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

Numerical semigroups:suitable sets of pseudo-Frobenius numbers

Aureliano M. Robles-PerezUniversidad de Granada

A talk based on joint works with Manuel Delgado,Pedro A. Garcıa-Sanchez and Jose Carlos Rosales

Meeting on Computer Algebra and Applications (EACA 2016)

22-24th June 2016, Logrono

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

Notation

∗ Z= {. . . ,−2,−1,0,1,2, . . .}

∗ N= {0,1,2, . . .}

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

DefinitionI 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}).


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].


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, Garcıa-Sanchez, Garcıa-Garcıa, Jimenez-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.


Problems to solve

Questions1. Find conditions on the set PF that ensure that S(PF) , ∅.

2. Find an algorithm to compute S(PF).

Solved casesI 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, Garcıa-Sanchez, R-P, LMS J. Comput. Math. 19(1), 2016)


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 thatPF(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}).


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− 20

2 = 7.I S =

(〈7〉∪

{202 + 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}).


Question 1 for n=2 and n=3

ExampleI 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)

ExampleI 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− 162 = 4, g∗3 = 16− 18

2 = 7)


General results: initial necessary condition

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

some f ∈ PF(S).

LemmaI 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 thatgk − (gi −g) ∈ S.

CorollaryI g1 ≥ gn −gn−1.

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


General results

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

PropositionI 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, thenS({g,αg +β,(α+ 1)g +β}) , ∅.

PropositionI If g1 + g2 ≥

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


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


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 assumingthat free integers are either gaps or elements.

ExampleI 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


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).

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

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

ExampleI 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}.


Starting forced gapsLemma

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

LemmaI 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 ∈ 〈PF〉,gi > g} .

CorollaryI Let PF be a set of positive integers. Then sfg(PF) ⊆ GF (PF).

ExampleI PF = {17,21} ⇒ {17,21}∪ {1,2}∪ {4} ⇒ sfg(PF) = {1,2,3,4,7,17,21}.


Forced elements: big forced elements

LemmaI If S is a numerical semigroup such that m(S) = m, then

F(S)− i ∈ S ∪PF(S) for all i ∈ {1, . . . ,m−1}.

ConsequenceI Let fg(PF) ⊆ GF (PF). If m = min(N \ (fg(PF)∪{0})), then

max(PF)− i ∈ EF (PF)∪PF for all i ∈ {1, . . . ,m−1}.

ExampleI PF = {17,21} ⇒ fg(PF) = sfg(PF) = {1,2,3,4,7,17,21} ⇒

bfe(PF) = {18,19,20}.I PF = {4,9}⇒ fg(PF) = sfg(PF) = {1,2,3,4,5,9}⇒ bfe(PF) = {5,6,7,8}⇒ Contradiction!


Forced elements: elements forced by exclusion (1)

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

some f ∈ PF(S).

ConsequenceI Let x ∈ fg(PF) ⊆ GF (PF). If there exists a unique f∗ ∈ PF(S) such that(

f∗−x > 0 and f∗−x < fg(PF)), then f∗−x ∈ efe1(PF) ⊆ EF (PF).


efe1(PF) = {13} (from x = 4).


Forced elements: elements forced by exclusion (2)Lemma

I If S is a numerical semigroup, then x ∈ S if and only if f −x < S for allf ∈ PF(S).

ConsequenceI Let x < fg(PF)∪bfe(PF)∪ efe1(PF) such that x ≤max(PF) + 1. If(

f −x < 0 or f −x ∈ fg(PF))

for all f ∈ PF, then x ∈ efe2(PF) ⊆ EF (PF).


bfe(PF)∪ efe1(PF) = {13,18,19,20} ⇒ efe2(PF) = {14,22}.

∗ fe(PF) = 〈bfe(PF)∪efe1(PF)∪efe2(PF)〉∩ [1,max(PF)+1]⊆EF (PF).I fe({17,21}) = 〈13,14,18,19,20,22〉∩ [1,22] ={0,13,14,18,19,20,22} ⊆ EF ({19,29}).


Further forced gaps

RemarkI Let S be a numerical semigroup. If e ∈ S and f ∈ gaps(S), then

f −e < 0 or f −e ∈ gaps(S).

ConsequenceI Let fg(PF) ⊆ GF (PF) and fe(PF) ⊆ EF (PF). Then

∆ = (fg(PF)−e)∩N ⊆ ffg(PF) ⊆ GF (PF) for every e ∈ fe(PF).

ExampleI PF = {17,21} ⇒ sfg(PF) = {1,2,3,4,7,17,21} ⇒

fe(PF) = {0,13,14,18,19,20,22} ⇒∆ = {8} (from e = 13).

∗ ffg(PF) = sfg(PF)∪{Positive divisors of ∆} ⊆ GF (PF).I PF = {17,21} ⇒ ffg(PF) = {1,2,3,4,7,17,21}∪ {1,2,4,8}={1,2,3,4,7,8,17,21}.


Algorithm

∗ PF

∗ PF⇒ sfg(PF)

∗ PF, sfg(PF)⇒ fe(PF)

∗ PF, sfg(PF), fe(PF)⇒ ffg1(PF)

∗ PF, ffg1(PF), fe(PF)⇒ ffe1(PF)

∗ PF, ffg1(PF), ffe1(PF)⇒ ffg2(PF)

∗ PF, ffg2(PF), ffe1(PF)⇒ ffe2(PF)...

∗ Repeat until ffgk+1(PF) = ffgk (PF) and ffek+1(PF) = ffek (PF)

∗∗ ffgk (PF) ⊆ GF (PF), ffek (PF) ⊆ EF (PF)

∗∗ [1,max(PF) + 1] \ (ffgk (PF)∪ffek (PF)) ⊇ Free(PF)


Example∗ PF = {17,31}

∗ PF⇒ sfg(PF) = {1,2,7,14,17,31}∗ PF, sfg(PF)⇒ fe(PF) = {0,3,6,9,12,15,18,21,24,27,29,30,32}∗ PF, sfg(PF), fe(PF)⇒ffg1(PF) = {1,2,4,5,7,8,10,11,13,14,16,17,19,22,25,28,31}

∗ PF, ffg1(PF), fe(PF)⇒ffe1(PF) = {0,3,6,9,12,15,18,20,21,23,24,26,27,29,30,32}

∗ PF, ffg1(PF), ffe1(PF)⇒ ffg2(PF) = ffg1(PF)

∗ PF, ffg2(PF), ffe1(PF)⇒ ffe2(PF) = ffe1(PF)

∗∗ ffg2(PF) = GF (PF) ={1,2,4,5,7,8,10,11,13,14,16,17,19,22,25,28,31}

∗∗ ffe2(PF) = EF (PF) ={0,3,6,9,12,15,18,20,21,23,24,26,27,29,30,32}

∗∗ Free(PF) = ∅


Example

∗ PF = {11,17}, GF (PF) = {1,2,3,4,6,7,11,12,17},

EF (PF) = {0,5,10,13,14,15,16,18}, Free(PF) = {8,9}

∗∗ Algorithm with ffg(PF) = {1,2,3,4,6,7,11,12,17} andffe(PF) = {0,5,8,10,13,14,15,16,18}

S = 〈5,8,14〉= {0,5,8,10,13,14,15,16,18,→}

∗∗ Algorithm with ffg(PF) = {1,2,3,4,6,7,8,11,12,17} andffe(PF) = {0,5,9,10,13,14,15,16,18}

S = 〈5,9,13,16〉= {0,5,9,10,13,14,15,16,18,→}


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

5 6 9 10 11 12 15 16

10 11

〈5,9,13〉

〈6,10

,13,14〉

〈9,10

,13,14

,15,16〉

〈9,11

,13,14

,15,16

,19〉

∅

〈10,12

,13,14

,15,16

,18,19〉

〈11,12

,13,14

,15,16

,18,19

,20〉

∅ ∅

STOP

Numerical semigroups with pseudo-Frobenius numbers {17,21}


Running times

gap> pf:=[11,17];;

gap> ans:=NumericalSemigroupsWithPseudoFrobeniusNumbers(pf);;

gap> time;Length(ans);

0

2

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



16

6

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



16

1


References

V. Blanco and J.C. Rosales.The tree of irreducible numerical semigroups with fixed Frobenius number.Forum Math. 25 (2013), 1249–1261.

M. Delgado, P.A. Garcıa-Sanchez and J. Morais.“NumericalSgps”, a GAP package for numerical semigroups, Version 1.0.1; 2015.Available via http://www.gap-system.org/

M. Delgado, P.A. Garcıa-Sanchez, and A.M. Robles-Perez.Numerical semigroups with a given set of pseudo-Frobenius numbers.LMS J. Comput. Math. 19(1) (2016), 186–205.

A.M. Robles-Perez and J.C. Rosales.The genus, the Frobenius number, and the pseudo-Frobenius numbers of numericalsemigroups with type two.To appear in Proc. Roy. Soc. Edinburgh Sect. A.

J.C. Rosales, P.A. Garcıa-Sanchez, J.I. Garcıa-Garcıa, and J.A. Jimenez-Madrid.Fundamental gaps in numerical semigroups.J. Pure Appl. Algebra 189 (2004), 301–313.


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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