‣ 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 ) |
‣ ReducedSetOfGeneratorsOfNumericalSemigroup( S[, bool, n] ) | ( 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 ReducedSetOfGeneratorsOfNumericalSemigroupNC Returns a set with possibly fewer generators than those recorded in S!.generators. It changes S!.generators to the set returned. The function has 1 to 3 arguments. One of them a numerical semigroup. Then an argument is a boolean (true means that all the elements not belonging to the Apery set with respect to the multiplicity are removed; the default is "false") and another argument is a positive integer n (meaning that generators that can be written as the sum of n or less generators are removed; the default is "2"). The boolean or the integer may not be present. If a minimal generating set for S is known or no generating set is known, then the minimal generating system is returned. 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 ] gap> ReducedSetOfGeneratorsOfNumericalSemigroup(NumericalSemigroup(5,7,9,10,25)); [ 5, 7, 9, 25 ] gap> ReducedSetOfGeneratorsOfNumericalSemigroup(true,NumericalSemigroup(5,7,9,10,25,28)); [ 5, 7, 9, 28 ] gap> ReducedSetOfGeneratorsOfNumericalSemigroup(NumericalSemigroup(5,7,9,10,25,28),3); [ 5, 7, 9 ]
‣ EmbeddingDimensionOfNumericalSemigroup( NS ) | ( attribute ) |
NS is a numerical semigroup. It returns the cardinality of its minimal generating system.
‣ 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 with respect to m. It contains for every i∈ {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 ]
‣ 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}⊆ 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).
‣ ConductorOfNumericalSemigroup( NS ) | ( attribute ) |
This is just a synonym of FrobeniusNumberOfNumericalSemigroup (NS)+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 ]
‣ TypeOfNumericalSemigroup( NS ) | ( attribute ) |
Stands for Length(PseudoFrobeniusOfNumericalSemigroup (NS)).
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 ]
‣ GenusOfNumericalSemigroup( NS ) | ( attribute ) |
NS is a numerical semigroup. It returns the number of gaps of NS.
‣ 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 ]
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