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Quotients of a numerical semigroup by a positiveinteger

J. C. Rosales

Porto 2008

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Notation

• Z denotes the set of integers

• N denotes the set of nonnegative integers

• 〈n1, . . . ,ne〉 = {λ1n1+ · · ·+λene | λ1, . . . ,λe ∈ N}

For a numerical semigroup S

• F(S) the largest integer not in S, the Frobenius number of S

• G(S) the set of nonnegative integers not in S, the gaps of S

• g(S) the cardinality of G(S), the gender S or singularitydegree of S

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Quotient

Let S be a numerical semigroup and let p be a positive integer

Sp= {x ∈ N | px ∈ S}

• This set is again a numerical semigroup

• S ⊆Sp

•Sp= N iff p ∈ S

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J. C. Rosales, J. M. Urbano, Proportionally modular diophantineinequalities and full semigroups, Semigroup Forum 72(2006), 362-374

Theorem

Let n1, n2 and p be positive integers with n1 and n2 relatively prime.

Then〈n1,n2〉

pis a proportionally modular numerical semigroup.

Every proportionally modular numerical semigroup is of this form

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• We have an algorithm that allows us to determine whether ornot a numerical semigroup is the quotient of an embeddingdimension two numerical semigroup by a positive integer

• If un2−vn1 = 1, then S([

n1up ,

n2vp

])=〈n1,n2〉

p. Thus by using

Bezout sequences, one can compute a minimal generating

system of〈n1,n2〉

p.

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Open problems

Find formulas for

1. the largest multiple of p not belonging to 〈n1,n2〉

2. the cardinality of the set of multiples of p not in 〈n1,n2〉

3. the least multiple of p in 〈n1,n2〉

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A. M. Robles, J. C. Rosales, Equivalent proportionally modular Dio-phantine inequalities, Archiv Math. 90 (2008), 24-30

Theorem

Every proportionally modular numerical semigroup is of the form

〈a,a +1〉p

with a and p positive integers

Open problem

n1 = a and n2 = a +1

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A. Toms, Strongly perforated K0-groups of simple C∗-algebra,Canad. Math. Bull. 46(2003), 457-472

Toms decomposition

A numerical semigroup S admits a Toms decomposition if thereexist positive integers q1, . . . ,qn, m1, . . . ,mn and L such that

1) gcd{qi ,mi} = gcd{L ,qi} = gcd{L ,mi} = 1 for all i

2) S =1L

⋂ni=1〈qi ,mi〉

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J. C. Rosales, P. A. Garcıa-Sanchez, Numerical semigroups havinga Toms decomposition, Canad. Math. Bull. 51 (2008), 134-139

Theorem

A numerical semigroup admits a Toms decomposition if and only ifit is the intersection of finitely many proportionally modularnumerical semigroups

M. Delgado, P. A. Garcıa-Sanchez, J. C. Rosales, J. M. Urbano-Blanco, Systems of proportionally modular Diophantine inequalities,Semigroup Forum

• Algorithm to detect whether or not a numerical semigroupadmits a Toms decomposition

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M. A. Moreno, J. Nicola, E. Pardo, H. Thomas, Numerical semi-groups that cannot we written as an intersection a d-squashedsemigroups, preprint

• There are numerical semigroups that are not the quotient ofan embedding dimension three numerical semigroup

Open problem

Find a procedure to determine if a numerical semigroup is thequotient of an embedding dimension three numerical semigroup

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J. C. Rosales, M. B. Branco, Irreducible numerical semigroups, Pa-cific J. Math. 209 (2003), 131-143

Irreducible numerical semigroup

A numerical semigroup is irreducible if it cannot be expressed asthe intersection of numerical semigroups properly containing it

• Every numerical semigroup is a finite intersection ofirreducible numerical semigroups

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• A numerical semigroup is irreducible if and only if it is maximalin the set of numerical semigroups with its same Frobeniusnumber

R. Froberg, C. Gottlieb, R. Haggvist, On numerical semigroups,Semigroup Forum 35(1987), 63-83

• A numerical semigroup is irreducible if and only if it is eithersymmetric or pseudo-symmetric

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J. C. Rosales, P. A. Garcıa-Sanchez, Every numerical semigroupis one half of symmetric numerical semigroup, Proc. Amer. Math.Soc. 136 (2008), 475-477

Theorem

Every numerical semigroup is one half of a symmetric numericalsemigroup

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J. C. Rosales, P. A. Garcıa-Sanchez, Every numerical semigroup isone half of infinitely many symmetric numerical semigroups, Comm.Algebra

Let S be a numerical semigroup

Pseudo-Frobenius number

The set of pseudo-Frobenius numbers of S is

PF(S) = {x ∈ Z \S | x +s ∈ S for all x ∈ S \ {0}}

The cardinality of PF(S) is the type of S, t(S)

• S is symmetric if and only if PF(S) = {F(S)} if and only t(S) = 1

• S is pseudo-symmetric if and only if PF(S) ={

F(S),F(S)

2

}

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Let S be a numerical semigroup with PF(S) = {f1, . . . , ft } and letx ∈ S

• x < S if and only if fi −x ∈ S for some i ∈ {1, . . . , t}

Theorem

Assume that {n1, . . . ,np} generates S. TFAE:

• T is a symmetric numerical semigroup with S =T2

• T = 〈2n1, . . . ,2np , f −2f1, . . . , f −2ft〉 for some odd integer fsuch that f − fi − fj ∈ S for all i, j ∈ {1, . . . , t}

• Every numerical semigroup is one half of infinitely manysymmetric numerical semigroups

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J. C. Rosales, One half of a pseudo-symmetric numerical semi-group, Bull. London Math. Soc.

• If S is a numerical semigroup and F(S) is even, then

F(S2

)=

F(S)2

• A numerical semigroup is not one half of infinitely manypseudo-symmetric numerical semigroups

• One half of a pseudo-symmetric numerical semigroup isalways irreducible

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Theorem

A numerical semigroup is irreducible if and only if if is one half of apseudo-symmetric numerical semigroup

• Every numerical semigroup is one fourth of apseudo-symmetric numerical semigroup

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A. M. Robles-Perez, J. C. Rosales, P. Vasco, The doubles of a nu-merical semigroup, preprint

Let S be a numerical semigroup

Doubles of S

D(S) ={

T | T is a numerical semigroup and S =T2

}

m-upper sets of gaps

A subset H of G(S) is an m-upper subset of G(S) if

1) (m+H)∩G(S) is empty

2) (m+H+H)∩G(S) is empty

3) if h ∈ H, then {g ∈ G(S) | g−h ∈ S} ⊆ H

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For S a numerical semigroup, m an odd integer in S and H anm-upper subset of G(S)

S(m,H) = (2S)∪ (m+2S)∪ (m+2H)

is a numerical semigroup

• g(S(m,H)) = 2g(S)+m−1

2−#H

• F(S(m,H)) ={

max{2F(S),m−2}, if H = G(S),max{2F(S),2max(G(S) \H)+m},otherwise

Theorem

D(S) ={

S(m,H) |m an odd integer in SH an m−upper subset of G(S)

}Moreover S(m1,H1) = S(m2,H2) if and only if (m1,H1) = (m2,H2)

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• If m is an odd integer in S greater than F(S), then

H = {x ∈ G(S) | F(S)−x ∈ G(S)}

is an m-upper subset of G(S) and S(m,H) is a symmetricnumerical semigroup

• Every numerical semigroup is one half of infinitely manysymmetric numerical semigroups

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Balanced numerical semigroup

A numerical semigroup is balanced if it has as many odd gaps aseven gaps

Let S be a numerical semigroup

• S is balanced if and only if g(S2

)=

g(S)2

Theorem

The set{T ∈ D(S) | T is balanced}

is not empty and has finitely many elements

• Every numerical semigroup is one half of finitely manybalanced numerical semigroups

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Theorem

Every symmetric numerical semigroup is one half of a balancedpseudo-symmetric numerical semigroup

• Every numerical semigroup is one fourth of infinitely manybalanced pseudo-symmetric numerical semigroups

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Open problem

Let S be a numerical semigroup. Find a formula, depending on S,for

min{g(T ) | T ∈ D(S)}