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Quotients of a numerical semigroup by a positive integer
J. C. Rosales
Porto 2008 I P T I B
Notation • Z denotes the set of integers • N denotes the set of nonnegative integers
• hn1,...,nei = {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 singularity degree of S I P T I B
Quotient Let S be a numerical semigroup and let p be a positive integer
S = {x ∈ N | px ∈ S} p
• This set is again a numerical semigroup S • S ⊆ p S • = N iff p ∈ S p I P T I B
J. C. Rosales, J. M. Urbano, Proportionally modular diophantine inequalities and full semigroups, Semigroup Forum 72(2006), 362- 374
Theorem
Let n1, n2 and p be positive integers with n1 and n2 relatively prime. hn ,n i Then 1 2 is a proportionally modular numerical semigroup. p Every proportionally modular numerical semigroup is of this form I P T I B
• We have an algorithm that allows us to determine whether or not a numerical semigroup is the quotient of an embedding dimension two numerical semigroup by a positive integer h i hn ,n i • If un − vn = 1, then S n1 , n2 = 1 2 . Thus by using 2 1 up vp p Bezout´ sequences, one can compute a minimal generating hn ,n i system of 1 2 . p I P T I B
Open problems Find formulas for
1. the largest multiple of p not belonging to hn1,n2i
2. the cardinality of the set of multiples of p not in hn1,n2i
3. the least multiple of p in hn1,n2i I P T I B
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
ha,a + 1i p
with a and p positive integers
Open problem
n1 = a and n2 = a + 1 I P T I B
∗ 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 there exist positive integers q1,...,qn, m1,...,mn and L such that
1) gcd{qi,mi} = gcd{L,qi} = gcd{L,mi} = 1 for all i 1 2) S = Tn hq ,m i L i=1 i i I P T I B
J. C. Rosales, P. A. Garc´ıa-Sanchez,´ Numerical semigroups having a Toms decomposition, Canad. Math. Bull. 51 (2008), 134-139
Theorem A numerical semigroup admits a Toms decomposition if and only if it is the intersection of finitely many proportionally modular numerical semigroups
M. Delgado, P. A. Garc´ıa-Sanchez,´ J. C. Rosales, J. M. Urbano- Blanco, Systems of proportionally modular Diophantine inequalities, Semigroup Forum
• Algorithm to detect whether or not a numerical semigroup admits a Toms decomposition I P T I B
M. A. Moreno, J. Nicola, E. Pardo, H. Thomas, Numerical semi- groups that cannot we written as an intersection a d-squashed semigroups, preprint
• There are numerical semigroups that are not the quotient of an embedding dimension three numerical semigroup
Open problem Find a procedure to determine if a numerical semigroup is the quotient of an embedding dimension three numerical semigroup I P T I B
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 as the intersection of numerical semigroups properly containing it
• Every numerical semigroup is a finite intersection of irreducible numerical semigroups I P T I B
• A numerical semigroup is irreducible if and only if it is maximal in the set of numerical semigroups with its same Frobenius number
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 either symmetric or pseudo-symmetric I P T I B
J. C. Rosales, P. A. Garc´ıa-Sanchez,´ Every numerical semigroup is one half of symmetric numerical semigroup, Proc. Amer. Math. Soc. 136 (2008), 475-477
Theorem Every numerical semigroup is one half of a symmetric numerical semigroup I P T I B
J. C. Rosales, P. A. Garc´ıa-Sanchez,´ Every numerical semigroup is one 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 ( ) F(S) • S is pseudo-symmetric if and only if PF(S) = F(S), 2 I P T I B
Let S be a numerical semigroup with PF(S) = {f1,...,ft } and let x ∈ S
• x < S if and only if fi − x ∈ S for some i ∈ {1,...,t}
Theorem
Assume that {n1,...,np} generates S. TFAE: T • T is a symmetric numerical semigroup with S = 2 • T = h2n1,...,2np,f − 2f1,...,f − 2ft i for some odd integer f such that f − fi − fj ∈ S for all i,j ∈ {1,...,t}
• Every numerical semigroup is one half of infinitely many symmetric numerical semigroups I P T I B
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 ! S F(S) F = 2 2 • A numerical semigroup is not one half of infinitely many pseudo-symmetric numerical semigroups • One half of a pseudo-symmetric numerical semigroup is always irreducible I P T I B
Theorem A numerical semigroup is irreducible if and only if if is one half of a pseudo-symmetric numerical semigroup
• Every numerical semigroup is one fourth of a pseudo-symmetric numerical semigroup I P T I B
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
( ) T D(S) = T | T is a numerical semigroup and S = 2
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 I P T I B
For S a numerical semigroup, m an odd integer in S and H an m-upper subset of G(S)
S(m,H) = (2S) ∪ (m + 2S) ∪ (m + 2H)
is a numerical semigroup m − 1 • g(S(m,H)) = 2g(S) + − #H 2 ( max{2F(S),m − 2},if H = G(S), • F(S(m,H)) = max{2F(S),2max(G(S) \ H) + m},otherwise
Theorem ( ) m an odd integer in S D(S) = S(m,H) | H an m − upper subset of G(S)
Moreover S(m1,H1) = S(m2,H2) if and only if (m1,H1) = (m2,H2) I P T I B
• 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 symmetric numerical semigroup • Every numerical semigroup is one half of infinitely many symmetric numerical semigroups I P T I B
Balanced numerical semigroup A numerical semigroup is balanced if it has as many odd gaps as even gaps
Let S be a numerical semigroup ! S g(S) • S is balanced if and only if g = 2 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 many balanced numerical semigroups I P T I B
Theorem Every symmetric numerical semigroup is one half of a balanced pseudo-symmetric numerical semigroup
• Every numerical semigroup is one fourth of infinitely many balanced pseudo-symmetric numerical semigroups I P T I B
Open problem Let S be a numerical semigroup. Find a formula, depending on S, for min{g(T) | T ∈ D(S)}