> 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 |
> 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 ] |
> 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 ] |
> 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 ] |
> 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 ] |
> 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).
> 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 ] ] |
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.
> 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 |
> FrobeniusNumber ( NS ) | ( attribute ) |
This is just a synonym of FrobeniusNumberOfNumericalSemigroup
(3.2-1).
> 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 ] |
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.
> 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 ] |
> 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 ] |
> 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 ] |
generated by GAPDoc2HTML