Modular Frobenius pseudo-varieties
aa r X i v : . [ m a t h . G R ] J a n Modular Frobenius pseudo-varieties
Aureliano M. Robles-P´erez ∗† and Jos´e Carlos Rosales ∗ ‡ Abstract If m ∈ N and A is a finite subset of S k ∈ N \{ , } { , . . . , m − } k , thenwe denote by C ( m, A ) = n S ∈ S m | s + · · · + s k − m ∈ S if ( s , . . . , s k ) ∈ S k and( s mod m, . . . , s k mod m ) ∈ A } . In this work we prove that C ( m, A ) is a Frobenius pseudo-variety. We alsoshow algorithms that allows us to establish whether a numerical semigroupbelongs to C ( m, A ) and to compute all the elements of C ( m, A ) with afixed genus. Moreover, we introduce and study three families of numericalsemigroups, called of second-level, thin and strong, and corresponding to C ( m, A ) when A = { , . . . , m − } , A = { (1 , , . . . , ( m − , m − } , and A = { , . . . , m − } \ { (1 , , . . . , ( m − , m − } , respectively. Keywords:
Modular pseudo-varieties, second-level numerical semigroups, thinnumerical semigroups, strong numerical semigroups, tree associated (to a mod-ular pseudo-variety).
Let N be the set of non-negative integers. A numerical semigroup is a subset S of N that is closed under addition, 0 ∈ S and N \ S is finite. The Frobeniusnumber of S , denoted by F( S ), is the greatest integer that does not belong to S . The cardinality of N \ S , denoted by g( S ), is the genus of S .A Frobenius pseudo-variety is a non-empty family P of numerical semigroupsthat fulfils the following conditions.1. P has a maximum element (with respect to the inclusion order). ∗ Both authors are supported by the project MTM2017-84890-P (funded by Ministerio deEconom´ıa, Industria y Competitividad and Fondo Europeo de Desarrollo Regional FEDER)and by the Junta de Andaluc´ıa Grant Number FQM-343. † Departamento de Matem´atica Aplicada, Universidad de Granada, 18071-Granada, Spain.E-mail: [email protected] ‡ Departamento de ´Algebra, Universidad de Granada, 18071-Granada, Spain.E-mail: [email protected]
1. If
S, T ∈ P , then S ∩ T ∈ P .3. If S ∈ P and S = max( P ), then S ∪ { F( S ) } ∈ P .The multiplicity of a numerical semigroup S , denoted by m( S ), is the leastpositive integer that belongs to S . If m is a positive integer, then we denote by S m = { S | S is a numerical semigroup with m( S ) = m } .Let m ∈ N \ { , } and let A be a finite subset of S k ∈ N \{ , } { , . . . , m − } k (where X k = X × ( k ) · · · × X = { ( x , . . . , x k ) | x , . . . , x k ∈ X } ). We denote by C ( m, A ) = (cid:8) S ∈ S m | s + · · · + s k − m ∈ S if ( s , . . . , s k ) ∈ S k and( s mod m, . . . , s k mod m ) ∈ A } . Our main purpose in this work is to study the set C ( m, A ).In Section 2 we show that C ( m, A ) is a Frobenius pseudo-variety with max-imum element given by ∆( m ) = { , m, →} (where the symbol → means thatevery integer greater than m belongs to ∆( m )). In fact, we say that such pseudo-varieties are modular (Frobenius pseudo-varieties) . Also, we give an algorithmthat allows us to establish whether a numerical semigroup belongs to C ( m, A ).In addition, with the help of the results in [7], we can arrange C ( m, A ) in arooted tree and we find an algorithm to compute all the elements of C ( m, A )with a fixed genus.If X is a non-empty subset of N , then we denote by h X i the submonoid of( N , +) generated by X , that is, h X i = (cid:8) λ x + · · · + λ n x n | n ∈ N \ { } , x , . . . , x n ∈ X, λ , . . . , λ n ∈ N (cid:9) . It is well known (see Lemma 2.1 of [9]) that h X i is a numerical semigroup ifand only if gcd( X ) = 1. If M is a submonoid of ( N , +) and M = h X i , thenwe say that X is a system of generators of M . Moreover, if M = h Y i for anysubset Y ( X , then we say that X is a minimal system of generators of M .In Corollary 2.8 of [9] it is shown that each submonoid of ( N , +) has a uniqueminimal system of generators and that such a system is finite. We denote bymsg( M ) the minimal system of generators of M . The cardinality of msg( M ),denoted by e( M ), is the embedding dimension of M .By applying Proposition 2.10 of [9], if S is a numerical semigroup, then weknow that e( S ) ≤ m( S ). A numerical semigroup S has maximal embeddingdimension if e( S ) = m( S ). This family of numerical semigroups has been ex-tensively studied (for instance, see [2] and [9]). Let us denote by M m the setformed by the numerical semigroups that have maximal embedding dimensionand with multiplicity m . It is easy to see that M m = C ( m, { , . . . , m − } ).On the other hand, we can observe that the modular Frobenius pseudo-varieties are associated with the nonhomogeneous patterns that involve in theirconstant parameter the multiplicity of the numerical semigroup (see [3]). Infact, let us take the pattern p = P ni =1 b i x i − m , where b i ≥ i . If wedenote by β j = P ji =1 b j , then the family of numerical semigroups which satisfies2hat pattern is equal to C ( m, A ), where A = (cid:8) ( a , . . . , a β n ) ∈ { , . . . , m − } β n | a = . . . = a β , a β +1 = . . . = a β , . . . ,a β n − +1 = . . . = a β n (cid:9) . In Sections 3, 4, and 5 we study the family C ( m, A ) for • A = { , . . . , m − } , • A = { (1 , , . . . , ( m − , m − } , • A = { , . . . , m − } \ { (1 , , . . . , ( m − , m − } ,respectively. C ( m, A ) In this section, m is an integer greater than or equal to 2 and A is a finite subsetof S k ∈ N \{ , } { , . . . , m − } k . Let us recall that C ( m, A ) = (cid:8) S ∈ S m | s + · · · + s k − m ∈ S if ( s , . . . , s k ) ∈ S k and( s mod m, . . . , s k mod m ) ∈ A } , where x mod m denotes the remainder after division of x by m .If S is a numerical semigroup and x ∈ S \ { } , then the Ap´ery set of x in S (see [1]) is Ap( S, x ) = { w (0) = 0 , w (1) , . . . , w ( x − } , where w ( i ) is the leastelement of S that is congruent with i modulus x . Observe that an integer s belongs to S if and only if there exists t ∈ N such that s = w ( s mod x ) + tx . Proposition 2.1.
Let S ∈ S m and Ap(
S, m ) = { w (0) = 0 , w (1) , . . . , w ( m − } .Then the following conditions are equivalent.1. S ∈ C ( m, A ) .2. w ( i ) + · · · + w ( i k ) − m ∈ S for all ( i , . . . , i k ) ∈ A .Proof. (1. ⇒ Since w ( i ) , . . . , w ( i k ) ∈ S and ( w ( i ) mod m, . . . , w ( i k ) mod m ) = ( i , . . . , i k ) ∈ A , then w ( i ) + · · · + w ( i k ) − m ∈ S .( ⇒ Let s , . . . , s k ∈ S such that( s mod m, . . . , s k mod m ) = ( i , . . . , i k ) ∈ A. Then there exist t , . . . , t k ∈ N such that s j = w ( i j ) + t j m , 1 ≤ j ≤ k , and thus, s + · · · + s k = ( w ( i ) + · · · + w ( i k ) − m ) + ( t + · · · + t k ) m ∈ S .By using the function AperyListOfNumericalSemigroupWRTElement(S,m) of [4], we can compute Ap(
S, m ) from a system of generators of S . Thereby, wehave the following algorithm to decide if a numerical semigroup S belongs ornot to C ( m, A ). 3 lgorithm 2.2. INPUT: A finite subset G of positive integers.OUTPUT: h G i ∈ C ( m, A ) or h G i / ∈ C ( m, A ).(1) If min(G) = m , return h G i / ∈ C ( m, A ).(2) If gcd( G ) = 1, return h G i / ∈ C ( m, A ).(3) Compute Ap( h G i , m ) = { w (0) , w (1) , . . . , w ( m − } .(4) If w ( i )+ · · · + w ( i k ) − m ∈ S for all ( i , . . . , i k ) ∈ A , return h G i ∈ C ( m, A ).(5) Return h G i / ∈ C ( m, A ).Let us illustrate the working of the previous algorithm through an example. Example 2.3.
Let us make use of Algorithm 2.2 with G = { , , } , m = 5,and A = { (1 , , (2 , } . • min( G ) = 5. • gcd( G ) = 1. • Ap( h G i ,
5) = { w (0) = 0 , w (1) = 16 , w (2) = 7 , w (3) = 18 , w (4) = 19 } . • w (1) + w (3) − ∈ h G i and w (2) + w (2) − ∈ h G i .Therefore, h G i = h , , i ∈ C (5 , { (1 , , (2 , } ).Let us recall that, if m is an integer greater than or equal to 2, then we denoteby ∆( m ) = { , m, →} . It is clear that ∆( m ) ∈ S m and that, if s , . . . , s k ∈ ∆( m ) \ { } and k ≥
2, then s + · · · + s k − m ∈ ∆( m ). Therefore, we have thenext result. Lemma 2.4.
Let m ∈ N \ { , } and A ⊆ S k ∈{ , →} { , . . . , m − } k ( A finite).Then ∆( m ) ∈ C ( m, A ) . The following result has an immediate proof. Therefore, we omit it.
Lemma 2.5.
Let m ∈ N \ { , } .1. If S, T ∈ S m , then S ∩ T ∈ S m .2. ∆( m ) is the maximum of S m .3. If s ∈ S m and S = ∆( m ) , then S ∪ { F( S ) } ∈ S m . Let us observe that, as an immediate consequence of the above lemma, wecan conclude that S m is a Frobenius pseudo-variety. Proposition 2.6. C ( m, A ) is a Frobenius pseudo-variety.Proof. Form Lemmas 2.4 and 2.5, we have that ∆( m ) is the maximum of C ( m, A ). It is also easy to see that, is S, T ∈ C ( m, A ), then S ∩ T ∈ C ( m, A ).In order to finish the proof, let us see that, if S ∈ C ( m, A ) and S = ∆( m ), then S ∪{ F( S ) } ∈ C ( m, A ). For that, we have to show that, if s , . . . , s k ∈ S ∪{ F( S ) } and ( s mod m, . . . , s k mod ) ∈ A , then s + · · · + s k − m ∈ S ∪ { F( S ) } . In effect,if F( S ) / ∈ { s , . . . , s k } , then the result is true because S ∈ C ( m, A ). On the oth-er hand, if F( S ) ∈ { s , . . . , s k } , then s + · · · + s k − m ≥ F( S ) and, consequently, s + · · · + s k − m ∈ S ∪ { F( S ) } . 4ur next purpose in this section is to show an algorithm that allows us tobuild all the elements of C ( m, A ) that have a fixed genus. To do that, we usethe concept of rooted tree.A graph G is a pair ( V, E ) where V is a non-empty set (whose elements arecalled vertices of G ) and E is a subset of { ( v, w ) ∈ V × V | v = w } (whoseelements are called edges of G ). A path (of length n ) connecting the vertices x and y of G is a sequence of different edges ( v , v ) , ( v , v ) , . . . , ( v n − , v n ) suchthat v = x and v n = y .We say that a graph G is a (rooted) tree if there exists a vertex r (known asthe root of G ) such that, for any other vertex x of G , there exists a unique pathconnecting x and r . If there exists a path connecting x and y , then we say that x is a descendant of y . In particular, if ( x, y ) is an edge of the tree, then we saythat x is a child of y . (See [10].)We define the graph G (cid:0) C ( m, A ) (cid:1) in the following way: C ( m, A ) is the setof vertices and ( S, S ′ ) ∈ C ( m, A ) × C ( m, A ) is an edge if S ∪ { F( S ) } = S ′ . Thefollowing result is a consequence of Lemma 4.2 and Theorem 4.3 of [7]. Theorem 2.7. G (cid:0) C ( m, A ) (cid:1) is a tree with root ∆( m ) . Moreover, the childrenset of S ∈ C ( m, A ) is (cid:8) S \ { x } | x ∈ msg( S ) , S \ { x } ∈ C ( m, A ) , x > F( S ) (cid:9) . Let S be a numerical semigroup and let x ∈ S . Then it is clear that S \ { x } is a numerical semigroup if and only if x ∈ msg( S ). Moreover, let us observethat, if x ∈ msg( S ) and m ∈ S \ { , x } , thenAp( S \ { x } , m ) = (cid:0) Ap(
S, m ) \ { x } (cid:1) ∪ { x + m } . In the following result we characterize the children of S ∈ C ( m, A ). Proposition 2.8.
Let S ∈ C ( m, A ) , Ap(
S, m ) = { w (0) , w (1) , . . . , w ( m − } and x ∈ msg( S ) \{ m } . Then S \{ x } ∈ C ( m, A ) if and only if w ( i )+ · · · + w ( i k ) = m + x for all ( i , . . . , i k ) ∈ A .Proof. Let us suppose that Ap( S \ { x } , m ) = (cid:0) Ap(
S, m ) \ { x } (cid:1) ∪ { x + m } = { w ′ (0) , w ′ (1) , . . . , w ′ ( m − } . (Necessity.) If w ( i ) + · · · + w ( i k ) = m + x , then m + x / ∈ { w ( i ) , . . . , w ( i k ) } and, therefore, w ( i ) = w ′ ( i ) , . . . , w ( i k ) = w ′ ( i k ). Moreover, w ( i ) + · · · + w ( i k ) − m = x / ∈ S \ { x } and, consequently, S \ { x } / ∈ C ( m, A ). (Sufficiency.) In order to prove that S \ { x } ∈ C ( m, A ), by Proposition 2.1,it is enough to see that, if ( i , . . . , i k ) ∈ A , then w ′ ( i )+ · · · + w ′ ( i k ) − m ∈ S \{ x } .Since S ∈ C ( m, A ), we easily deduce that w ′ ( i ) + · · · + w ′ ( i k ) − m ∈ S .Now, if w ′ ( i ) + · · · + w ′ ( i k ) − m / ∈ S \ { x } , then w ′ ( i ) + · · · + w ′ ( i k ) = m + x .Thus, w ( i ) = w ′ ( i ) , . . . , w ( i k ) = w ′ ( i k ) and w ( i ) + · · · + w ( i k ) = m + x , wherethe last equality is in contradiction with S ∈ C ( m, A ).Let us observe that a tree can be building in a recurrent way starting fromits root and connecting each vertex with its children. Let us also observe thatthe elements of C ( m, A ) with genus equal to g + 1 are precisely the children ofthe elements of C ( m, A ) with genus equal to g .We are ready to show the above announced algorithm.5 lgorithm 2.9. INPUT: A positive integer g .OUTPUT: { S ∈ C ( m, A ) | g( S ) = g } .(1) If g < m −
1, return ∅ .(2) X = { ∆( m ) } and i = m − i = g , return X .(4) For each S ∈ X , compute the set B S = (cid:8) x ∈ msg( S ) | x > F( S ) , x = m, S \ { x } ∈ C ( m, A ) (cid:9) . (5) If S S ∈ X B S = ∅ , return ∅ .(6) X = S S ∈ X (cid:8) S \ { x } | x ∈ B S (cid:9) , i = i + 1, and go to (3).For computing item (4) in the above algorithm, we use Proposition 2.6. Thenext result is a reformulation of Corollary 18 of [6] and it is useful for computingthe minimal system of generators of S \ { x } starting from the minimal systemof generators of S . Lemma 2.10.
Let S be a numerical semigroup with msg( S ) = { n < n < · · · < n e } . If i ∈ { , . . . , e } and n i > F( S ) , then msg( S \ { n i } ) = { n , . . . , n e } \ { n i } , if there exists j ∈ { , . . . , i − } such that n i + n − n j ∈ S ; (cid:0) { n , . . . , n e } \ { n i } (cid:1) ∪ { n i + n } , in other case. We illustrate the functioning of Algorithm 2.9 with an example.
Example 2.11.
Let us compute all the elements of C (5 , { (1 , , (1 , } ) withgenus equal to 6. • X = {h , , , , i} and i = 4. • B h , , , , i = { , } . • X = {h , , , , i , h , , , i} and i = 5. • B h , , , , i = { , , , } and B h , , , i = ∅ . • X = {h , , , , i , h , , , , i , h , , , i , h , , , i} and i = 6. • Return {h , , , , i , h , , , , i , h , , , i , h , , , i} .Taking advantage of the above example, we finish this section building thefirst three levels of the tree G (cid:0) C (5 , { (1 , , (1 , } ) (cid:1) . h , , , , i ✏✏✏✏✏✶ PPPPP✐ h , , , , i h , , , i ✘✘✘✘✘✘✿✚✚❃ ❩❩⑥ ❳❳❳❳❳❳② h , , , , i h , , , , i h , , , i h , , , i Observe that the number which appear next to each edge ( S ′ , S ) is the minimalgenerator x of S such that S ′ = S \ { x } .6 Second-level numerical semigroups
We say that a numerical semigroup S is of second-level if x + y + z − m( S ) ∈ S for all ( x, y, z ) ∈ ( S \ { } ) . We denote by L ,m the set of all the second-levelnumerical semigroups with multiplicity equal to m . Proposition 3.1.
Let S be a numerical semigroup with minimal system ofgenerators given by { m = n < n < · · · < n e } . Then the following twoconditions are equivalents.1. S ∈ L ,m .2. If ( i, j, k ) ∈ { , . . . , e } , then n i + n j + n k − m ∈ S .Proof. (1. ⇒ It is evident from the definition of second-level numericalsemigroup. (2. ⇒ Let ( x, y, z ) ∈ ( S \ { } ) . If 0 ∈ { x mod m, y mod m, z mod m } ,then it is clear that x + y + z − m ∈ S . Now, if 0 / ∈ { x mod m, y mod m, z mod m } , then we easily deduce that there exist ( i, j, k ) ∈ { , . . . , e } and ( s , s , s ) ∈ S such that ( x, y, z ) = ( n i , n j , n k ) + ( s , s , s ). Therefore, x + y + z − m =( n i + n j + n k − m ) + s + s + s ∈ S . Consequently, S ∈ L ,m .The above proposition allows us to easily decide whether a numerical semi-group is of second-level or not. Example 3.2.
Let us see that S = h , , i ∈ L ,m . In effect, it is clear that { − , − , ,
16 + 16 + 16 − } = { , , , } ⊆ S .Therefore, by Proposition 3.1, we have that S ∈ L ,m .Now our intention is to show that L ,m is a modular Frobenius pseudo-variety. Proposition 3.3.
Let m ∈ N \ { , } . Then L ,m = C ( m, { , . . . , m − } ) .Proof. Let S ∈ L ,m and Ap( S, m ) = { w (0) , w (1) , . . . , w ( m − } . If ( i, j, k ) ∈{ , . . . , m − } , then ( w ( i ) , w ( j ) , w ( k )) ∈ ( S \{ } ) and, therefore, w ( i )+ w ( j )+ w ( k ) − m ∈ S . By applying Proposition 2.1, we have that S ∈ C ( m, { , . . . , m − } ).In order to see the other inclusion, let S ∈ C ( m, { , . . . , m − } ) and( x, y, z ) ∈ ( S \ { } ) . Firstly, if 0 ∈ { x mod m, y mod m, z mod m } , then itis clear that x + y + z − m ∈ S . Secondly, if 0 / ∈ { x mod m, y mod m, z mod m } ,then there exist ( i, j, k ) ∈ { , . . . , m − } and ( p, q, r ) ∈ N such that x = w ( i ) + pm , y = w ( j ) + qm , and z = w ( k ) + rm . Thus, x + y + z − m =( w ( i ) + w ( j ) + w ( k ) − m ) + ( p + q + r ) m ∈ S and, therefore, S ∈ L ,m .As an immediate consequence of Propositions 2.6 and 3.3, we have the fol-lowing result. Corollary 3.4.
Let m ∈ N \ { , } . Then L ,m is a modular pseudo-Frobeniusvariety and, in addition, ∆( m ) is the maximum of L ,m . L ,m .To do this, we should characterize the possible children of each element in L ,m .Let S be a numerical semigroup with msg( S ) = { n , . . . , n e } . If s ∈ S , thenwe denote by L S ( s ) = max { a + · · · + a e | ( a , . . . , a e ) ∈ N e and a n + · · · + a e n e = s } . Proposition 3.5.
Let m ∈ N \ { , } , S ∈ L ,m , and x ∈ msg( S ) \ { m } . Then S \ { x } ∈ L ,m if and only if L S \{ x } ( x + m ) ≤ .Proof. (Necessity.) Let us suppose that L S \{ x } ( x + m ) ≥
3. Then there exists( a, b, c ) ∈ ( S \{ , x } ) such that x + m = a + b + c . Thus, a + b + c − m = x / ∈ S \{ x } and, therefore, S \ { x } / ∈ L ,m . (Sufficiency.) If ( a, b, c ) ∈ ( S \ { , x } ) , then a + b + c − m ∈ S , since S ∈ L ,m . Now, if a + b + c − m = x , then L S \{ x } ( x + m ) ≥
3. Therefore, a + b + c − m = x and, consequently, a + b + c − m ∈ S \ { x } . Thus, we concludethat S \ { x } ∈ L ,m .By applying Theorem 2.7, Propositions 3.3 and 3.5, and Lemma 2.10, wecan easily build the tree G( L ,m ). Example 3.6.
The first four levels of G( L , ) appear in the following figure. h , , , i ✏✏✏✏✏✶ ✻ PPPPP✐ h , , , i h , , i h , , i ✟✟✟✯ ✻ ❍❍❍❨ ❍❍❍❨ h , , , i h , , , i h , , i h , , i ✚✚❃ ✻ ❩❩⑥ ❅❅■ ❳❳❳❳❳❳② h , , , i h , , , i h , , i h , , , i h , , i The Frobenius problem (see [5]) consists in finding formulas that allow usto compute the Frobenius number and the genus of a numerical semigroup interms of the minimal system of generators of such a numerical semigroup. Thisproblem was solved in [12] for numerical semigroups with embedding dimensiontwo. At present, the Frobenius problem is open for embedding dimension greaterthan or equal to 3. However, if we know the Ap´ery set Ap(
S, x ) for some x ∈ S \ { } , then we have solved the Frobenius problem for S because we havethe following result from [11]. Lemma 3.7.
Let S be a numerical semigroup and let x ∈ S \ { } . Then1. F( S ) = (max(Ap( S, x ))) − x ,2. g( S ) = x ( P w ∈ Ap(
S,x ) w ) − x − . The knowledge of Ap(
S, x ) = { w (0) , w (1) , . . . , w ( x − } also allows us todetermine the membership of an integer to the numerical semigroup S . In fact,if n ∈ N , then n ∈ S if and only if n ≥ w ( n mod x ).8ow our purpose is to show that, if S ∈ L ,m , then is rather easy to computeAp( S, m ). We need the following easy result.
Lemma 3.8.
Let m ∈ N \ { , } , S ∈ L ,m , msg( S ) = { n = m, n , . . . , n e } .Then { , n , . . . , n e } ⊆ Ap(
S, m ) ⊆ { , n , . . . , n e }∪{ n i + n j | ( i, j ) ∈ { , . . . , e } } . As an immediate consequence of the above lemma we can formulate thefollowing result.
Proposition 3.9.
Let m ∈ N \{ , } , S ∈ L ,m , msg( S ) = { n = m, n , . . . , n e } .Then Ap(
S, m ) = { w (0) , w (1) , . . . , w ( m − } where w ( i ) is the smallest elementof { , n , . . . , n e } ∪ { n i + n j | ( i, j ) ∈ { , . . . , e } } that is congruent to i modulo m . Corollary 3.10.
Let m ∈ N \ { , } and S ∈ L ,m . Then m = m( S ) ≤ e( S )(e( S )+1)2 . Following the notation introduced in [8], we say that x ∈ Z \ S is a pseudo-Frobenius number of S if x + s ∈ S for all s ∈ S \ { } . We denote by PF( S )the set of all the pseudo-Frobenius numbers of S . The cardinality of PF( S ) isan important invariant of S (see [2]) that is the so-called type of S and it isdenoted by t( S ).Let S be a numerical semigroup. Then we define the following binary relationover Z : a ≤ S b if b − a ∈ S . In [9] it is shown that ≤ S is a partial order (thatis, reflexive, transitive, and antisymmetric). Moreover, Proposition 2.20 of [9]is the next result. Proposition 3.11.
Let S be a numerical semigroup and x ∈ S \ { } . Then PF( S ) = { w − x | w ∈ Maximals ≤ S (Ap( S, x )) } Let us observe that, if w, w ′ ∈ Ap(
S, x ), then w ′ − w ∈ S if and only if w ′ − w ∈ Ap(
S, x ). Therefore, Maximals ≤ S (Ap( S, x )) is the set { w ∈ Ap(
S, x ) | w ′ − w / ∈ Ap(
S, x ) \ { } for all w ′ ∈ Ap(
S, x ) } . We finish this section with an example that illustrates the above results.
Example 3.12.
Having in mind that S = h , , i is a second-level numericalsemigroup, it is easy to compute Ap( S, S, ⊆ { , , } ∪ { , , } and, by Proposition 3.9, weconclude that Ap( S,
5) = { , , , , } . On the other hand, by Lemma 3.7,we know that F( S ) = 19 − S ) = (6 + 12 + 13 + 19) − − = 8.Finally, since Maximals ≤ S (Ap( S, { , } , Proposition 3.11 allows us toclaim that PF( S ) = { , } and t( S ) = 2. Remark . The definition of second-level numerical semigroup can be easilygeneralize to greater levels. Thus, we say that a numerical semigroup is of n th-level if x + · · · + x n +1 − m( S ) ∈ S for all ( x , . . . , x n +1 ) ∈ ( S \{ } ) n +1 and denote9y L n,m the set of all the n th-level numerical semigroups with multiplicity equalto m .It is clear that for n th-level we obtain similar results to those of second-level.In particular, L n,m = C ( m, { , . . . , m − } n +1 ) and Proposition 3.5 remains truetaking L S \{ x } ( x + m ) ≤ n .On the other hand, having in mind that L ,m is the family of numericalsemigroups with maximal embedding dimension, we can observe that L ,m ( L ,m ( · · · ( L n,m ( . . . ( S m . We say that a numerical semigroup S is thin if 2 x − m( S ) ∈ S for all x ∈ S \ { } .We denote by T m the set of all the thin numerical semigroups with multiplicityequal to m . Proposition 4.1.
Let S be a numerical semigroup with minimal system ofgenerators given by { m = n < n < · · · < n e } . Then the following twoconditions are equivalents.1. S ∈ T m .2. n i − m ∈ S for all i ∈ { , . . . , e } .Proof. (1. ⇒ It follows from the definition of thin numerical semigroup. (2. ⇒ Let x ∈ S \{ } . If x ≡ m ), then it is clear that 2 x − m ∈ S .Now, if x m ), then we have that there exist i ∈ { , . . . , e } and s ∈ S such that x = n i + s . Therefore, 2 x − m = (2 n i − m ) + 2 s ∈ S and, consequently, S ∈ T m .The above proposition allows us to easily decide whether a numerical semi-group is thin or not. Example 4.2.
Let us see that S = h , , i ∈ T . In effect, it is clear that { · − , · − } = { , } ⊆ S . Therefore, by Proposition 4.1, we have that S ∈ T m .Now we want to show that T m is a modular Frobenius pseudo-variety. Proposition 4.3. T m = C ( m, { (1 , , (2 , . . . , ( m − , m − } ) for all m ∈ N \ { , } .Proof. If S ∈ T m and Ap( S, m ) = { w (0) , w (1) , . . . , w ( m − } , then it is clearthat { w (1)+ w (1) − m, . . . , w ( m − w ( m − − m } ⊆ S . Therefore, by applyingProposition 2.1, we have that S ∈ C ( m, { (1 , , (2 , . . . , ( m − , m − } ).To see the other inclusion, let S ∈ C ( m, { (1 , , (2 , . . . , ( m − , m − } ) and x ∈ S \{ } . On the one hand, if x ≡ m ), then it is clear that 2 x − m ∈ S .On the other hand, if x m ), then there exist i ∈ { , . . . , m − } and t ∈ N such that x = w ( i )+ tm . Therefore, 2 x − m = ( w ( i )+ w ( i ) − m )+ 2 tm ∈ S and in consequence S ∈ T m . 10rom Propositions 2.6 and 4.3, we get the following result. Corollary 4.4.
Let m ∈ N \ { , } . Then T m is a modular pseudo-Frobeniusvariety and, in addition, ∆( m ) is the maximum of S ∈ T m . In order to build the tree associated with the pseudo-variety S ∈ T m , we aregoing to characterize the possible children of each S ∈ T m . Proposition 4.5.
Let m ∈ N \ { , } , S ∈ T m , and x ∈ msg( S ) \ { m } . Then S \ { x } ∈ T m if and only if x + m / ∈ S .Proof. (Necessity.) If x + m ∈ S , then x + m ∈ S \ { , x } and 2 x + m − m = x / ∈ S \ { x } . Therefore, S \ { x } / ∈ T m . (Sufficiency.) If a ∈ S \ { , x } , then 2 a − m ∈ S because S ∈ T m . Now,if 2 a − m = x , then x + m = a ∈ S . Therefore, 2 a − m = x and, consequently,2 a − m ∈ S \ { x } . Thereby, S \ { x } ∈ T m .By applying Theorem 2.7, Propositions 4.3 and 4.5, and Lemma 2.10, wecan build the tree G( T m ). Let us see an example. Example 4.6.
In the next figure we have the first four levels of G( T ). h , , , i ✏✏✏✏✏✶ PPPPP✐ h , , , i h , , i ✘✘✘✘✘✘✿ ✻ ❳❳❳❳❳❳② h , , , i h , , , i h , , i ✑✑✑✸ ◗◗◗❦ ✻ ❍❍❍❨ h , , , i h , , , i h , , , i h , , i We finish this section describing the Ap´ery set for S ∈ T m . Proposition 4.7.
Let m ∈ N \ { } , S ∈ T m , msg( S ) = { m = n , n , . . . , n e } .Then Ap(
S, m ) = { w (0) , w (1) , . . . , w ( m − } , where w ( i ) is the least elementof the set { a n + . . . + a e n e | ( a , . . . , a e ) ∈ { , } e − } that is congruent to i modulo m . Corollary 4.8.
Let m ∈ N \ { } and S ∈ T m . Then m = m( S ) ≤ e( S ) − . We say that a numerical semigroup S is strong if x + y − m( S ) ∈ S for all( x, y ) ∈ ( S \ { } ) such that x y (mod m( S )). We denote by R m the set ofall the strong numerical semigroups with multiplicity equal to m . Proposition 5.1.
Let S be a numerical semigroup with minimal system ofgenerators given by { m = n < n < · · · < n e } . Then the following twoconditions are equivalents. . S ∈ R m .2. { n i + n j − m, n i − m } ⊆ S for all i ∈ { , . . . , e } and for all ( i, j ) ∈{ , . . . , e } such that i = j .Proof. (1. ⇒ It is enough to observe that n i n j (mod m ) and that2 n i n i (mod m ). (2. ⇒ Let x, y ∈ S \ { } such that x y (mod m). If x ≡ m ) or y ≡ m ), then it is clear that x + y − m ∈ S . Now, if x m ) and y m ), then there exists ( i, j ) ∈ { , . . . , m − } , with i = j , and thereexists ( p, q ) ∈ N such that x = w ( i )+ pm and y = w ( j )+ qm . Moreover, if thereexists ( a, b ) ∈ { , . . . , e } such that a = b and w ( i ) − n a , w ( j ) − n b ∈ S , then itis easy to see that x + y − m ∈ S . In other case, there exists ( r, t ) ∈ ( N \ { } ) such that w ( i ) = r · n a and w ( j ) = t · n a for some a ∈ { , . . . , e } . Then, since i = j , we deduce that r + t ≥ x + y − m ∈ S . In conclusion, S ∈ R m .The above proposition allows us to easily decide whether a numerical semi-group is strong or not. Example 5.2. If S = h , , i , then { − , · − , · − } = { , , } ⊆ S .Therefore, by Proposition 5.1, we have that S ∈ R m .Now we want to show that R m is a modular Frobenius pseudo-variety. Letus denote A = { , . . . , m − } \ { (1 , , (2 , . . . , ( m − , m − } . Proposition 5.3. R m = C ( m, A ) for all m ∈ N \ { , } .Proof. Let S ∈ R m , Ap( S, m ) = { w (0) , w (1) , . . . , w ( m − } , and ( i, j ) ∈ A .Then { ( w ( i ) , w ( j ) } ∈ ( S \ { } ) and w ( i ) w ( j ) (mod m ). Therefore, w ( i ) − w ( j ) − m ∈ S and, by applying Proposition 2.1, we have that S ∈ C ( m, A ).To see the other inclusion, let S ∈ C ( m, A ) and ( x, y ) ∈ ( S \ { } ) . Onthe one hand, if 0 ∈ { x mod m, y mod m } , then it is clear that x + y − m ∈ S .On the other hand, if 0 / ∈ { x mod m, y mod m } , then there exist ( i, j ) ∈ A and ( p, q ) ∈ N such that x = w ( i ) + pm and y = w ( j ) + qm . Therefore, x + y − m = ( w ( i ) + w ( i ) − m ) + ( p + q ) m ∈ S and, consequently, S ∈ R m .From Propositions 2.6 and 5.3, we get the following result. Corollary 5.4.
Let m ∈ N \ { , } . Then R m is a modular pseudo-Frobeniusvariety and, in addition, ∆( m ) is the maximum of S ∈ R m . We are now interested in the description of the tree associated with thepseudo-variety S ∈ R m . In order to do that, we are going to characterize thechildren of an arbitrary S ∈ R m . Proposition 5.5.
Let m ∈ N \ { , } , S ∈ R m , and x ∈ msg( S ) \ { m } . Then S \ { x } ∈ R m if and only if x + m / ∈ { a + b | a, b ∈ msg( S ) \ { m, x } , a = b } ∪ { a | a ∈ msg( S ) \ { m, x }} . roof. (Necessity.) If a, b ∈ msg( S ) \{ m, x } and a = b , then ( a, b ) ∈ ( S \{ , x } ) and a b (mod m ). Since S \ { x } ∈ R m , we have that a + b − m ∈ S \ { x } and,therefore, a + b − x = x . Thus, x + m / ∈ { a + b | a, b ∈ msg( S ) \ { m, x } , a = b } .On the other hand, if a ∈ msg( S ) \ { m, x } , then ( a, a ) ∈ ( S \ { , x } ) and a a (mod m ). Once again, since S \{ x } ∈ R m , we have that 3 a − m ∈ S \{ x } and, therefore, a + b − x = x . Thus, x + m / ∈ { a | a ∈ msg( S ) \ { m, x }} . (Sufficiency.) Let a, b ∈ S \ { , x } such that a = b . Since S ∈ R m , we havethat a + b − m ∈ S and 3 a − m ∈ S . Now, by Lemma 2.10, we know thatmsg( S ) \ { x } ⊆ msg( S \ { x } ) ⊆ (msg( S ) \ { x } ) ∪ { x + m } . Thus, from this factand the hypothesis, it easily follows that a + b − m = x and 3 a − m = x , thatis, a + b − m, a − m ∈ S \ { x } . By applying Proposition 5.1, we conclude that S \ { x } ∈ R m .From Theorem 2.7, Propositions 5.3 and 5.5, and Lemma 2.10, we can buildthe tree G( R m ). Let us see an example. Example 5.6.
In the next figure we have the first four levels of G( R ). h , , , i ✟✟✟✯ ❍❍❍❨ h , , , i h , , i ✏✏✏✏✏✶ PPPPP✐ ❅❅■ h , , , i h , , , i h , , i ✟✟✟✯ ✻ ❍❍❍❨ ❅❅■ h , , , i h , , , i h , , i h , , , i We finish this section describing the Ap´ery set for S ∈ R m . Proposition 5.7.
Let m ∈ N \{ , } , S ∈ R m , msg( S ) = { m = n , n , . . . , n e } .Then Ap( S ) = { w (0) , w (1) , . . . , w ( n ) } , where w ( i ) is the least element of the set { , n , . . . , n e } ∪ { n , . . . , n e } that is congruent to i modulo m . Corollary 5.8.
Let m ∈ N \ { , } and S ∈ R m . Then m = m( S ) ≤ S ) − . It is interesting to observe that, under the hypotheses of Proposition 5.7, n i ∈ Maximals ≤ S (Ap( S, m )) if and only if 2 n i / ∈ Ap(
S, m ). In addition,2 n i ∈ Ap(
S, m ) if and only if 2 n i ∈ Maximals ≤ S (Ap( S, m )). Therefore, fromPropositions 3.11 and 5.7, we get the next result.
Corollary 5.9.
Let m ∈ N \ { , } and S ∈ R m . Then t( S ) = e( S ) − . It is well known that, if S is a numerical semigroup, then e( S ) ≤ m( S ) andt( S ) ≤ m( S ) − Corollary 5.10.
Let m ∈ N \ { , } and S ∈ R m . Then m( S ) − ≤ t( S ) ≤ m( S ) − (or, equivalently, m( S )+12 ≤ e( S ) ≤ m( S ) ). cknowledgement Both authors are supported by the project MTM2017-84890-P (funded by Minis-terio de Econom´ıa, Industria y Competitividad and Fondo Europeo de Desarrol-lo Regional FEDER) and by the Junta de Andaluc´ıa Grant Number FQM-343.
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