Numerical semigroups: suitable sets of pseudo Numerical semigroups: suitable sets of pseudo-Frobenius numbers Aureliano M. Robles-Perez´ Universidad de Granada A talk based on joint

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• Numerical semigroups: suitable sets of pseudo-Frobenius numbers

A talk based on joint works with Manuel Delgado, Pedro A. Garcı́a-Sánchez and José Carlos Rosales

Meeting on Computer Algebra and Applications (EACA 2016)

22-24th June 2016, Logroño

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 1 / 25

• Notation

∗ Z= {. . . ,−2,−1,0,1,2, . . .} ∗ N= {0,1,2, . . .}

Definition I S numerical semigroup = S cofinite submonoid of (N,+).

Definition I x < S is the Frobenius number of S if x + s ∈ S for all s ∈ N \ {0}. I x < S is a pseudo-Frobenius number of S if x + s ∈ S for all s ∈ S \ {0}.

∗ PF(S) = {Pseudo-Frobenius numbers of S}. ∗ F(S) = max(PF(S)) = max(Z \S) = Frobenius number of S. ∗ m(S) = min(S \ {0}).

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 2 / 25

• Remark

∗ t(S) = ]PF(S) = type of S.

∗∗ Let K[x] be a polynomial ring in one variable over a field K. We define the semigroup ring of S as the monomial subring

K[S] := K[ts | s ∈ S].

Then t(S) coincides with the Cohen-Macaulay type of K[S].

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 3 / 25

• Background

∗ f ∈ N \ {0}. ∗ S(f) = {S numerical semigroup | F(S) = f }. ∗ S(f) , ∅ for all f ∈ N \ {0}. ∗ There are algorithms to compute S(f)

(Rosales, Garcı́a-Sánchez, Garcı́a-Garcı́a, Jiménez-Madrid, J. Pure Appl. Algebra 189, 2004)

∗∗ PF = {g1,g2, . . . ,gn−1,gn} ⊆ N \ {0}. ∗∗ S(PF) = {S numerical semigroup | PF(S) = PF}. ∗∗ S(PF) may be empty if n ≥ 1. ∗∗ S(PF) is finite when it is non-empty.

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 4 / 25

• Problems to solve

Questions 1. Find conditions on the set PF that ensure that S(PF) , ∅. 2. Find an algorithm to compute S(PF).

Solved cases I PF = {f }: symmetric numerical semigroups.

(Blanco, Rosales, Forum Math. 25, 2013)

I PF = {f , f2 }: pseudo-symmetric numerical semigroups. (Blanco, Rosales, Forum Math. 25, 2013)

I Question 1 for PF = {g1,g2}. (R-P, Rosales, article to appear in Proc. Roy. Soc. Edinburgh Sect. A.)

I Question 2 for general PF. (Delgado, Garcı́a-Sánchez, R-P, LMS J. Comput. Math. 19(1), 2016)

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 5 / 25

• Question 1 for n=2 (1) Proposition

I Let g1,g2 be two positive integers such that g1 < g2 and g2 is odd. Let m = 2g1−g2. There exists a numerical semigroup S such that PF(S) = {g1,g2} if and only if m > 0, m - g1, m - g2, m - (g2−g1).

S = ( 〈m〉∪

{ g2 + 1

2 ,→

}) \ ({

g1−km | k ∈ N } ∪

{ g2−km | k ∈ N

}) Example

I PF = {17,21} is an admissible set of pseudo-Frobenius numbers. I m = 2 ·17−21 = 13. I S =

( 〈13〉∪

{ 21+1

2 ,→ } ) \ ( {17,4}∪ {21,8}

) = {0,11,→}\ {17,21}=

{0,9,15,16,17,18,21,22,23,24,25,26,27,28,30→}= 〈11,12,13,14,15,16,18,19,20〉 ∈ S({17,21}).

I S = 〈5,9,13〉 ∈ S({17,21}).

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 6 / 25

• Question 1 for n=2 (2) Proposition

I Let g1,g2 be two positive integers such that g1 < g2 and g2 is even. Let m = g1− g22 , 0. There exists a numerical semigroup S such that PF(S) = {g1,g2} if and only if m > 0, m - g1, m - g2, m - (g2−g1).

S = ( 〈m〉∪

{g2 2

+ 1,→ }) \ ({

g1−km | k ∈ N } ∪

{ g2−km | k ∈ N

}) Example

I PF = {17,20} is an admissible set of pseudo-Frobenius numbers. I m = 17− 202 = 7. I S =

( 〈7〉∪

{ 20 2 + 1,→

} ) \ ( {17}∪ {20,13}

) = {0,7,11,→}\ {13,17,20}=

{0,7,11,12,14,15,16,18,19,21,→}= 〈7,11,12,15,16〉 ∈ S({17,20}).

I S = 〈7,8,11〉 ∈ S({17,20}).

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 7 / 25

• Question 1 for n=2 and n=3

Example I PF = {9,15} is not a set of pseudo-Frobenius numbers.

(g∗ = 2×9−15 = 3) I PF = {9,10,15} is a set of pseudo-Frobenius numbers.

S = 〈4,13,14,19〉. (g∗1 = 2×9−15 = 3, g∗2 = 2×9−10 = 8, g∗3 = 2×10−15 = 5)

Example I PF = {12,18} is not a set of pseudo-Frobenius numbers.

(g∗ = 12− 182 = 3) I PF = {12,16,18} is a set of pseudo-Frobenius numbers.

S = 〈7,10,13,15,19〉. (g∗1 = 12−

18 2 = 3, g

∗ 2 = 12−

16 2 = 4, g

∗ 3 = 16−

18 2 = 7)

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 8 / 25

• General results: initial necessary condition

Lemma I If S is a numerical semigroup, then x ∈ Z \S if and only if f −x ∈ S for

some f ∈ PF(S).

Lemma I Let PF(S) = PF = {g1 < · · · < gn}, with n ≥ 2. If i ∈ {2, . . . ,n} and

g ∈ 〈PF〉 \ {0} with g < gi , then there exists k ∈ {1, . . . ,n} such that gk − (gi −g) ∈ S.

Corollary I g1 ≥ gn −gn−1.

Example I PF = {17,41} ⇒ 17 ≤ 41−17⇒S(PF) = ∅.

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 9 / 25

• General results

Lemma I If S({g1,g2,g3 = g1 + g2}) , ∅, then g3 is odd.

Proposition I Let α,β,g be positive integers.

• S({g,αg,βg}) , ∅ if and only if g is odd, α = 2 and β = 3. • If (α+ 1)g +β is odd and α > β, then S({g,αg +β,(α+ 1)g +β}) , ∅.

Proposition I If g1 + g2 ≥ 3g3+12 and g3 is odd, then S({g1,g2,g3}) , ∅.

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 10 / 25

• A simple idea to compute S(PF): by filtering S(max(PF))

gap> pf := [17,21];;

gap> nsf21 := NumericalSemigroupsWithFrobeniusNumber(21);;time;

Length(nsf21);

437

1828

gap> nspf1721 := Filtered(nsf21, s -> PseudoFrobeniusOfNumerical

Semigroup(s) = pf);;time;Length(nspf1721);

110

6

gap> pf := [17,31];;

gap> nsf31 := NumericalSemigroupsWithFrobeniusNumber(31);;time;

Length(nsf31);

88937

70854

gap> nspf1731 := Filtered(nsf31, s -> PseudoFrobeniusOfNumerical

Semigroup(s) = pf);;time;Length(nspf1731);

4906

1

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 11 / 25

• Alternative

∗ Determine the elements and the gaps of all numerical semigroups (“forced” integers) with the given set of pseudo-Frobenius numbers, obtaining a list of “free” integers.

∗ We then construct a tree in which branches correspond to assuming that free integers are either gaps or elements.

Example I PF = {17,21}

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 12 / 25

• Forced and free integers

∗ PF = {g1,g2, . . . ,gn−1,gn} a given set (⊆ N \ {0}). ∗ ` ∈ GF (PF) if ` is a gap of any S ∈ S(PF). ∗ ` ∈ EF (PF) if ` ≤ gn + 1 and it is an element of any S ∈ S(PF).

Proposition I S(PF) , ∅ if and only if GF (PF)∩EF (PF) = ∅.

∗∗ Free(PF) = {1, . . . ,gn} \ (GF (PF)∪EF (PF)).

Example I PF = {17,21}. I GF (PF) = {1,2,3,4,7,8,17,21}. I EF (PF) = {0,13,14,18,19,20,22}. I Free(PF) = {5,6,9,10,11,12,15,16}.

A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 13 / 25

• Starting forced gaps Lemma

I If S is a numerical semigroup, then m(S) ≥ t(S) + 1.

Lemma I Let PF(S) = {g1 < g2 < · · · < gn−1 < gn}, with n > 1. Let i ∈ {2, . . . , t(S)}

and g ∈ 〈PF(S)〉 with g < gi . Then gi −g ∈ gaps(S) = N \S.

∗ PF = {g1 < g2 < · · · < gn−1 < gn} (n > 1). ∗ Let sfg(PF) be the set of positive divisors of

PF∪{x ∈ N | 1 ≤ x ≤ n}∪ {gi −g | i ∈ {2, . . . ,n},g ∈ 〈P

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