Goto Chapter: Top 1 2 3 4 5 6 7 8 9 A B C Bib Ind
 Top of Book   Previous Chapter   Next Chapter 

3 Basic operations with numerical semigroups
 3.1 The definitions
  3.1-1 MultiplicityOfNumericalSemigroup

  3.1-2 GeneratorsOfNumericalSemigroup

  3.1-3 SmallElementsOfNumericalSemigroup

  3.1-4 FirstElementsOfNumericalSemigroup

  3.1-5 AperyListOfNumericalSemigroupWRTElement

  3.1-6 DrawAperyListOfNumericalSemigroup

  3.1-7 AperyListOfNumericalSemigroupAsGraph
 3.2 Frobenius Number
  3.2-1 FrobeniusNumberOfNumericalSemigroup

  3.2-2 FrobeniusNumber

  3.2-3 PseudoFrobeniusOfNumericalSemigroup
 3.3 Gaps
  3.3-1 GapsOfNumericalSemigroup

  3.3-2 FundamentalGapsOfNumericalSemigroup

  3.3-3 SpecialGapsOfNumericalSemigroup

3 Basic operations with numerical semigroups

3.1 The definitions

3.1-1 MultiplicityOfNumericalSemigroup
> MultiplicityOfNumericalSemigroup( NS )( attribute )

NS is a numerical semigroup. Returns the multiplicity of NS, which is the smallest positive integer belonging to NS.

gap> S := NumericalSemigroup("modular", 7,53);
<Modular numerical semigroup satisfying 7x mod 53 <= x >
gap> MultiplicityOfNumericalSemigroup(S);
8

3.1-2 GeneratorsOfNumericalSemigroup
> GeneratorsOfNumericalSemigroup( S )( function )
> GeneratorsOfNumericalSemigroupNC( S )( function )
> MinimalGeneratingSystemOfNumericalSemigroup( S )( attribute )

S is a numerical semigroup. GeneratorsOfNumericalSemigroup returns a set of generators of S, which may not be minimal. GeneratorsOfNumericalSemigroupNC returns the set of generators recorded in S!.generators, which may not be minimal. MinimalGeneratingSystemOfNumericalSemigroup returns the minimal set of generators of S.

gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> GeneratorsOfNumericalSemigroup(S);
[ 11, 12, 13, 32, 53 ]
gap> S := NumericalSemigroup(3, 5, 53);
<Numerical semigroup with 3 generators>
gap> GeneratorsOfNumericalSemigroup(S);
[ 3, 5, 53 ]
gap> MinimalGeneratingSystemOfNumericalSemigroup(S);
[ 3, 5 ]

3.1-3 SmallElementsOfNumericalSemigroup
> SmallElementsOfNumericalSemigroup( NS )( attribute )

NS is a numerical semigroup. It returns the list of small elements of NS. Of course, the time consumed to return a result may depend on the way the semigroup is given.

gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 0, 3, 5 ]

3.1-4 FirstElementsOfNumericalSemigroup
> FirstElementsOfNumericalSemigroup( n, NS )( function )

NS is a numerical semigroup. It returns the list with the first n elements of NS.

gap> FirstElementsOfNumericalSemigroup(2,NumericalSemigroup(3,5,7));
[ 0, 3 ]
gap> FirstElementsOfNumericalSemigroup(10,NumericalSemigroup(3,5,7));
[ 0, 3, 5, 6, 7, 8, 9, 10, 11, 12 ]

3.1-5 AperyListOfNumericalSemigroupWRTElement
> AperyListOfNumericalSemigroupWRTElement( S, m )( operation )

S is a numerical semigroup and m is a positive element of S. Computes the Apéry list of S wrt m. It contains for every iin {0,...,m-1}, in the i+1th position, the smallest element in the semigroup congruent with i modulo m.

gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> AperyListOfNumericalSemigroupWRTElement(S,12);
[ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]

3.1-6 DrawAperyListOfNumericalSemigroup
> DrawAperyListOfNumericalSemigroup( ap )( function )

ap is the Apéry list of a numerical semigroup. This function draws the graph (ap, E) where the edge u -> v is in E iff v - u is in ap. To use this function, Graphviz (http://www.graphviz.org) should be installed and also Evince (http://www.gnome.org/projects/evince/) or ggv (http://directory.fsf.org/ggv.html).

3.1-7 AperyListOfNumericalSemigroupAsGraph
> AperyListOfNumericalSemigroupAsGraph( ap )( function )

ap is the Apéry list of a numerical semigroup. This function returns the adjacency list of the graph (ap, E) where the edge u -> v is in E iff v - u is in ap. The 0 is ignored.

gap> s:=NumericalSemigroup(3,7);
<Numerical semigroup with 2 generators>
gap> AperyListOfNumericalSemigroupWRTElement(s,10);
[ 0, 21, 12, 3, 14, 15, 6, 7, 18, 9 ]
gap> AperyListOfNumericalSemigroupAsGraph(last);
[ ,, [ 3, 6, 9, 12, 15, 18, 21 ],,, [ 6, 9, 12, 15, 18, 21 ],
[ 7, 14, 21 ],, [ 9, 12, 15, 18, 21 ],,, [ 12, 15, 18, 21 ],,
[ 14, 21 ], [ 15, 18, 21 ],,, [ 18, 21 ],,, [ 21 ] ]

3.2 Frobenius Number

The largest nonnegative integer not belonging to a numerical semigroup S is the Frobenius number of S. If S is the set of nonnegative integers, then clearly its Frobenius number is -1, otherwise its Frobenius number coincides with the maximum of the gaps (or fundamental gaps) of S. An integer z is a pseudo-Frobenius number of S if z+S\{0}subseteq S.

3.2-1 FrobeniusNumberOfNumericalSemigroup
> FrobeniusNumberOfNumericalSemigroup( NS )( attribute )

NS is a numerical semigroup. It returns the Frobenius number of NS. Of course, the time consumed to return a result may depend on the way the semigroup is given or on the knowledge already produced on the semigroup.

gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
4

3.2-2 FrobeniusNumber
> FrobeniusNumber( NS )( attribute )

This is just a synonym of FrobeniusNumberOfNumericalSemigroup (3.2-1).

3.2-3 PseudoFrobeniusOfNumericalSemigroup
> PseudoFrobeniusOfNumericalSemigroup( S )( attribute )

S is a numerical semigroup. It returns set of pseudo-Frobenius numbers of S.

gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> PseudoFrobeniusOfNumericalSemigroup(S);
[ 21, 40, 41, 42 ]

3.3 Gaps

A gap of a numerical semigroup S is a nonnegative integer not belonging to S. The fundamental gaps of S are those gaps that are maximal with respect to the partial order induced by division in N. The special gaps of a numerical semigroup S, are those fundamental gaps such that if they are added to the given numerical semigroup, then the resulting set is again a numerical semigroup.

3.3-1 GapsOfNumericalSemigroup
> GapsOfNumericalSemigroup( NS )( attribute )

NS is a numerical semigroup. It returns the set of gaps of NS.

gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 1, 2, 4 ]

3.3-2 FundamentalGapsOfNumericalSemigroup
> FundamentalGapsOfNumericalSemigroup( S )( attribute )

S is a numerical semigroup. It returns the set of fundamental gaps of S.

gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> FundamentalGapsOfNumericalSemigroup(S);
[ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ]
gap> GapsOfNumericalSemigroup(S);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29,
  30, 31, 40, 41, 42 ]

3.3-3 SpecialGapsOfNumericalSemigroup
> SpecialGapsOfNumericalSemigroup( S )( attribute )

S is a numerical semigroup. It returns the special gaps of S.

gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> SpecialGapsOfNumericalSemigroup(S);
[ 40, 41, 42 ]
 Top of Book   Previous Chapter   Next Chapter 
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 A B C Bib Ind

generated by GAPDoc2HTML