , 4 , 6 , 7 , 4 , 5 , 11 } .

3. Elements in L 2 ( m ) 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 L 2 ( f , m ) = { S L 2 ( m ) F ( S ) = f } . The objective of this section is to obtain an algorithm to compute L 2 ( f , m ) .
Proposition 3. 
Take f N { 0 } . Then, L 2 ( f , m ) if and only if m 1 f and m f .
Proof. 
Necessity. If  S L 2 ( f , m ) , then m 1 S , so m 1 f . Moreover, since f S , then f m . Hence, m f .
Sufficiency. Observe that m { f + 1 , } L 2 ( f , m ) .    □
If f = m 1 and S L 2 ( f , m ) , then S = { 0 , f + 1 , } . In the rest of this section, we assume { m , f } N , such that 2 m < f and m f .
Let S be a numerical semigroup, such that S N . The ratio of S is r ( S ) = min { s S m ( S ) s } . Note that r ( S ) = min ( msg ( S ) { m ( S ) } ) .
A ratio-covariety is a family R of numerical semigroups satisfying the following properties:
  • The family R has a minimum element (with respect to the inclusion order);
  • If { S , T } R , then S T R ;
  • If S R and S min ( R ) , then S { r ( S ) } R .
The following result is easy to prove:
Lemma 4. 
Let S , T be two numerical semigroups. Then, S T is also a numerical semigroup, and  F ( S T ) = max { F ( S ) , F ( T ) } .
We denote Δ ( f , m ) = m { f + 1 , } .
Proposition 4. 
L 2 ( f , m ) is a ratio-covariety and Δ ( f , m ) is its minimum.
Proof. 
It is clear that Δ ( f , m ) is the minimum of L 2 ( f , m ) . If  { S , T } L 2 ( f , m ) , then using Lemma 4, we deduce S T L 2 ( f , m ) . To finish the proof, we see that if S L 2 ( f , m ) such that S Δ ( f , m ) , then S { r ( S ) } L 2 ( f , m ) . As  S Δ ( f , m ) , then r ( S ) < f , and we obtain that S { r ( S ) } is a numerical semigroup with multiplicity m and a Frobenius number f using Lemma 2. On the other hand, if  { x , y , z } S { r ( S ) , 0 } , then x + y + z m S . Furthermore, x + y + z m r ( S ) , so x + y + z m S { r ( S ) } . From this, we have S { r ( S ) } L 2 ( f , m ) .    □
Let S be a numerical semigroup, and  x Z . We say that x is a special gap of S if x S and S { x } is a numerical semigroup. We denote by SG ( S ) the set of special gaps of S.
We define the graph G ( L 2 ( f , m ) ) as follows: The set of vertices is L 2 ( f , m ) , and  ( S , T ) L 2 ( f , m ) × L 2 ( f , m ) is an edge if and only if T = S { r ( S ) } .
Using Proposition 4, [13], and Propositions 3 and 4, we obtain the following result:
Theorem 2. 
G ( L 2 ( f , m ) ) is a tree whose root is Δ ( f , m ) . Moreover, the set of children of a vertex S is { S { x } x SG ( S ) , m < x < r ( S ) , a n d S { x } L 2 ( f , m ) } .
The following result is Proposition 3.1 of [11].
Proposition 5. 
Let S be a numerical semigroup with multiplicity m. Then, S L 2 ( m ) if and only if a + b + c m S for every { a , b , c } msg ( S ) { m } .
Corollary 1. 
Consider S L 2 ( f , m ) , and  x SG ( S ) , such that m < x < f . Then, S { x } L 2 ( f , m ) if and only if { 3 x m , 2 x + a m , x + a + b m } S for every { a , b } msg ( S ) { m } .
Proof. 
To apply Proposition 5, we need to observe that msg ( S { x } ) msg ( S ) { x } .    □
Let S be a numerical semigroup, and  n S { 0 } , we define (in honor of [21]) the Apéry set of n in S as Ap ( S , n ) = { s S s n S } . The following result appears in [1], Lemma 2.4:
Lemma 5. 
Let S be a numerical semigroup and n S { 0 } . Then, Ap ( S , n ) = { w ( 0 ) = 0 , w ( 1 ) , , w ( n 1 ) } , where w ( i ) is the smallest element in S congruent with i modulo n.
Remark 1. 
Let S be a numerical semigroup and assume that Ap ( S , m ( S ) ) = { w ( 0 ) , w ( 1 ) , , w ( m ( S ) 1 ) } 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 x w ( x mod m ( S ) ) .
2. 
We can obtain the special gaps of S. From ([22], Remark 1), we have an easy way to compute them as SG ( S ) = { x PF ( S ) 2 x PF ( S ) }  [22], Lemma 3.4, where PF ( S ) = { w m ( S ) w Maximals S Ap ( S , m ( S ) ) }  [23], Lemma 10. Here, S is the partial order defined on Z as a S b , if and only if b a S .
3. 
From ([22], Lemma 3.5), we have Ap ( S { x } , m ( S ) ) = ( Ap ( S , m ( S ) ) { x + m ( S ) } ) { x } , for every x SG ( S ) .
4. 
Finally, as  msg ( S ) { m ( S ) } = { w Ap ( S , m ( S ) ) { 0 } w w Ap ( S , m ( S ) ) w Ap ( S , m ( S ) ) { 0 , w } } , we can obtain msg ( S { x } ) from (3).
Using the above ideas, we present Algorithm 2. It allow to compute L 2 ( f , m ) being f and m integers under certain conditions.
Algorithm 2
Input: Integers f and m such that 2 m < f and m f .
Output: L 2 ( f , m ) .
  • L 2 ( f , m ) = { Δ ( f , m ) } and B = { Δ ( f , m ) } .
  •  Compute Ap ( Δ ( f , m ) , m ) .
  •  For every S B , compute θ ( S ) = { x SG ( S ) m < x < r ( S ) , x f , { 3 x m , 2 x + a m , x + a + b m } S f o r e v e r y { a , b } msg ( S ) { m } } .
  •  If S B θ ( S ) = , then return L 2 ( f , m ) , and the algorithm is finished.
  • C = S B { S { x } x θ ( S ) } .
  • L 2 ( f , m ) : = L 2 ( f , m ) C and B : = C .
  •  For every S B , compute Ap ( S , m ) and go back to (3).
Note that the computation of Ap ( S , m ) in line 7 is used to compute the set SG ( S ) in a new loop, as we point out in Remark 1.
Example 2. 
Using Algorithm 2, we compute L 2 ( 7 , 4 ) .
First loop of the algorithm is:
(1) 
L 2 ( 7 , 4 ) = { Δ ( 7 , 4 ) } and B = { Δ ( 7 , 4 ) } .
(2) 
Ap ( Δ ( 7 , 4 ) , 4 ) = { 0 , 9 , 10 , 11 } .
Therefore, Maximals ( Ap ( Δ ( 7 , 4 ) , 4 ) ) = { 9 , 10 , 11 } . PF ( Δ ( 7 , 4 ) ) = { 5 , 6 , 7 } .
Moreover, SG ( Δ ( 7 , 4 ) ) = { 5 , 6 , 7 } .
Meanwhile, msg ( Δ ( 7 , 4 ) ) = { 4 , 9 , 10 , 11 } and r ( Δ ( 7 , 4 ) ) = 9 .
Then, we have
(3) 
θ ( Δ ( 7 , 4 ) ) = { 5 , 6 } , as  7 = f .
(5) 
C = { Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } } .
(6) 
L 2 ( 7 , 4 ) = { Δ ( 7 , 4 ) , Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } } and B = { Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } } .
(7) 
Ap ( Δ ( 7 , 4 ) { 5 } , 4 ) = { 0 , 5 , 10 , 11 } and Ap ( Δ ( 7 , 4 ) { 6 } , 4 ) = { 0 , 6 , 9 , 11 } .
Therefore,
(a) 
Maximals ( Ap ( Δ ( 7 , 4 ) { 5 } , 4 ) ) = { 10 , 11 } . PF ( Δ ( 7 , 4 ) { 5 } ) = { 6 , 7 } .
Moreover, SG ( Δ ( 7 , 4 ) { 5 } ) = { 6 , 7 } .
Meanwhile, msg ( Δ ( 7 , 4 ) { 5 } ) = { 4 , 5 , 11 } and r ( Δ ( 7 , 4 ) { 5 } ) = 5 .
(b) 
Maximals ( Ap ( Δ ( 7 , 4 ) { 6 } , 4 ) ) = { 6 , 9 , 11 } . PF ( Δ ( 7 , 4 ) { 6 } ) = { 2 , 5 , 7 } .
Moreover, SG ( Δ ( 7 , 4 ) { 6 } ) = { 2 , 5 , 7 } .
Meanwhile, msg ( Δ ( 7 , 4 ) { 6 } ) = { 4 , 6 , 9 , 11 } and r ( Δ ( 7 , 4 ) { 6 } ) = 6 .
Then, second loop is
(3) 
θ ( Δ ( 7 , 4 ) { 5 } ) = , as  r ( Δ ( 7 , 4 ) { 5 } ) < 6 < 7
and θ ( Δ ( 7 , 4 ) { 6 } ) = { 5 } , as  2 < m < 5 < r ( Δ ( 7 , 4 ) { 6 } ) < 7 .
(5) 
C = { Δ ( 7 , 4 ) { 5 , 6 } } .
(6) 
L 2 ( 7 , 4 ) = { Δ ( 7 , 4 ) , Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } , Δ ( 7 , 4 ) { 5 , 6 } } and B = { Δ ( 7 , 4 ) { 5 , 6 } } .
(7) 
Ap ( Δ ( 7 , 4 ) { 5 , 6 } ) , 4 ) = { 0 , 5 , 6 , 11 } .
Therefore, Maximals ( Ap ( Δ ( 7 , 4 ) { 5 , 6 } , 4 ) ) = { 11 } . PF ( Δ ( 7 , 4 ) { 5 , 6 } ) = { 7 } .
Moreover, SG ( Δ ( 7 , 4 ) { 5 , 6 } ) = { 7 } .
Meanwhile, msg ( Δ ( 7 , 4 ) { 5 , 6 } ) = { 4 , 5 , 6 } and r ( Δ ( 7 , 4 ) { 5 , 6 } ) = 5 .
Then, the third loop is
(3) 
θ ( Δ ( 7 , 4 ) { 5 , 6 } ) = , as  7 = f .
(4) 
L 2 ( 7 , 4 ) = { Δ ( 7 , 4 ) , Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } , Δ ( 7 , 4 ) { 5 , 6 } } and the algorithm is finished.
Algorithm 2 returns L 2 ( 7 , 4 ) = { Δ ( 7 , 4 ) , Δ ( 7 , 4 ) { 5 } , Δ ( 7 , 4 ) { 6 } , Δ ( 7 , 4 ) { 5 , 6 } } .

4. L 2 ( m ) -Systems of Generators

It is clear that L 2 ( 1 ) = { N } and L 2 ( 2 ) = { 2 , 2 k + 1 k N { 0 } } . 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 L 2 ( m ) is again an element in L 2 ( m ) . However, when we consider an infinite number of them, the intersection is generally not an element in L 2 ( m ) , as shown in the following example:
Example 3. 
For any k { m , } , we consider S ( k ) = m { k , } . Clearly, S ( k ) L 2 ( m ) . However, k { m , } S ( k ) = m L 2 ( m ) .
The finite or infinite intersection of elements in L 2 ( m ) is always a submonoid of ( N , + ) . This fact allows us to give the following definition. A  L 2 ( m ) -monoid is a submonoid of ( N , + ) that can be written as an intersection of elements in L 2 ( m ) .
The following result is easy to check:
Lemma 6. 
The intersection of L 2 ( m ) -monoids is again a L 2 ( m ) -monoid.
Consider A Δ ( m ) . We denote by L 2 ( m ) [ A ] the intersection of all the L 2 ( m ) -monoids containing A. As a consequence of Lemma 6, we find that L 2 ( m ) [ A ] is the smallest (with respect to the inclusion) L 2 ( m ) -monoid containing A.
We can easily reach the following result:
Lemma 7. 
Let A be a subset of Δ ( m ) . Then, L 2 ( m ) [ A ] is equal to the intersection of all the elements in L 2 ( m ) containing A.
If M = L 2 ( m ) [ A ] , then we will say that A is a L 2 ( m ) -system of generators of M. Furthermore, if  M L 2 ( m ) [ B ] for any B A , then A is called a minimal L 2 ( m ) -system of generators of M.
The following result is easy to deduce from Proposition 2 and [12], Corollary 1.
Proposition 6. 
Every L 2 ( m ) -monoid has a unique minimal L 2 ( m ) -system of generators. Furthermore, such a minimal L 2 ( m ) -system of generators is finite.
Let M be a L 2 ( m ) -monoid. We denote L 2 ( m ) - msg ( M ) as the minimal L 2 ( m ) -system of generators of M. We define the L 2 ( m ) -rank of M as the cardinality of this minimal L 2 ( m ) -system of generators.
Proposition 7. 
Let M be a submonoid of ( N , + ) . Then, the following statements are equivalent.
  • M is a L 2 ( m ) -monoid.
  • m = min ( M { 0 } ) and x + y + z m M for every { x , y , z } M { 0 } .
  • m = min ( M { 0 } ) and x + y + z m M for every { x , y , z } msg ( M ) { m } .
Proof. 
( 1 ) ( 2 ) . If M is a L 2 ( M ) -monoid, then there exists a family { S i } i I of elements in L 2 ( m ) , such that M = i I S i . Hence, min ( M { 0 } ) = m . On the other hand, if  { x , y , z } M { 0 } , then { x , y , z } S i { 0 } for every i I , so x + y + z m S i for every i I , and then x + y + z m M .
( 2 ) ( 3 ) . Trivial.
( 3 ) ( 1 ) . For every k { m , } , we consider S ( k ) = M { k , } . From Proposition 5, we easily deduce that S ( k ) L 2 ( m ) for every k { m , } . It is clear that M = k { m , } S ( k ) , so M is a L 2 ( m ) -monoid.    □
Proposition 7 allows us to present Algorithm 3 which compute L 2 ( m ) [ A ] for any A Δ ( m ) .
Algorithm 3
Input: A finite subset A of Δ ( m ) .
Output: L 2 ( m ) [ A ]
  • M = A { m } .
  • B = { x + y + z m { x , y , z } msg ( M ) { m } a n d x + y + z m M } .
  •  If B = , then return M, and the algorithm is finished.
  • M : = msg ( M ) B and go back to (2).
Now, we present two different examples to show how Algorithm 3 works. In the first example, we calculate L 2 ( 5 ) [ { 7 } ] , and we will see that L 2 ( 5 ) [ { 7 } ] belongs to L 2 ( 5 ) . In the second example, we calculate L 2 ( 8 ) [ { 10 } ] , and we will show that L 2 ( 8 ) [ { 10 } ] does not belong to L 2 ( 8 ) . This raises the following question: What is the condition for L 2 ( m ) [ A ] to belong to L 2 ( m ) ?
Example 4. 
Let us calculate L 2 ( 5 ) [ { 7 } ] using Algorithm 3.
  • M = 5 , 7 .
  • B = { 16 } .
  • M = 5 , 7 , 16 .
  • B = , and the algorithm is finished.
Algorithm 3 returns L 2 ( 5 ) [ { 7 } ] = 5 , 7 , 16 .
Example 5. 
Let us calculate L 2 ( 8 ) [ { 10 } ] using Algorithm 3.
  • M = 8 , 10 .
  • B = { 22 } .
  • M = 8 , 10 , 22 .
  • B = , and the algorithm is finished.
Algorithm 3 returns L 2 ( 8 ) [ { 10 } ] = 8 , 10 , 22 . Since gcd { 8 , 10 , 22 } 1 , then 8 , 10 , 12 is not a numerical semigroup; therefore, 8 , 10 , 22 L 2 ( 8 ) .
Proposition 8. 
Consider A Δ ( m ) . Then, L 2 ( m ) [ A ] L 2 ( m ) if and only if gcd ( A { m } ) = 1 .
Proof. 
Necessity. If gcd ( A { m } ) = d 1 , then using Proposition 7, we deduce M = { 0 } { m + k d k N } is a L 2 ( m ) -monoid. It is clear that A M and L 2 ( m ) [ A ] M . From this, we have that L 2 ( m ) [ A ] is not a numerical semigroup. Consequently, L 2 ( m ) [ A ] L 2 ( m ) .
Sufficiency. We have that L 2 ( m ) [ A ] is a submonoid of ( N , + ) and A { m } L 2 ( m ) [ A ] . As gcd ( A { m } ) = 1 , we deduce gcd ( L 2 ( m ) [ A ] ) = 1 , and then L 2 ( m ) [ A ] is a numerical semigroup. Consequently, L 2 ( m ) [ A ] belongs to L 2 ( m ) . □

5. Elements in L 2 ( m ) with L 2 ( m ) -Rank Equal to 1

In this section, m denotes an integer greater than or equal to 3.
Proposition 9. 
Consider S L 2 ( m ) . Then, the following statements are equivalent:
1. 
The L 2 ( m ) -rank of S is equal to 1.
2. 
S = L 2 ( m ) [ { r } ] for some integer r, such that m < r and gcd ( m , r ) = 1 .
Proof. 
( 1 ) ( 2 ) . If L 2 ( m ) -rank  ( S ) = 1 , then there exists r Δ ( m ) such that L 2 ( m ) -msg  ( S ) = { r } . Hence, r m and S = L 2 ( m ) [ { r } ] . Using Proposition 8, we have gcd ( m , r ) = 1 , which implies m < r .
( 2 ) ( 1 ) . It is clear that { r } = L 2 ( m ) -msg  ( S ) . Then, L 2 ( m ) -rank  ( S ) = 1 . □
In the rest of this section, r will be an integer greater than m such that gcd ( m , r ) = 1 . We also denote A ( m , r ) = { ( 2 k + 1 ) r k m k N } .
Proposition 10. 
L 2 ( m ) [ { r } ] = A ( m , r ) { m } .
Proof. 
First of all, we will show, by induction on k, that ( 2 k + 1 ) r k m L 2 ( m ) [ { r } ] . For k = 0 , the result is clear because ( 2 · 0 + 1 ) r 0 · m = r L 2 ( m ) [ { r } ] . We assume the statement to be true for k; then, we prove it for k + 1 . As { ( 2 k + 1 ) r k m , r } L 2 ( m ) [ { r } ] { 0 } and L 2 ( m ) [ { r } ] L 2 ( m ) , then ( 2 k + 1 ) r k m + r + r m L 2 ( m ) [ { r } ] and so ( 2 ( k + 1 ) + 1 ) r ( k + 1 ) m L 2 ( m ) [ { r } ] .
Therefore, we have that A ( m , r ) { m } L 2 ( m ) [ { r } ] and consequently A ( m , r ) { m } L 2 ( m ) [ { r } ] .
Now, we will see that L 2 ( m ) [ { r } ] A ( m , r ) { m } . For this, we only need to prove A ( m , r ) { m } L 2 ( m ) .
Consider { i , j , k } N . Then, ( 2 i + 1 ) r i m + ( 2 j + 1 ) r j m + ( 2 k + 1 ) r k m m A ( m , r ) { m } as ( 2 i + 1 ) r i m + ( 2 j + 1 ) r j m + ( 2 k + 1 ) r k m m = ( 2 ( i + j + k + 1 ) + 1 ) r ( i + j + k + 1 ) m A ( m , r ) . Then, Proposition 5 ensures that A ( m , r ) { m } L 2 ( m ) .
Consequently, we have L 2 ( m ) [ { r } ] = A ( m , r ) { m } . □
As a consequence of Proposition 10, we have Ap ( L 2 ( m ) [ { r } ] , m ) A ( m , r ) . We also have L 2 ( m ) [ { r } ] L 2 ( m ) . Then, x + y + z m L 2 ( m ) [ { r } ] for every { x , y , z } A ( m , r ) . Then, we have the following result.
Lemma 8. 
If w Ap ( L 2 ( m ) [ { r } ] , m ) , then we have one of the following asserts:
1. 
w = ( 2 i + 1 ) r i m for some i N .
2. 
w = ( 2 i + 1 ) r i m + ( 2 j + 1 ) r j m for some { i , j } N .
As gcd ( m , r ) = 1 , we have
{ ( 1 · r ) mod m , ( 2 · r ) mod m , , ( ( m 1 ) · r ) mod m } = { 1 , 2 , , m 1 } .
Theorem 3. 
If Ap ( L 2 ( m ) [ { r } ] , m ) = { w ( 0 ) , w ( 1 ) , , w ( m 1 ) } , then
w ( k r mod m ) = k r k 1 2 m i f k i s o d d k r k 2 2 m i f k i s e v e n
for every k { 1 , , m 1 } .
Proof. 
If k is odd, then there exists s N such that k = 2 s + 1 , so k r k 1 2 m = ( 2 s + 1 ) r s m A ( m , r ) L 2 ( m ) [ { r } ] . However, if k is even, then k = 2 s + 2 for some s N , and we have k r k 2 2 m = ( 2 s + 2 ) r s m = ( 2 s + 1 ) r s m + r L 2 ( m ) [ { r } ] because { ( s k + 1 ) r s m , r } A ( m , r ) .
Now, we will first see that in the case in which k is odd, we have k r k 1 2 m m L 2 ( m ) [ { r } ] . Assume the contrary. By Lemma 8, we consider two cases:
  • There exist t N { 0 } and i N such that k r k 1 2 m t m = ( 2 i + 1 ) r i m . Then, k r ( 2 i + 1 ) r ( mod m ) . However, since gcd ( m , r ) = 1 , we have k 2 i + 1 ( mod m ) . Hence, there exists l N such that 2 i + 1 = k + l m . Consequently, k r k 1 2 m t m = ( k + l m ) r k + l m 1 2 m and then k + l m 1 2 k 1 2 t m = l m r . Therefore, l m 2 t = l r . Hence, l m 2 > l r , and m 2 > r , which is a contradiction, since r > m . Observe that l 0 . Otherwise, we have t = 0 , which is impossible.
  • There exist t N { 0 } and { i , j } N , such that k r k 1 2 m t m = ( 2 i + 1 ) r i m + ( 2 j + 1 ) r j m . Then, k ( 2 i + 2 j + 2 ) ( mod m ) . Hence, 2 i + 2 j + 2 = k + l m for some l N . Therefore, k r k 1 2 m t m = ( 2 i + 2 j + 2 ) r ( i + j ) m = ( k + l m ) r k + l m 2 2 m . Then, following the same reasoning as in (1), we have l m 1 2 t = l r . Hence, l m 2 > l r . Consequently, m 2 > r , which is again a contradiction. Observe that l 0 . Otherwise, we have t = 1 2 , which is impossible.
We then focus on the case where k is even and try to prove that k r k 2 2 m m L 2 ( m ) [ { r } ] . As above, we assume the contrary, and using Lemma 8 we consider two cases:
  • There exist t N { 0 } and i N such that k r k 2 2 m t m = ( 2 i + 1 ) r i m . Then, k r ( 2 i + 1 ) r ( mod m ) , since gcd ( m , r ) = 1 we have k 2 i + 1 ( mod m ) , and so there exists l N such that 2 i + 1 = k + l m . Then, k r k 2 2 m t m = ( k + l m ) r k + l m 1 2 m and then k + l m 1 2 k 2 2 t m = l m r . Therefore, l m + 1 2 t = l r , which implies that l m 2 > l r and m 2 > r , and we obtain the same contradiction. As we have commented in the above paragraph, l must be different from 0. Otherwise, t = 1 2 .
  • There exist t N { 0 } and { i , j } N such that k r k 2 2 m t m = ( 2 i + 1 ) r i m + ( 2 j + 1 ) r j m . Then, k ( 2 i + 2 j + 2 ) ( mod m ) , and we can write 2 i + 2 j + 2 = k + l m for some l N . Therefore, k r k 2 2 m t m = ( 2 i + 2 j + 2 ) r ( i + j ) m = ( k + l m ) r k + l m 2 2 m . Consequently, l m 2 t = l r . Then, l m 2 > l r , allowing us to obtain m 2 > r , which is a contradiction. Again, we can ensure l 0 since t 0 .
We give the following example to illustrate Theorem 3:
Example 6. 
Using Theorem 3, we calculate Ap ( L 2 ( 10 ) [ { 13 } ] , 10 ) .
w ( 1 · 13 mod 10 ) = w ( 3 ) = 1 · 13 1 1 2 · 10 = 13 w ( 2 · 13 mod 10 ) = w ( 6 ) = 2 · 13 2 2 2 · 10 = 26
w ( 3 · 13 mod 10 ) = w ( 9 ) = 3 · 13 3 1 2 · 10 = 29 w ( 4 · 13 mod 10 ) = w ( 2 ) = 4 · 13 4 2 2 · 10 = 42
w ( 5 · 13 mod 10 ) = w ( 5 ) = 5 · 13 5 1 2 · 10 = 45 w ( 6 · 13 mod 10 ) = w ( 8 ) = 6 · 13 6 2 2 · 10 = 58
w ( 7 · 13 mod 10 ) = w ( 1 ) = 7 · 13 7 1 2 · 10 = 61 w ( 8 · 13 mod 10 ) = w ( 4 ) = 8 · 13 8 2 2 · 10 = 74
w ( 9 · 13 mod 10 ) = w ( 7 ) = 9 · 13 9 1 2 · 10 = 77
The following result is easy to prove:
Lemma 9. 
Let S be a numerical semigroup, and n S { 0 } . Then, S = ( Ap ( S , n ) { 0 } ) { n } .
Lemma 10. 
L 2 ( m ) [ { r } ] = { m } { ( 2 k + 1 ) r k m k { 0 , 1 , , m 2 2 } } .
Proof. 
Applying Lemma 9 and Theorem 3, we deduce that L 2 ( m ) [ { r } ] = { m } { j r j 1 2 m j { 1 , , m 1 } a n d j o d d } { j r j 2 2 m j { 1 , , m 1 } a n d j e v e n } . However, when j is even, we have j r j 2 2 m = ( j 1 ) r ( j 1 ) 1 2 m + r , so j r j 2 2 m msg ( L 2 ( m ) [ { r } ] ) . Therefore, L 2 ( m ) [ { r } ] = { m } { j r j 1 2 m j { 1 , , m 1 } a n d j o d d } = { m } { ( 2 k + 1 ) r k m k { 0 , 1 , , m 2 2 } } . □
Proposition 11. 
The embedding dimension of L 2 ( m ) [ { r } ] is m 2 2 + 2 . Moreover, msg ( L 2 ( m ) [ { r } ] ) = { m } { ( 2 k + 1 ) r k m k { 0 , 1 , , m 2 2 } } .
Proof. 
Suppose, on the contrary, that ( 2 k + 1 ) r k m msg ( L 2 ( m ) [ { r } ] ) for some k { 0 , 1 , , m 2 2 } . Then, there exists some g msg ( L 2 ( m ) [ { r } ] ) such that ( 2 k + 1 ) r k m g L 2 ( m ) [ { r } ] { 0 } . Theorem 3 ensures that ( 2 k + 1 ) r k m Ap ( L 2 ( m ) [ { r } ] , m ) for our k, which ensures that g m . Then, using Lemma 10, we can assume g = ( 2 i + 1 ) r i m for some i < k . Therefore, we have the following: ( 2 k + 1 ) r k m ( ( 2 i + 1 ) r i m ) = 2 ( k i ) r ( k i ) m L 2 ( m ) [ { r } ] and we also have 2 ( k i ) r ( k i ) m 2 ( k i ) r ( mod m ) . Furthermore, we have 2 ( k i ) < 2 ( k + 1 ) m , since k m 2 2 . So, by Theorem 3, we can take w ( 2 ( k i ) r mod m ) Ap ( L 2 ( m ) [ { r } ] , m ) . Thus, using Theorem 3, we have 2 ( k i ) r ( k i ) m w ( 2 ( k i ) r mod m ) = 2 ( k i ) r 2 ( k i ) 2 2 m . However, this implies 2 ( k i ) 2 2 ( k i ) , which is absurd, and the proof is finished. □
Example 7. 
Using Proposition 11, we can compute msg ( L 2 ( 10 ) [ { 13 } ] ) = { 10 } { ( 2 k + 1 ) 13 k 10 k { 0 , 1 , 2 , 3 , 4 } } = { 10 , 13 , 29 , 45 , 61 , 77 } .
Now, we are going to solve the Frobenius problem for elements in L 2 ( m ) with the L 2 ( m ) -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 n S { 0 } . Then, we have the following:
1. 
F ( S ) = max ( Ap ( S , n ) ) n .
2. 
g ( S ) = 1 n ( w Ap ( S , n ) w ) n 1 2 .
Proposition 12. 
F ( L 2 ( m ) [ { r } ] ) = ( m 1 ) r m 2 2 i f m i s e v e n ( m 1 ) r m ( m 1 ) 2 i f m i s o d d
Proof. 
Using Theorem 3, we deduce the following:
max ( Ap ( L 2 ( m ) [ { r } ] ) , m ) = ( m 1 ) r ( m 2 ) m 2 i f m 1 i s o d d ( m 1 ) r ( m 3 ) m 2 i f m 1 i s e v e n
Moreover, with Lemma 11, we finish the proof. □
Example 8. 
1. 
F ( L 2 ( 10 ) [ { 11 } ] ) = 9 · 11 10 · 10 2 = 49 .
2. 
F ( L 2 ( 9 ) [ { 13 } ] ) = 8 · 13 9 · 8 2 = 68 .
Lemma 12. 
If a j = j 1 2 for all j { 1 , , m 1 } , then we have
( a 1 , , a m 1 ) = ( 0 , 0 , 1 , 1 , , m 3 2 , m 3 2 ) i f m i s o d d ( 0 , 0 , 1 , 1 , , m 4 2 , m 4 2 , m 2 2 ) i f m i s e v e n
Proposition 13. 
g ( L 2 ( m ) [ { r } ] ) = ( m 1 ) r 2 ( m 1 ) 2 4 i f m i s o d d ( m 1 ) r 2 ( m 1 ) 2 + 1 4 i f m i s e v e n
Proof. 
By Theorem 3, we can write the elements in Ap ( L 2 ( m ) [ { r } ] ) as w ( j r mod m ) = j r j 1 2 m for all j { 1 , , m 1 } . Moreover, from Lemmas 11 and 12, we have
  • If m is odd: g ( L 2 ( m ) [ { r } ] ) = 1 m r j = 1 m 1 j 2 m i = 1 m 3 2 i m 1 2 = ( m 1 ) r 2 ( m 1 ) 2 4 .
  • If m is even: g ( L 2 ( m ) [ { r } ] ) = 1 m r j = 1 m 1 j 2 m i = 1 m 4 2 i m 2 2 m m 1 2 = ( m 1 ) r 2 ( m 1 ) 2 + 1 4 .
Example 9. 
1. 
g ( L 2 ( 10 ) [ { 11 } ] ) = 9 · 11 2 9 · 9 + 1 4 = 99 2 41 2 = 58 2 = 29 .
2. 
g ( L 2 ( 9 ) [ { 13 } ] ) = 8 · 13 2 8 · 8 4 = 52 16 = 36 .
As we pointed out in Remark 1, the next result appears in [23], Lemma 10.
Lemma 13. 
Let S be a numerical semigroup, and n S { 0 } . Then,
PF ( S ) = { w n w Maximals S ( Ap ( S , n ) ) }
Proposition 14. 
Maximals L 2 ( m ) [ { r } ] ( Ap ( L 2 ( m ) [ { r } ] , m ) ) = { w ( r j mod m ) j i s e v e n } { F ( L 2 ( m ) [ { r } ] ) + m } . For j { 1 , , m 1 } .
Proof. 
If j < m is odd, then w ( j r mod m ) + r = j r j 1 2 m + r = ( j + 1 ) r ( j + 1 ) 2 2 m = w ( ( j + 1 ) r mod m ) , so w ( j r mod m ) is not a maximal element in Ap ( L 2 ( m ) [ { r } ] , m ) .
On the other hand, if j < m is even, from Proposition 11, we have that every element in msg ( L 2 ( m ) [ { r } ] ) { m } can be written as w ( i r mod m ) for any i odd number, such that i < m . Therefore, using Theorem 3, we have w ( j r mod m ) + w ( i r mod m ) = j r j 2 2 m + i r i 1 2 m = ( i + j ) r ( j 2 ) + ( i 1 ) 2 m = ( i + j ) r ( i + j 3 ) 2 m Ap ( L 2 ( m ) [ { r } ] , m ) , which is also stated in Theorem 3. Furthermore, w ( j r mod m ) + m = j r j 2 2 m + m = j r j 4 2 m Ap ( L 2 ( m ) [ { r } ] , m ) for every j < m , such that j is even. Therefore, w ( j r mod m ) is maximal in Ap ( L 2 ( m ) [ { r } ] , m ) when j is even. Clearly, F ( L 2 ( m ) [ { r } ] ) + m is also maximal in Ap ( L 2 ( m ) [ { r } ] , m ) . □
Corollary 2. 
PF ( L 2 ( m ) [ { r } ] ) = { k ( 2 r m ) k { 1 , , m 2 2 } } { F ( L 2 ( m ) [ { r } ] ) } .
Example 10. 
Using Corollary 2, we can compute PF ( L 2 ( 10 ) [ { 13 } ] ) = { 16 k k { 1 , 2 , 3 , 4 } } { F ( L 2 ( 10 ) [ { 13 } ] ) } = { 16 , 32 , 48 , 64 , 67 } .
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 L 2 ( 10 ) [ { 13 } ] . 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 L 2 ( 11 ) [ { 13 } ] .
The Apéry set Ap ( L 2 ( 11 ) [ { 13 } ] , 11 ) is
w ( 1 · 13 mod 11 ) = w ( 3 ) = 1 · 13 1 1 2 · 11 = 13 w ( 2 · 13 mod 11 ) = w ( 6 ) = 2 · 13 2 2 2 · 11 = 26
w ( 3 · 13 mod 11 ) = w ( 9 ) = 3 · 13 3 1 2 · 11 = 28 w ( 4 · 13 mod 11 ) = w ( 2 ) = 4 · 13 4 2 2 · 11 = 41
w ( 5 · 13 mod 11 ) = w ( 5 ) = 5 · 13 5 1 2 · 11 = 43 w ( 6 · 13 mod 11 ) = w ( 8 ) = 6 · 13 6 2 2 · 11 = 56
w ( 7 · 13 mod 11 ) = w ( 1 ) = 7 · 13 7 1 2 · 11 = 58 w ( 8 · 13 mod 11 ) = w ( 4 ) = 8 · 13 8 2 2 · 11 = 71
w ( 9 · 13 mod 11 ) = w ( 7 ) = 9 · 13 9 1 2 · 11 = 73 w ( 10 · 13 mod 11 ) = w ( 4 ) = 10 · 13 10 2 2 · 11 = 86
From the above, we have msg ( L 2 ( 11 ) [ { 13 } ] ) = { 11 , 13 , 28 , 43 , 58 , 73 } , and PF ( L 2 ( 11 ) [ { 13 } ] ) = { 15 , 30 , 45 , 60 , 75 } .
Corollary 3. 
t ( L 2 ( m ) [ { r } ] ) = m 2 if m is even or t ( L 2 ( m ) [ { r } ] ) = m 1 2 if m is odd. Summarizing t ( L 2 ( m ) [ { r } ] ) = m 2 .
The following result comes from Proposition 11 and Corollary 3:
Corollary 4. 
We have e ( S ) = t ( S ) + 1 for every S L 2 ( m ) with the L 2 ( m ) -rank being equal to 1.
On page 15 of [1], it is shown that F ( S ) + 1 ( t ( S ) + 1 ) n ( S ) 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 L 2 ( m ) with the L 2 ( m ) -rank being equal to 1 satisfies Wilf’s conjecture.

Author Contributions

All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Proyecto de Excelencia de la Junta de Andalucía (ProyExcel 00868), and by the Junta de Andalucía Grant Number FQM-343.

Data Availability Statement

The original contributions presented in this study are included in this article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments, which improved the quality of the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

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