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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
#A testall.tst NumericalSgps package Manuel Delgado
## Pedro A. Garcia-Sanchez
## (based on the cooresponding file of the 'example' package,
## by Alexander Konovalov)
##
## To create a test file, place GAP prompts, input and output exactly as
## they must appear in the GAP session. Do not remove lines containing
## START_TEST and STOP_TEST statements.
##
## The first line starts the test. START_TEST reinitializes the caches and
## the global random number generator, in order to be independent of the
## reading order of several test files. Furthermore, the assertion level
## is set to 2 by START_TEST and set back to the previous value in the
## subsequent STOP_TEST call.
##
## The argument of STOP_TEST may be an arbitrary identifier string.
##
gap> START_TEST("NumericalSgps package: testall.tst");
## Set info level to 0 as suggested by Alexander Konovalov
gap> INFO_NSGPS:=InfoLevel(InfoNumSgps);;
gap> SetInfoLevel( InfoNumSgps, 0);
# Note that you may use comments in the test file
# and also separate parts of the test by empty lines
# First load the package without banner (the banner must be suppressed to
# avoid reporting discrepancies in the case when the package is already
# loaded)
gap> LoadPackage("numericalsgps",false);
true
# Check that the data are consistent
gap> ns := NumericalSemigroup([10..30]);
<Numerical semigroup with 10 generators>
gap> IsNumericalSemigroup(ns);
true
gap> MinimalGeneratingSystemOfNumericalSemigroup(ns);
[ 10 .. 19 ]
gap> GapsOfNumericalSemigroup(ns);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
#############################################################################
# Some more elaborated tests
gap> NumericalSemigroupsWithFrobeniusNumber(7);
[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup> ]
gap> List(last, s -> MinimalGeneratingSystemOfNumericalSemigroup(s));
[ [ 3, 5 ], [ 4, 5, 11 ], [ 4, 5, 6 ], [ 5, 6, 8, 9 ], [ 2, 9 ],
[ 3, 8, 10 ], [ 5, 8, 9, 11, 12 ], [ 4, 9, 10, 11 ], [ 4, 6, 9, 11 ],
[ 6, 8, 9, 10, 11, 13 ], [ 8 .. 15 ] ]
gap> NumericalSemigroupsWithGenus(5);
[ <Numerical semigroup with 6 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 2 generators> ]
gap> List(last, s -> MinimalGeneratingSystemOfNumericalSemigroup(s));
[ [ 6 .. 11 ], [ 5, 7, 8, 9, 11 ], [ 5, 6, 8, 9 ], [ 5, 6, 7, 9 ],
[ 5, 6, 7, 8 ], [ 4, 6, 7 ], [ 4, 7, 9, 10 ], [ 4, 6, 9, 11 ],
[ 4, 5, 11 ], [ 3, 8, 10 ], [ 3, 7, 11 ], [ 2, 11 ] ]
gap> ns := NumericalSemigroup([10..30]);
<Numerical semigroup with 10 generators>
gap> IsNumericalSemigroup(ns);
true
gap> MinimalGeneratingSystemOfNumericalSemigroup(ns);
[ 10 .. 19 ]
gap> GapsOfNumericalSemigroup(ns);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
gap> genus8 := NumericalSemigroupsWithGenus(8);;
gap> List(genus8, s -> Length(GapsOfNumericalSemigroup(s)));
[ 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8 ]
gap> List(genus8, s -> IsIrreducibleNumericalSemigroup(s));
[ false, false, false, false, false, false, true, true, true, false, false,
false, false, false, false, false, false, false, false, false, false,
false, false, false, false, true, false, false, false, false, true, false,
false, false, false, false, true, false, false, false, false, false, false,
false, false, false, false, false, true, false, false, false, false, true,
false, false, false, false, true, false, true, false, true, false, false,
true, true ]
gap> List(genus8, s -> WilfNumber(s)+EliahouNumber(s));
[ 0, 12, 15, 24, 35, 48, 63, 80, 64, 8, 15, 18, 28, 40, 10, 12, 21, 32, 12,
21, 32, 21, 32, 32, 29, 40, 4, 10, 18, 29, 48, 5, 18, 20, 6, 20, 32, 5, 6,
6, 6, 4, 13, 20, 20, 0, 6, 20, 26, 8, 6, 8, 5, 12, 6, 18, 2, 6, 12, 6, 16,
16, 16, 7, 0, 10, 0 ]
gap> a := AffineSemigroupByGenerators([5,3,1],[2,7,4],[3,1,5]);
<Affine semigroup in 3 dimensional space, with 3 generators>
gap> MinimalPresentationOfAffineSemigroup(a);
[ ]
gap> BettiElementsOfAffineSemigroup(a);
[ ]
gap> pf := [ 83, 169, 173, 214, 259 ];;
gap> ns := ANumericalSemigroupWithPseudoFrobeniusNumbers(pf);;
gap> PseudoFrobeniusOfNumericalSemigroup(ns) = pf;
true
gap> s:=NumericalSemigroup(4,5,7);;
gap> Set(MinimalPresentation(s),p->Set(p));
[ [ [ 0, 0, 2 ], [ 1, 2, 0 ] ], [ [ 0, 1, 1 ], [ 3, 0, 0 ] ],
[ [ 0, 3, 0 ], [ 2, 0, 1 ] ] ]
gap> a:=AsAffineSemigroup(s);;
gap> Set(MinimalPresentation(a),p->Set(p));
[ [ [ 0, 0, 2 ], [ 1, 2, 0 ] ], [ [ 0, 1, 1 ], [ 3, 0, 0 ] ],
[ [ 0, 3, 0 ], [ 2, 0, 1 ] ] ]
gap> CatenaryDegreeOfAffineSemigroup(a)=CatenaryDegreeOfNumericalSemigroup(s);
true
#############################################################################
#############################################################################
# Examples from the manual
# (These examples use at least a funtion from each file)
#Generating_Numerical_Semigroups.xml
gap> s1 := NumericalSemigroupByGenerators(3,5,7);
<Numerical semigroup with 3 generators>
gap> s2 := NumericalSemigroupByGenerators([3,5,7]);
<Numerical semigroup with 3 generators>
gap> s3 := NumericalSemigroup("generators",3,5,7);
<Numerical semigroup with 3 generators>
gap> s4 := NumericalSemigroup("generators",[3,5,7]);
<Numerical semigroup with 3 generators>
gap> s5 := NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> s6 := NumericalSemigroup([3,5,7]);
<Numerical semigroup with 3 generators>
gap> s1=s2;s2=s3;s3=s4;s4=s5;s5=s6;
true
true
true
true
true
gap> s := NumericalSemigroupBySubAdditiveFunction([5,4,2,0]);
<Numerical semigroup>
gap> t := NumericalSemigroup("subadditive",[5,4,2,0]);;
gap> s=t;
true
gap> s:=NumericalSemigroup(3,11);;
gap> ap := AperyListOfNumericalSemigroupWRTElement(s,20);
[ 0, 21, 22, 3, 24, 25, 6, 27, 28, 9, 30, 11, 12, 33, 14, 15, 36, 17, 18, 39 ]
gap> t:=NumericalSemigroupByAperyList(ap);;
gap> r := NumericalSemigroup("apery",ap);;
gap> s=t;t=r;
true
true
gap> s:=NumericalSemigroup(3,11);;
gap> se := SmallElements(s);
[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ]
gap> t := NumericalSemigroupBySmallElements(se);;
gap> r := NumericalSemigroup("elements",se);;
gap> s=t;t=r;
true
true
# gap> e := [ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ];
# [ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ]
# gap> NumericalSemigroupBySmallElements(e);
# Error, The argument does not represent a numerical semigroup called from
# <function "NumericalSemigroupBySmallElements">( <arguments> )
# called from read-eval loop at line 35 of *stdin*
# you can 'quit;' to quit to outer loop, or
# you can 'return;' to continue
# brk>
gap> g := [ 1, 2, 4, 5, 7, 8, 10, 13, 16 ];;
gap> s := NumericalSemigroupByGaps(g);;
gap> t := NumericalSemigroup("gaps",g);;
gap> s=t;
true
# gap> h := [ 1, 2, 5, 7, 8, 10, 13, 16 ];;
# gap> NumericalSemigroupByGaps(h);
# Error, The argument does not represent the gaps of a numerical semigroup called
# from
# <function "NumericalSemigroupByGaps">( <arguments> )
# called from read-eval loop at line 34 of *stdin*
# you can 'quit;' to quit to outer loop, or
# you can 'return;' to continue
# brk>
gap> fg := [ 11, 14, 17, 20, 23, 26, 29, 32, 35 ];;
gap> NumericalSemigroupByFundamentalGaps(fg);
<Numerical semigroup>
gap> NumericalSemigroup("fundamentalgaps",fg);
<Numerical semigroup>
gap> last=last2;
true
gap> gg := [ 11, 17, 20, 22, 23, 26, 29, 32, 35 ];; #22 is not fundamental
gap> NumericalSemigroup("fundamentalgaps",fg);
<Numerical semigroup>
gap> s:=NumericalSemigroupByAffineMap(3,1,3);
<Numerical semigroup with 3 generators>
gap> SmallElements(s);
[ 0, 3, 6, 9, 10, 12, 13, 15, 16, 18 ]
gap> t:=NumericalSemigroup("affinemap",3,1,3);;
gap> s=t;
true
gap> ModularNumericalSemigroup(3,7);
<Modular numerical semigroup satisfying 3x mod 7 <= x >
gap> NumericalSemigroup("modular",3,7);
<Modular numerical semigroup satisfying 3x mod 7 <= x >
gap> ProportionallyModularNumericalSemigroup(3,7,12);
<Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x >
gap> NumericalSemigroup("propmodular",3,7,12);
<Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x >
gap> NumericalSemigroup("propmodular",67,98,1);
<Modular numerical semigroup satisfying 67x mod 98 <= x >
gap> NumericalSemigroupByInterval(7/5,5/3);
<Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x >
gap> NumericalSemigroup("interval",[7/5,5/3]);
<Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x >
gap> SmallElements(last);
[ 0, 3, 5 ]
gap> NumericalSemigroupByOpenInterval(7/5,5/3);
<Numerical semigroup>
gap> NumericalSemigroup("openinterval",[7/5,5/3]);
<Numerical semigroup>
gap> SmallElements(last);
[ 0, 3, 6, 8 ]
##Some_basic_tests.xml
gap> s:=NumericalSemigroup(3,7);
<Numerical semigroup with 2 generators>
gap> AperyListOfNumericalSemigroupWRTElement(s,30);;
gap> t:=NumericalSemigroupByAperyList(last);
<Numerical semigroup>
gap> IsNumericalSemigroupByGenerators(s);
true
gap> IsNumericalSemigroupByGenerators(t);
false
gap> IsNumericalSemigroupByAperyList(s);
false
gap> IsNumericalSemigroupByAperyList(t);
true
gap> L:=[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ];
[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ]
gap> RepresentsSmallElementsOfNumericalSemigroup(L);
true
gap> L:=[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ];
[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ]
gap> RepresentsSmallElementsOfNumericalSemigroup(L);
false
gap> s:=NumericalSemigroup(3,7);
<Numerical semigroup with 2 generators>
gap> L:=GapsOfNumericalSemigroup(s);
[ 1, 2, 4, 5, 8, 11 ]
gap> RepresentsGapsOfNumericalSemigroup(L);
true
gap> IsAperyListOfNumericalSemigroup([0,21,7,28,14]);
true
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> T:=NumericalSemigroup(2,3);
<Numerical semigroup with 2 generators>
gap> IsSubsemigroupOfNumericalSemigroup(T,S);
true
gap> IsSubsemigroupOfNumericalSemigroup(S,T);
false
gap> ns1 := NumericalSemigroup(5,7);;
gap> ns2 := NumericalSemigroup(5,7,11);;
gap> IsSubset(ns1,ns2);
false
gap> IsSubset(ns2,[5,15]);
true
gap> IsSubset(ns1,[5,11]);
false
gap> IsSubset(ns2,ns1);
true
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> BelongsToNumericalSemigroup(15,S);
false
gap> 15 in S;
false
gap> SmallElementsOfNumericalSemigroup(S);
[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> BelongsToNumericalSemigroup(13,S);
true
gap> 13 in S;
true
##The_definitions.xml
gap> S := NumericalSemigroup("modular", 7,53);
<Modular numerical semigroup satisfying 7x mod 53 <= x >
gap> MultiplicityOfNumericalSemigroup(S);
8
gap> NumericalSemigroup(3,5);
<Numerical semigroup with 2 generators>
gap> Multiplicity(last);
3
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> MinimalGeneratingSystem(S)=MinimalGeneratingSystemOfNumericalSemigroup(S);
true
gap> s := NumericalSemigroup(3,5,7,15);
<Numerical semigroup with 4 generators>
gap> HasGenerators(s);
true
gap> HasMinimalGenerators(s);
false
gap> MinimalGenerators(s);
[ 3, 5, 7 ]
gap> Generators(s);
[ 3, 5, 7, 15 ]
gap> s := NumericalSemigroup(3,5,7,15);
<Numerical semigroup with 4 generators>
gap> EmbeddingDimension(s);
3
gap> EmbeddingDimensionOfNumericalSemigroup(s);
3
gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 0, 3, 5 ]
gap> SmallElements(NumericalSemigroup(3,5,7));
[ 0, 3, 5 ]
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 ]
gap> S := NumericalSemigroup(7,8,17);;
gap> RthElementOfNumericalSemigroup(S,53);
68
gap> S := NumericalSemigroup("modular", 5,53);;
gap> AperyListOfNumericalSemigroupWRTElement(S,12);
[ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
gap> AperyList(S,12);
[ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
gap> S := NumericalSemigroup("modular", 5,53);;
gap> AperyListOfNumericalSemigroup(S);
[ 0, 12, 13, 25, 26, 38, 39, 51, 52, 53, 32 ]
gap> AperyList(NumericalSemigroup(5,7,11));
[ 0, 11, 7, 18, 14 ]
gap> s:=NumericalSemigroup(10,13,19,27);;
gap> AperyListOfNumericalSemigroupWRTInteger(s,11);
[ 0, 10, 13, 19, 20, 23, 26, 27, 29, 32, 33, 36, 39, 42, 45, 46, 52, 55 ]
gap> Length(last);
18
gap> AperyListOfNumericalSemigroupWRTInteger(s,10);
[ 0, 13, 19, 26, 27, 32, 38, 45, 51, 54 ]
gap> AperyListOfNumericalSemigroupWRTElement(s,10);
[ 0, 51, 32, 13, 54, 45, 26, 27, 38, 19 ]
gap> Length(last);
10
gap> AperyList(s,10);
[ 0, 51, 32, 13, 54, 45, 26, 27, 38, 19 ]
gap> s:=NumericalSemigroup(3,7);;
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 ] ]
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> KunzCoordinatesOfNumericalSemigroup(s);
[ 2, 1 ]
gap> KunzCoordinatesOfNumericalSemigroup(s,5);
[ 1, 1, 0, 1 ]
gap> KunzPolytope(3);
[ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 2, -1, 0 ], [ -1, 2, 1 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> CocycleOfNumericalSemigroupWRTElement(s,3);
[ [ 0, 0, 0 ], [ 0, 3, 4 ], [ 0, 4, 1 ] ]
gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
4
gap> FrobeniusNumber(NumericalSemigroup(3,5,7));
4
gap> ConductorOfNumericalSemigroup(NumericalSemigroup(3,5,7));
5
gap> Conductor(NumericalSemigroup(3,5,7));
5
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> PseudoFrobeniusOfNumericalSemigroup(S);
[ 21, 40, 41, 42 ]
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> Type(S);
4
gap> TypeOfNumericalSemigroup(S);
4
gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 1, 2, 4 ]
gap> Gaps(NumericalSemigroup(5,7,11));
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> DesertsOfNumericalSemigroup(s);
[ [ 1, 2 ], [ 4 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> IsOrdinary(s);
false
gap> s:=NumericalSemigroup(3,5,7);;
gap> IsAcute(s);
true
gap> s:=NumericalSemigroup(3,5);;
gap> Holes(s);
[ ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> HolesOfNumericalSemigroup(s);
[ 2 ]
gap> s:=NumericalSemigroup(16,17,71,72);;
gap> LatticePathAssociatedToNumericalSemigroup(s,16,17);
[ [ 0, 14 ], [ 1, 13 ], [ 2, 12 ], [ 3, 11 ], [ 4, 10 ], [ 5, 9 ], [ 6, 8 ],
[ 7, 7 ], [ 8, 6 ], [ 9, 5 ], [ 10, 4 ], [ 11, 3 ], [ 12, 2 ], [ 13, 1 ],
[ 14, 0 ] ]
gap> s:=NumericalSemigroup(16,17,71,72);;
gap> GenusOfNumericalSemigroup(s);
80
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> Genus(S);
26
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 ]
gap> Gaps(NumericalSemigroup(5,7,11));
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
gap> FundamentalGaps(NumericalSemigroup(5,7,11));
[ 6, 8, 9, 13 ]
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> SpecialGaps(S);
[ 40, 41, 42 ]
gap> SpecialGapsOfNumericalSemigroup(S);
[ 40, 41, 42 ]
##Wilf.xml
gap> l:=NumericalSemigroupsWithGenus(10);;
gap> Filtered(l, s->WilfNumberOfNumericalSemigroup(s)<0);
[ ]
gap> Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s)));
70
gap> s := NumericalSemigroup(13,25,37);;
gap> WilfNumber(s);
96
gap> s:=NumericalSemigroup(5,7,9);;
gap> TruncatedWilfNumberOfNumericalSemigroup(s);
4
gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
gap> EliahouNumber(s);
-1
gap> s:=NumericalSemigroup(5,7,9);;
gap> ProfileOfNumericalSemigroup(s);
[ 2, 1 ]
gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
gap> ProfileOfNumericalSemigroup(s);
[ 3, 0, 0 ]
gap> s:=NumericalSemigroup(5,7,9);;
gap> EliahouSlicesOfNumericalSemigroup(s);
[ [ 5, 7 ], [ 9, 10, 12 ] ]
gap> SmallElements(s);
[ 0, 5, 7, 9, 10, 12, 14 ]
##Presentations_of_Numerical_Semigroups.xml
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> MinimalPresentationOfNumericalSemigroup(s);
[ [ [ 0, 2, 0 ], [ 1, 0, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ],
[ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> GraphAssociatedToElementInNumericalSemigroup(10,s);
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> BettiElementsOfNumericalSemigroup(s);
[ 10, 12, 14 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> PrimitiveElementsOfNumericalSemigroup(s);
[ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> ShadedSetOfElementInNumericalSemigroup(10,s);
[ [ ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> IsUniquelyPresentedNumericalSemigroup(s);
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> IsGenericNumericalSemigroup(s);
true
##Adding_and_removing_elements_of_a_numerical_semigroup.xml
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> RemoveMinimalGeneratorFromNumericalSemigroup(7,s);
<Numerical semigroup with 3 generators>
gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
[ 3, 5 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> s2:=RemoveMinimalGeneratorFromNumericalSemigroup(5,s);
<Numerical semigroup with 3 generators>
gap> s3:=AddSpecialGapOfNumericalSemigroup(5,s2);
<Numerical semigroup>
gap> SmallElementsOfNumericalSemigroup(s) =
> SmallElementsOfNumericalSemigroup(s3);
true
gap> s=s3;
true
#Operations_Numerical_Semigroups.xml
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> T := NumericalSemigroup(2,17);
<Numerical semigroup with 2 generators>
gap> SmallElementsOfNumericalSemigroup(S);
[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> SmallElementsOfNumericalSemigroup(T);
[ 0, 2, 4, 6, 8, 10, 12, 14, 16 ]
gap> IntersectionOfNumericalSemigroups(S,T);
<Numerical semigroup>
gap> SmallElementsOfNumericalSemigroup(last);
[ 0, 12, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> s:=NumericalSemigroup(3,29);
<Numerical semigroup with 2 generators>
gap> SmallElementsOfNumericalSemigroup(s);
[ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42,
44, 45, 47, 48, 50, 51, 53, 54, 56 ]
gap> t:=QuotientOfNumericalSemigroup(s,7);
<Numerical semigroup>
gap> SmallElementsOfNumericalSemigroup(t);
[ 0, 3, 5, 6, 8 ]
gap> u := s / 7;
<Numerical semigroup>
gap> SmallElementsOfNumericalSemigroup(u);
[ 0, 3, 5, 6, 8 ]
gap> N:=NumericalSemigroup(1);;
gap> s:=MultipleOfNumericalSemigroup(N,4,20);;
gap> SmallElements(s);
[ 0, 4, 8, 12, 16, 20 ]
gap> ns1 := NumericalSemigroup(5,7);;
gap> ns2 := NumericalSemigroup(7,11,12);;
gap> Difference(ns1,ns2);
[ 5, 10, 15, 17, 20, 27 ]
gap> Difference(ns2,ns1);
[ 11, 18, 23 ]
gap> DifferenceOfNumericalSemigroups(ns2,ns1);
[ 11, 18, 23 ]
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> e:=6+s;
<Ideal of numerical semigroup>
gap> ndup:=NumericalDuplication(s,e,3);
<Numerical semigroup with 4 generators>
gap> SmallElements(ndup);
[ 0, 6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 ]
gap> s:=InductiveNumericalSemigroup([4,2],[5,23]);;
gap> SmallElements(s);
[ 0, 8, 16, 24, 32, 40, 42, 44, 46 ]
##Constructing_sets_of_numerical_semigroups.xml
gap> OverSemigroupsNumericalSemigroup(NumericalSemigroup(3,5,7));
[ <The numerical semigroup N>, <Numerical semigroup with 2 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 3 generators> ]
gap> List(last,s->MinimalGeneratingSystemOfNumericalSemigroup(s));
[ [ 1 ], [ 2, 3 ], [ 3 .. 5 ], [ 3, 5, 7 ] ]
gap> Length(NumericalSemigroupsWithFrobeniusNumber(15));
200
gap> NumericalSemigroupsWithGenus(5);
[ <Numerical semigroup with 6 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 2 generators> ]
gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
[ [ 6 .. 11 ], [ 5, 7, 8, 9, 11 ], [ 5, 6, 8, 9 ], [ 5, 6, 7, 9 ],
[ 5, 6, 7, 8 ], [ 4, 6, 7 ], [ 4, 7, 9, 10 ], [ 4, 6, 9, 11 ],
[ 4, 5, 11 ], [ 3, 8, 10 ], [ 3, 7, 11 ], [ 2, 11 ] ]
gap> pf := [ 58, 64, 75 ];
[ 58, 64, 75 ]
gap> ForcedIntegersForPseudoFrobenius(pf);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 11, 15, 16, 17, 25, 29, 32, 58, 64, 75 ],
[ 0, 59, 60, 67, 68, 69, 70, 71, 72, 73, 74, 76 ] ]
gap> pf := [ 15, 20, 27, 35 ];;
gap> fint := ForcedIntegersForPseudoFrobenius(pf);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 20, 27, 35 ],
[ 0, 19, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36 ] ]
gap> free := Difference([1..Maximum(pf)],Union(fint));
[ 11, 13, 14, 17, 18, 21, 22, 24 ]
gap> SimpleForcedIntegersForPseudoFrobenius(fint[1],Union(fint[2],[free[1]]),pf);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 20, 24, 27, 35 ],
[ 0, 11, 19, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36 ] ]
gap> pf := [ 58, 64, 75 ];
[ 58, 64, 75 ]
gap> Length(NumericalSemigroupsWithPseudoFrobeniusNumbers(pf));
561
gap> pf := [11,19,22];;
gap> NumericalSemigroupsWithPseudoFrobeniusNumbers(pf);
[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup> ]
gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
[ [ 7, 9, 17, 20 ], [ 7, 10, 13, 16, 18 ], [ 9, 12, 14, 15, 16, 17, 20 ],
[ 10, 13, 14, 15, 16, 17, 18, 21 ],
[ 12, 13, 14, 15, 16, 17, 18, 20, 21, 23 ] ]
##Irreducible_numerical_semigroups.xml
gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,9));
true
gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,7,9));
false
gap> IsSymmetricNumericalSemigroup(NumericalSemigroup(10,23));
true
gap> IsSymmetricNumericalSemigroup(NumericalSemigroup(10,11,23));
false
gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(6,7,8,9,11));
true
gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(4,6,9));
false
gap> FrobeniusNumber(AnIrreducibleNumericalSemigroupWithFrobeniusNumber(28));
28
gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(19));
20
gap> DecomposeIntoIrreducibles(NumericalSemigroup(5,6,8));
[ <Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators> ]
##Complete_Intersections.xml
gap> s := NumericalSemigroup( 10, 15, 16 );
<Numerical semigroup with 3 generators>
gap> AsGluingOfNumericalSemigroups(s);
[ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ]
gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
<Numerical semigroup with 8 generators>
gap> AsGluingOfNumericalSemigroups(s);
[ ]
gap> s := NumericalSemigroup( 10, 15, 16 );
<Numerical semigroup with 3 generators>
gap> IsACompleteIntersectionNumericalSemigroup(s);
true
gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
<Numerical semigroup with 8 generators>
gap> IsACompleteIntersectionNumericalSemigroup(s);
false
gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));
34
gap> IsFreeNumericalSemigroup(NumericalSemigroup(10,15,16));
true
gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));
false
gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57));
33
gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));
false
gap> IsFreeNumericalSemigroup(NumericalSemigroup(4,11,14));
true
gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));
20
gap> ns := NumericalSemigroup(4,11,14);;
gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
false
gap> ns := NumericalSemigroup(4,11,19);;
gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
true
gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));
7
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsAperySetGammaRectangular(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsAperySetGammaRectangular(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsAperySetBetaRectangular(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsAperySetBetaRectangular(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsAperySetAlphaRectangular(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsAperySetAlphaRectangular(s);
true
##Almost_symmetric.xml
gap> AlmostSymmetricNumericalSemigroupsFromIrreducible(NumericalSemigroup(5,8,9,11));
[ <Numerical semigroup with 4 generators>,
<Numerical semigroup with 5 generators>,
<Numerical semigroup with 5 generators> ]
gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
[ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]
gap> IsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17));
true
gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12));
15
gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(12));
2
##Ideals_of_numerical_semigroups.xml
gap> IdealOfNumericalSemigroup([3,5],NumericalSemigroup(9,11));
<Ideal of numerical semigroup>
gap> [3,5]+NumericalSemigroup(9,11);
<Ideal of numerical semigroup>
gap> last=last2;
true
gap> 3+NumericalSemigroup(5,9);
<Ideal of numerical semigroup>
gap> I:=[1..7]+NumericalSemigroup(7,19);;
gap> IsIdealOfNumericalSemigroup(I);
true
gap> IsIdealOfNumericalSemigroup(2);
false
gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
[ 3 ]
gap> MinimalGeneratingSystem(I);
[ 3 ]
gap> MinimalGenerators([3,5]+NumericalSemigroup(2,11));
[ 3 ]
gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
gap> GeneratorsOfIdealOfNumericalSemigroup(I);
[ 3, 5, 9 ]
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
[ 3 ]
gap> Generators(I);
[ 3, 5, 9 ]
gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
gap> AmbientNumericalSemigroupOfIdeal(I);
<Numerical semigroup with 2 generators>
gap> s:=NumericalSemigroup(3,7,5);;
gap> IsIntegralIdealOfNumericalSemigroup(4+s);
false
gap> IsIntegralIdealOfNumericalSemigroup(10+s);
true
gap> IsIntegral(10+s);
true
gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
gap> SmallElementsOfIdealOfNumericalSemigroup(I);
[ 3, 5, 7, 9, 11, 13 ]
gap> SmallElements(I) = SmallElementsOfIdealOfNumericalSemigroup(I);
true
gap> J:=[2,11]+NumericalSemigroup(2,11);;
gap> SmallElementsOfIdealOfNumericalSemigroup(J);
[ 2, 4, 6, 8, 10 ]
gap> s:=NumericalSemigroup(3,7,5);;
gap> ConductorOfIdealOfNumericalSemigroup(10+s);
15
gap> Conductor(10+s);
15
gap> J:=[2,11]+NumericalSemigroup(2,11);;
gap> Minimum(J);
2
gap> J:=[2,11]+NumericalSemigroup(2,11);;
gap> BelongsToIdealOfNumericalSemigroup(9,J);
false
gap> 9 in J;
false
gap> BelongsToIdealOfNumericalSemigroup(10,J);
true
gap> 10 in J;
true
gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
gap> J:=[2,11]+NumericalSemigroup(2,11);;
gap> I+J;
<Ideal of numerical semigroup>
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
[ 5, 14 ]
gap> SumIdealsOfNumericalSemigroup(I,J);
<Ideal of numerical semigroup>
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
[ 5, 14 ]
gap> I:=[0,1]+NumericalSemigroup(3,5,7);;
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(2*I);
[ 0, 1, 2 ]
gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);;
gap> I:=[0,1]+S;;
gap> II:=S-I;;
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
[ 0, 1 ]
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(II);
[ 14, 20 ]
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I+II);
[ 14, 15, 20, 21 ]
gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);;
gap> I:=[0,1]+S;
<Ideal of numerical semigroup>
gap> 2*I-2*I;
<Ideal of numerical semigroup>
gap> I-I;
<Ideal of numerical semigroup>
gap> ii := 2*I-2*I;
<Ideal of numerical semigroup>
gap> i := I-I;
<Ideal of numerical semigroup>
gap> DifferenceOfIdealsOfNumericalSemigroup(last2,last);
[ 26, 27, 37, 38 ]
gap> Difference(ii,i);
[ 26, 27, 37, 38 ]
gap> Difference(i,ii);
[ ]
gap> s:=NumericalSemigroup(13,23);;
gap> l:=List([1..6], _ -> Random([8..34]));;
gap> I:=IdealOfNumericalSemigroup(l, s);;
gap> It:=TranslationOfIdealOfNumericalSemigroup(7,I);
<Ideal of numerical semigroup>
gap> It2:=7+I;
<Ideal of numerical semigroup>
gap> It2=It;
true
gap> i:=IdealOfNumericalSemigroup([75,89],s);;
gap> j:=IdealOfNumericalSemigroup([115,289],s);;
gap> IntersectionIdealsOfNumericalSemigroup(i,j);
<Ideal of numerical semigroup>
gap> MaximalIdealOfNumericalSemigroup(NumericalSemigroup(3,7));
<Ideal of numerical semigroup>
gap> s:=NumericalSemigroup(4,6,11);;
gap> m:=MaximalIdealOfNumericalSemigroup(s);;
gap> c:=CanonicalIdealOfNumericalSemigroup(s);
<Ideal of numerical semigroup>
gap> c-(c-m)=m;
true
gap> id:=3+s;
<Ideal of numerical semigroup>
gap> c-(c-id)=id;
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> c:=3+CanonicalIdealOfNumericalSemigroup(s);;
gap> c-(c-(3+s))=3+s;
true
gap> IsCanonicalIdealOfNumericalSemigroup(c);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> TypeSequenceOfNumericalSemigroup(s);
[ 13, 3, 4, 4, 7, 3, 3, 3, 2, 2, 2, 3, 3, 2, 4, 3, 2, 1, 3, 2, 1, 1, 2, 2, 1,
1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,
1, 1, 1 ]
gap> s:=NumericalSemigroup(4,6,11);;
gap> TypeSequenceOfNumericalSemigroup(s);
[ 1, 1, 1, 1, 1, 1, 1 ]
gap> I:=[6,9,11]+NumericalSemigroup(6,9,11);;
gap> List([1..7],n->HilbertFunctionOfIdealOfNumericalSemigroup(n,I));
[ 3, 5, 6, 6, 6, 6, 6 ]
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
gap> BlowUpIdealOfNumericalSemigroup(I);;
gap> SmallElementsOfIdealOfNumericalSemigroup(last);
[ 0, 2, 4, 6, 8 ]
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
gap> ReductionNumberIdealNumericalSemigroup(I);
2
gap> MaximalIdealOfNumericalSemigroup(NumericalSemigroup(3,7));
<Ideal of numerical semigroup>
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> BlowUpOfNumericalSemigroup(s);
<Numerical semigroup with 10 generators>
gap> SmallElementsOfNumericalSemigroup(last);
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
40, 41, 42, 44 ]
gap> m:=MaximalIdealOfNumericalSemigroup(s);
<Ideal of numerical semigroup>
gap> BlowUpIdealOfNumericalSemigroup(m);
<Ideal of numerical semigroup>
gap> SmallElementsOfIdealOfNumericalSemigroup(last);
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
40, 41, 42, 44 ]
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> LipmanSemigroup(s);
<Numerical semigroup with 10 generators>
gap> SmallElementsOfNumericalSemigroup(last);
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
40, 41, 42, 44 ]
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
gap> RatliffRushNumberOfIdealOfNumericalSemigroup(I);
1
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
gap> RatliffRushClosureOfIdealOfNumericalSemigroup(I);
<Ideal of numerical semigroup>
gap> MinimalGenerators(last);
[ 0, 2, 4 ]
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
gap> AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup(I);
2
gap> s:=NumericalSemigroup(3,5);
<Numerical semigroup with 2 generators>
gap> MultiplicitySequenceOfNumericalSemigroup(s);
[ 3, 2, 1 ]
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> bu:=BlowUpOfNumericalSemigroup(s);;
gap> ap:=AperyListOfNumericalSemigroupWRTElement(s,30);;
gap> apbu:=AperyListOfNumericalSemigroupWRTElement(bu,30);;
gap> (ap-apbu)/30;
[ 0, 4, 4, 3, 2, 1, 3, 4, 4, 3, 2, 3, 1, 4, 4, 3, 3, 1, 4, 4, 4, 3, 2, 4, 2,
5, 4, 3, 3, 2 ]
gap> MicroInvariantsOfNumericalSemigroup(s)=last;
true
gap> s:=NumericalSemigroup(10,11,13);;
gap> i:=[12,14]+s;;
gap> AperyListOfIdealOfNumericalSemigroupWRTElement(i,10);
[ 40, 51, 12, 23, 14, 25, 36, 27, 38, 49 ]
gap> s:=NumericalSemigroup(10,11,13);;
gap> AperyTableOfNumericalSemigroup(s);
[ [ 0, 11, 22, 13, 24, 35, 26, 37, 48, 39 ],
[ 10, 11, 22, 13, 24, 35, 26, 37, 48, 39 ],
[ 20, 21, 22, 23, 24, 35, 26, 37, 48, 39 ],
[ 30, 31, 32, 33, 34, 35, 36, 37, 48, 39 ],
[ 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> StarClosureOfIdealOfNumericalSemigroup([0,2]+s,[[0,4]+s]);;
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
[ 0, 2, 4 ]
gap> IsAdmissiblePattern([1,1,-1]);
true
gap> IsAdmissiblePattern([1,-2]);
false
gap> IsAdmissiblePattern([1,-1]);
true
gap> IsStronglyAdmissiblePattern([1,-1]);
false
gap> IsStronglyAdmissiblePattern([1,1,-1]);
true
gap> s:=NumericalSemigroup(3,7,5);;
gap> t:=NumericalSemigroup(10,11,14);;
gap> AsIdealOfNumericalSemigroup(10+s,t);
fail
gap> AsIdealOfNumericalSemigroup(100+s,t);
<Ideal of numerical semigroup>
gap> BoundForConductorOfImageOfPattern([1,1,-1],10);
10
gap> s:=NumericalSemigroup(3,7,5);;
gap> i:=10+s;;
gap> ApplyPatternToIdeal([1,1,-1],i);
[ 1, <Ideal of numerical semigroup> ]
gap> s:=NumericalSemigroup(3,7,5);;
gap> ApplyPatternToNumericalSemigroup([1,1,-1],s);
[ 1, <Ideal of numerical semigroup> ]
gap> SmallElements(last[2]);
[ 0, 3, 5 ]
gap> s:=NumericalSemigroup(3,7,5);;
gap> i:=[3,5]+s;;
gap> IsAdmittedPatternByIdeal([1,1,-1],i,i);
false
gap> IsAdmittedPatternByIdeal([1,1,-1],i,0+s);
true
gap> IsAdmittedPatternByNumericalSemigroup([1,1,-1],s,s);
true
gap> IsArfNumericalSemigroup(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsGradedAssociatedRingNumericalSemigroupCM(s);
false
gap> MicroInvariantsOfNumericalSemigroup(s);
[ 0, 4, 4, 3, 2, 1, 3, 4, 4, 3, 2, 3, 1, 4, 4, 3, 3, 1, 4, 4, 4, 3, 2, 4, 2,
5, 4, 3, 3, 2 ]
gap> List(AperyListOfNumericalSemigroupWRTElement(s,30),
> w->MaximumDegreeOfElementWRTNumericalSemigroup (w,s));
[ 0, 1, 4, 1, 2, 1, 3, 1, 4, 3, 2, 3, 1, 1, 4, 3, 3, 1, 4, 1, 4, 3, 2, 4, 2,
5, 4, 3, 1, 2 ]
gap> last=last2;
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsGradedAssociatedRingNumericalSemigroupCM(s);
true
gap> MicroInvariantsOfNumericalSemigroup(s);
[ 0, 2, 1, 1 ]
gap> List(AperyListOfNumericalSemigroupWRTElement(s,4),
> w->MaximumDegreeOfElementWRTNumericalSemigroup(w,s));
[ 0, 2, 1, 1 ]
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsGradedAssociatedRingNumericalSemigroupBuchsbaum(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> TorsionOfAssociatedGradedRingNumericalSemigroup(s);
[ 181, 153, 157, 193, 169, 148 ]
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup(s);
1
gap> IsGradedAssociatedRingNumericalSemigroupBuchsbaum(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsMpureNumericalSemigroup(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsMpureNumericalSemigroup(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsPureNumericalSemigroup(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsPureNumericalSemigroup(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsGradedAssociatedRingNumericalSemigroupGorenstein(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsGradedAssociatedRingNumericalSemigroupGorenstein(s);
true
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsGradedAssociatedRingNumericalSemigroupCI(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsGradedAssociatedRingNumericalSemigroupCI(s);
true
##Numerical_semigroups_with_maximal_embedding_dimension.xml
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5,7));
true
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5));
false
gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5));
<Numerical semigroup>
gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
[ 3, 5, 7 ]
gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup(
> NumericalSemigroup(3,5,7));
[ 3, 5 ]
gap> IsArfNumericalSemigroup(NumericalSemigroup(3,5,7));
true
gap> IsArfNumericalSemigroup(NumericalSemigroup(3,7,11));
false
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,7,11));
true
gap> ArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11));
<Numerical semigroup>
gap> MinimalGenerators(last);
[ 3, 7, 8 ]
gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup(
> NumericalSemigroup(3,7,8));
[ 3, 7 ]
gap> ArfNumericalSemigroupsWithFrobeniusNumber(10);
[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ]
gap> Set(last,MinimalGenerators);
[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],
[ 6, 11, 13, 14, 15, 16 ], [ 7, 9, 11, 12, 13, 15, 17 ],
[ 7, 11, 12, 13, 15, 16, 17 ], [ 8, 11, 12, 13, 14, 15, 17, 18 ],
[ 9, 11, 12, 13, 14, 15, 16, 17, 19 ], [ 11 .. 21 ] ]
gap> Length(ArfNumericalSemigroupsWithFrobeniusNumberUpTo(10));
46
gap> Length(ArfNumericalSemigroupsWithGenus(10));
21
gap> Length(ArfNumericalSemigroupsWithGenusUpTo(10));
86
gap> ArfNumericalSemigroupsWithGenusAndFrobeniusNumber(10,13);
[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,
<Numerical semigroup>, <Numerical semigroup> ]
gap> List(last,MinimalGenerators);
[ [ 8, 10, 12, 14, 15, 17, 19, 21 ], [ 6, 10, 14, 15, 17, 19 ],
[ 5, 12, 14, 16, 18 ], [ 6, 9, 14, 16, 17, 19 ], [ 4, 14, 15, 17 ] ]
gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(4,6,9,11));
true
gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(8, 9, 12, 13, 15, 19 ));
false
gap> SaturatedNumericalSemigroupClosure(NumericalSemigroup(8, 9, 12, 13, 15));
<Numerical semigroup>
gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
[ 8 .. 15 ]
gap> SaturatedNumericalSemigroupsWithFrobeniusNumber(10);
[ <Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 6 generators>,
<Numerical semigroup with 6 generators>,
<Numerical semigroup with 7 generators>,
<Numerical semigroup with 8 generators>,
<Numerical semigroup with 9 generators>,
<Numerical semigroup with 11 generators> ]
gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],
[ 6, 11, 13, 14, 15, 16 ], [ 7, 11, 12, 13, 15, 16, 17 ],
[ 8, 11, 12, 13, 14, 15, 17, 18 ], [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ],
[ 11 .. 21 ] ]
##catenary-tame.xml
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
[ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> FactorizationsElementWRTNumericalSemigroup(1100,s);
[ [ 0, 8, 1, 0, 0, 0 ], [ 0, 0, 0, 2, 2, 0 ], [ 5, 1, 1, 0, 0, 1 ],
[ 0, 2, 3, 0, 0, 1 ] ]
gap> s:=NumericalSemigroup(10,11,13);
<Numerical semigroup with 3 generators>
gap> FactorizationsElementListWRTNumericalSemigroup([100,101,103],s);
[ [ [ 0, 2, 6 ], [ 1, 7, 1 ], [ 3, 4, 2 ], [ 5, 1, 3 ], [ 10, 0, 0 ] ],
[ [ 0, 8, 1 ], [ 1, 0, 7 ], [ 2, 5, 2 ], [ 4, 2, 3 ], [ 9, 1, 0 ] ],
[ [ 0, 7, 2 ], [ 2, 4, 3 ], [ 4, 1, 4 ], [ 7, 3, 0 ], [ 9, 0, 1 ] ] ]
gap> s:=NumericalSemigroup(10,11,19,23);;
gap> BettiElementsOfNumericalSemigroup(s);
[ 30, 33, 42, 57, 69 ]
gap> FactorizationsElementWRTNumericalSemigroup(69,s);
[ [ 5, 0, 1, 0 ], [ 2, 1, 2, 0 ], [ 0, 0, 0, 3 ] ]
gap> RClassesOfSetOfFactorizations(last);
[ [ [ 2, 1, 2, 0 ], [ 5, 0, 1, 0 ] ], [ [ 0, 0, 0, 3 ] ] ]
gap> s:=NumericalSemigroup(4,6,9);;
gap> LShapesOfNumericalSemigroup(s);
[ [ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 2, 0 ], [ 1, 1 ], [ 0, 2 ], [ 2, 1 ],
[ 1, 2 ], [ 2, 2 ] ],
[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 2, 0 ], [ 1, 1 ], [ 3, 0 ], [ 2, 1 ],
[ 4, 0 ], [ 5, 0 ] ] ]
gap> s:=NumericalSemigroup(101,113,195,272,278,286);;
gap> DenumerantOfElementInNumericalSemigroup(1311,s);
6
gap> LengthsOfFactorizationsIntegerWRTList(100,[11,13,15,19]);
[ 6, 8 ]
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(1100,s);
[ 4, 6, 8, 9 ]
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> ElasticityOfFactorizationsElementWRTNumericalSemigroup(1100,s);
9/4
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> ElasticityOfNumericalSemigroup(s);
286/101
gap> LengthsOfFactorizationsIntegerWRTList(100,[11,13,15,19]);
[ 6, 8 ]
gap> DeltaSetOfSetOfIntegers(last);
[ 2 ]
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> DeltaSetOfFactorizationsElementWRTNumericalSemigroup(1100,s);
[ 1, 2 ]
gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetPeriodicityBoundForNumericalSemigroup(s);
60
gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetPeriodicityStartForNumericalSemigroup(s);
21
gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetListUpToElementWRTNumericalSemigroup(31,s);
[ [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],
[ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ 2 ], [ ], [ ], [ 2 ], [ ],
[ 2 ], [ ], [ 2 ], [ 2 ], [ ] ]
gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetUnionUpToElementWRTNumericalSemigroup(60,s);
[ 2 ]
gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetOfNumericalSemigroup(s);
[ 2 ]
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);
9
gap> s:=NumericalSemigroup(101,113,196,272,278,286);;
gap> MaximalDenumerantOfElementInNumericalSemigroup(1100,s);
1
gap> MaximalDenumerantOfElementInNumericalSemigroup(1311,s);
2
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
[ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> MaximalDenumerantOfSetOfFactorizations(last);
6
gap> s:=NumericalSemigroup(101,113,196,272,278,286);;
gap> MaximalDenumerantOfNumericalSemigroup(s);
4
gap> s:=NumericalSemigroup(101,113,196,272,278,286);;
gap> AdjustmentOfNumericalSemigroup(s);
[ 0, 12, 24, 36, 48, 60, 72, 84, 95, 96, 107, 108, 119, 120, 131, 132, 143,
144, 155, 156, 167, 168, 171, 177, 179, 180, 183, 185, 189, 190, 191, 192,
195, 197, 201, 203, 204, 207, 209, 213, 215, 216, 219, 221, 225, 227, 228,
231, 233, 237, 239, 240, 243, 245, 249, 251, 252, 255, 257, 261, 263, 264,
266, 267, 269, 273, 275, 276, 279, 280, 281, 285, 287, 288, 292, 293, 299,
300, 304, 305, 311, 312, 316, 317, 323, 324, 328, 329, 335, 336, 340, 341,
342, 347, 348, 352, 353, 354, 356, 359, 360, 361, 362, 364, 365, 366, 368,
370, 371, 372, 374, 376, 377, 378, 380, 382, 383, 384, 388, 389, 390, 394,
395, 396, 400, 401, 402, 406, 407, 408, 412, 413, 414, 418, 419, 420, 424,
425, 426, 430, 431, 432, 436, 437, 438, 442, 444, 448, 450, 451, 454, 456,
460, 465, 466, 472, 477, 478, 484, 489, 490, 496, 501, 502, 508, 513, 514,
519, 520, 525, 526, 527, 531, 532, 533, 537, 539, 543, 545, 549, 551, 555,
561, 567, 573, 579, 585, 591, 597, 603, 609, 615, 621, 622, 627, 698, 704,
710, 716, 722 ]
gap> l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(31);;
gap> Length(l);
109
gap> Length(Filtered(l,IsAdditiveNumericalSemigroup));
20
gap> l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(31);;
gap> Length(l);
109
gap> Length(Filtered(l,IsSuperSymmetricNumericalSemigroup));
7
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
[ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> CatenaryDegreeOfSetOfFactorizations(last);
5
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
[ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> AdjacentCatenaryDegreeOfSetOfFactorizations(last);
5
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
[ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> EqualCatenaryDegreeOfSetOfFactorizations(last);
2
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
[ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> MonotoneCatenaryDegreeOfSetOfFactorizations(last);
5
gap> CatenaryDegreeOfElementInNumericalSemigroup(157,NumericalSemigroup(13,18));
0
gap> CatenaryDegreeOfElementInNumericalSemigroup(1157,NumericalSemigroup(13,18));
18
gap> FactorizationsIntegerWRTList(100,[11,13,15,19]);
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
[ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
gap> TameDegreeOfSetOfFactorizations(last);
4
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> CatenaryDegreeOfNumericalSemigroup(s);
8
# to slow without normaliz or 4ti2
#gap> s:=NumericalSemigroup(3,5,7);;
#gap> EqualPrimitiveElementsOfNumericalSemigroup(s);
#[ 3, 5, 7, 10 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> EqualCatenaryDegreeOfNumericalSemigroup(s);
2
# to slow without normaliz or 4ti2
# gap> s:=NumericalSemigroup(3,5,7);;
# gap> MonotonePrimitiveElementsOfNumericalSemigroup(s);
# [ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
# to slow without normaliz or 4ti2
#gap> s:=NumericalSemigroup(10,23,31,44);;
#gap> CatenaryDegreeOfNumericalSemigroup(s);
#9
# to slow without normaliz or 4ti2
#gap> MonotoneCatenaryDegreeOfNumericalSemigroup(s);
#21
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> TameDegreeOfNumericalSemigroup(s);
14
gap> s:=NumericalSemigroup(10,11,13);
<Numerical semigroup with 3 generators>
gap> TameDegreeOfElementInNumericalSemigroup(100,s);
5
gap> s:=NumericalSemigroup(10,11,13);
<Numerical semigroup with 3 generators>
gap> OmegaPrimalityOfElementInNumericalSemigroup(100,s);
13
gap> s:=NumericalSemigroup(10,11,13);;
gap> l:=FirstElementsOfNumericalSemigroup(100,s);;
gap> List(l,x->OmegaPrimalityOfElementInNumericalSemigroup(x,s));
[ 0, 4, 5, 5, 4, 6, 7, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 8, 7, 8, 9, 8, 8, 8,
8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 10, 10, 10, 10, 10,
10, 10, 10, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 12, 12, 12, 12,
12, 12, 12, 12, 13, 14, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14,
14, 14, 14, 14, 14, 15, 16, 15, 15, 15, 15, 15, 15, 15, 15 ]
gap> OmegaPrimalityOfElementListInNumericalSemigroup(l,s);
[ 0, 4, 5, 5, 4, 6, 7, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 8, 7, 8, 9, 8, 8, 8,
8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 10, 10, 10, 10, 10,
10, 10, 10, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 12, 12, 12, 12,
12, 12, 12, 12, 13, 14, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14,
14, 14, 14, 14, 14, 15, 16, 15, 15, 15, 15, 15, 15, 15, 15 ]
gap> s:=NumericalSemigroup(10,11,13);
<Numerical semigroup with 3 generators>
gap> OmegaPrimalityOfNumericalSemigroup(s);
5
gap> s:=NumericalSemigroup(10,11,13);;
gap> BelongsToHomogenizationOfNumericalSemigroup([10,23],s);
true
gap> BelongsToHomogenizationOfNumericalSemigroup([1,23],s);
false
gap> s:=NumericalSemigroup(10,11,13);;
gap> FactorizationsInHomogenizationOfNumericalSemigroup([20,230],s);
[ [ 0, 0, 15, 5 ], [ 0, 2, 12, 6 ], [ 0, 4, 9, 7 ], [ 0, 6, 6, 8 ],
[ 0, 8, 3, 9 ], [ 0, 10, 0, 10 ], [ 1, 1, 7, 11 ], [ 1, 3, 4, 12 ],
[ 1, 5, 1, 13 ], [ 2, 0, 2, 16 ] ]
gap> FactorizationsElementWRTNumericalSemigroup(230,s);
[ [ 23, 0, 0 ], [ 12, 10, 0 ], [ 1, 20, 0 ], [ 14, 7, 1 ], [ 3, 17, 1 ],
[ 16, 4, 2 ], [ 5, 14, 2 ], [ 18, 1, 3 ], [ 7, 11, 3 ], [ 9, 8, 4 ],
[ 11, 5, 5 ], [ 0, 15, 5 ], [ 13, 2, 6 ], [ 2, 12, 6 ], [ 4, 9, 7 ],
[ 6, 6, 8 ], [ 8, 3, 9 ], [ 10, 0, 10 ], [ 1, 7, 11 ], [ 3, 4, 12 ],
[ 5, 1, 13 ], [ 0, 2, 16 ] ]
gap> s:=NumericalSemigroup(10,17,19);;
gap> BettiElementsOfNumericalSemigroup(s);
[ 57, 68, 70 ]
gap> HomogeneousBettiElementsOfNumericalSemigroup(s);
[ [ 5, 57 ], [ 5, 68 ], [ 6, 95 ], [ 7, 70 ], [ 9, 153 ] ]
gap> s:=NumericalSemigroup(10,17,19);;
gap> CatenaryDegreeOfNumericalSemigroup(s);
7
gap> HomogeneousCatenaryDegreeOfNumericalSemigroup(s);
9
gap> s:=NumericalSemigroup(3,5,7);;
gap> MoebiusFunctionAssociatedToNumericalSemigroup(s,10);
2
gap> MoebiusFunctionAssociatedToNumericalSemigroup(s,34);
25
gap> s:=NumericalSemigroup(5,7,11);;
gap> DivisorsOfElementInNumericalSemigroup(s,20);
[ 0, 5, 10, 15, 20 ]
gap> DivisorsOfElementInNumericalSemigroup(20,s);
[ 0, 5, 10, 15, 20 ]
gap> S := NumericalSemigroup(7,9,17);;
gap> FengRaoDistance(S,6,100);
86
gap> S := NumericalSemigroup(7,8,17);;
gap> FengRaoNumber(S,209);
224
gap> FengRaoNumber(209,S);
224
##polynomial.xml
gap> x:=X(Rationals,"x");;
gap> s:=NumericalSemigroup(5,7,9);;
gap> NumericalSemigroupPolynomial(s,x);
x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1
gap> x:=X(Rationals,"x");;
gap> s:=NumericalSemigroup(5,6,7,8);;
gap> f:=NumericalSemigroupPolynomial(s,x);
x^10-x^9+x^5-x+1
gap> IsNumericalSemigroupPolynomial(f);
true
gap> x:=X(Rationals,"x");;
gap> s:=NumericalSemigroup(5,6,7,8);;
gap> f:=NumericalSemigroupPolynomial(s,x);
x^10-x^9+x^5-x+1
gap> NumericalSemigroupFromNumericalSemigroupPolynomial(f)=s;
true
gap> x:=X(Rationals,"x");;
gap> s:=NumericalSemigroup(5,7,9);;
gap> HilbertSeriesOfNumericalSemigroup(s,x);
(x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1)/(-x+1)
gap> x:=Indeterminate(Rationals,1);; SetName(x,"x");
gap> GraeffePolynomial(x^2-1);
x^2-2*x+1
gap> CyclotomicPolynomial(Rationals,3);
x^2+x+1
gap> IsCyclotomicPolynomial(last);
true
gap> x:=X(Rationals,"x");;
gap> s:=NumericalSemigroup(3,5,7);;
gap> t:=NumericalSemigroup(4,6,9);;
gap> p:=NumericalSemigroupPolynomial(s,x);
x^5-x^4+x^3-x+1
gap> q:=NumericalSemigroupPolynomial(t,x);
x^12-x^11+x^8-x^7+x^6-x^5+x^4-x+1
gap> IsKroneckerPolynomial(p);
false
gap> IsKroneckerPolynomial(q);
true
gap> l:=CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(21);;
gap> ForAll(l,IsCyclotomicNumericalSemigroup);
true
gap> l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(13);;
gap> x:=X(Rationals,"x");;
gap> ForAll(l, s->
> IsSelfReciprocalUnivariatePolynomial(NumericalSemigroupPolynomial(s,x)));
true
# Semigroup of values of algebraic curves
gap> x:=Indeterminate(Rationals,1);; SetName(x,"x");
gap> y:=Indeterminate(Rationals,2);; SetName(y,"y");
gap> f:=((y^3-x^2)^2-x*y^2)^4-(y^3-x^2);;
gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f,"all");
[ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
gap> IsDeltaSequence([24,16,28,7]);
true
gap> DeltaSequencesWithFrobeniusNumber(21);
[ [ 8, 6, 11 ], [ 10, 4, 15 ], [ 12, 8, 6, 11 ], [ 14, 4, 11 ],
[ 15, 10, 4 ], [ 23, 2 ] ]
gap> CurveAssociatedToDeltaSequence([24,16,28,7]);
y^24-8*x^2*y^21+28*x^4*y^18-56*x^6*y^15-4*x*y^20+70*x^8*y^12+24*x^3*y^17-56*x^\
10*y^9-60*x^5*y^14+28*x^12*y^6+80*x^7*y^11+6*x^2*y^16-8*x^14*y^3-60*x^9*y^8-24\
*x^4*y^13+x^16+24*x^11*y^5+36*x^6*y^10-4*x^13*y^2-24*x^8*y^7-4*x^3*y^12+6*x^10\
*y^4+8*x^5*y^9-4*x^7*y^6+x^4*y^8-y^3+x^2
gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(last,"all");
[ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
# NEEDS singular
# gap> x:=X(Rationals,"x");; y:=X(Rationals,"y");;
# gap> f:= y^4-2*x^3*y^2-4*x^5*y+x^6-x^7;
# -x^7+x^6-4*x^5*y-2*x^3*y^2+y^4
# gap> SemigroupOfValuesOfPlaneCurve(f);
# <Numerical semigroup with 3 generators>
# gap> MinimalGenerators(last);
# [ 4, 6, 13 ]
# gap> f:=(y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)*(y^2-x^3);;
# gap> SemigroupOfValuesOfPlaneCurve(f);
# <Good semigroup>
# gap> MinimalGenerators(last);
# [ [ 4, 2 ], [ 6, 3 ], [ 13, 15 ], [ 29, 13 ] ]
gap> x:=Indeterminate(Rationals,"x");;
gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13]);
<Numerical semigroup with 4 generators>
gap> MinimalGeneratingSystem(last);
[ 4, 6, 13, 15 ]
gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], "basis");
[ x^4, x^7+x^6, x^13, x^15 ]
gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], 20);
x^20
gap> x:=Indeterminate(Rationals,"x");;
gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13]);
<Numerical semigroup with 3 generators>
gap> MinimalGeneratingSystem(last);
[ 4, 7, 13 ]
gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],"basis");
[ x^4, x^7+x^6, x^13 ]
gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],12);
x^12
gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],6);
fail
gap> t:=Indeterminate(Rationals,"t");;
gap> A:=[t^6+t,t^4];;
gap> M:=[t^3,t^4];;
gap> GeneratorsModule_Global(A,M);
[ t^3, t^4, t^5, t^6 ]
gap> t:=Indeterminate(Rationals,"t");;
gap> GeneratorsKahlerDifferentials([t^3,t^4]);
[ t^2, t^3 ]
gap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,7));
true
gap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,11));
false
##affine.xml
gap> s1 := AffineSemigroupByGenerators([1,3],[7,2],[1,5]);
<Affine semigroup in 2 dimensional space, with 3 generators>
gap> s2 := AffineSemigroupByGenerators([[1,3],[7,2],[1,5]]);;
gap> s3 := AffineSemigroup("generators",[[1,3],[7,2],[1,5]]);;
gap> s4 := AffineSemigroup([1,3],[7,2],[1,5]);;
gap> s5 := AffineSemigroup([[1,3],[7,2],[1,5]]);;
gap> Length(Set([s1,s2,s3,s4,s5]));
1
gap> s1 := AffineSemigroupByEquations([[[-2,1]],[3]]);
<Affine semigroup>
gap> s2 := AffineSemigroup("equations",[[[-2,1]],[3]]);
<Affine semigroup>
gap> s1=s2;
true
gap> a1:=AffineSemigroupByInequalities([[2,-1],[-1,3]]);
<Affine semigroup>
gap> a2:=AffineSemigroup("inequalities",[[2,-1],[-1,3]]);
<Affine semigroup>
gap> a1=a2;
true
gap> a:=AffineSemigroup([[1,0],[0,1],[1,1]]);
<Affine semigroup in 2 dimensional space, with 3 generators>
gap> Generators(a);
[ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ] ]
gap> a:=AffineSemigroup([[1,0],[0,1],[1,1]]);
<Affine semigroup in 2 dimensional space, with 3 generators>
gap> MinimalGenerators(a);
[ [ 0, 1 ], [ 1, 0 ] ]
gap> s:=NumericalSemigroup(1310,1411,1546,1601);
<Numerical semigroup with 4 generators>
gap> a:=AsAffineSemigroup(s);
<Affine semigroup in 1 dimensional space, with 4 generators>
gap> GeneratorsOfAffineSemigroup(a);
[ [ 1310 ], [ 1411 ], [ 1546 ], [ 1601 ] ]
gap> a1:=AffineSemigroup([[3,0],[2,1],[1,2],[0,3]]);
<Affine semigroup in 2 dimensional space, with 4 generators>
gap> IsAffineSemigroupByEquations(a1);
false
gap> IsAffineSemigroupByGenerators(a1);
true
gap> a:=AffineSemigroup([[2,0],[0,2],[1,1]]);;
gap> BelongsToAffineSemigroup([5,5],a);
true
gap> BelongsToAffineSemigroup([1,2],a);
false
gap> [5,5] in a;
true
gap> [1,2] in a;
false
gap> a:=AffineSemigroup("equations",[[[1,1,1],[0,0,2]],[2,2]]);;
gap> IsFullAffineSemigroup(a);
true
gap> HilbertBasisOfSystemOfHomogeneousEquations([[1,0,1],[0,1,-1]],[2]);
[ [ 0, 2, 2 ], [ 1, 1, 1 ], [ 2, 0, 0 ] ]
gap> HilbertBasisOfSystemOfHomogeneousInequalities([[2,-3],[0,1]]);
[ [ 1, 0 ], [ 2, 1 ], [ 3, 2 ] ]
gap> EquationsOfGroupGeneratedBy([[1,2,0],[2,-2,2]]);
[ [ [ 0, 0, -1 ], [ -2, 1, 3 ] ], [ 2 ] ]
gap> BasisOfGroupGivenByEquations([[0,0,1],[2,-1,-3]],[2]);
[ [ -1, -2, 0 ], [ -2, 2, -2 ] ]
gap> a1:=AffineSemigroup([[2,0],[0,2]]);
<Affine semigroup in 2 dimensional space, with 2 generators>
gap> a2:=AffineSemigroup([[1,1]]);
<Affine semigroup in 2 dimensional space, with 1 generators>
gap> GluingOfAffineSemigroups(a1,a2);
<Affine semigroup in 2 dimensional space, with 3 generators>
gap> Generators(last);
[ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ] ]
gap> M:=[[3],[5],[7]];;
gap> CanonicalBasisOfKernelCongruence(M,MonomialLexOrdering());
[ [ [ 0, 7, 0 ], [ 0, 0, 5 ] ], [ [ 1, 0, 1 ], [ 0, 2, 0 ] ],
[ [ 1, 5, 0 ], [ 0, 0, 4 ] ], [ [ 2, 3, 0 ], [ 0, 0, 3 ] ],
[ [ 3, 1, 0 ], [ 0, 0, 2 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
gap> CanonicalBasisOfKernelCongruence(M,MonomialGrlexOrdering());
[ [ [ 0, 7, 0 ], [ 0, 0, 5 ] ], [ [ 1, 0, 1 ], [ 0, 2, 0 ] ],
[ [ 1, 5, 0 ], [ 0, 0, 4 ] ], [ [ 2, 3, 0 ], [ 0, 0, 3 ] ],
[ [ 3, 1, 0 ], [ 0, 0, 2 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
gap> CanonicalBasisOfKernelCongruence(M,MonomialGrevlexOrdering());
[ [ [ 0, 2, 0 ], [ 1, 0, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ],
[ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
gap> gr:=GraverBasis([[3,5,7]]);
[ [ -7, 0, 3 ], [ -5, 3, 0 ], [ -4, 1, 1 ], [ -3, -1, 2 ], [ -2, -3, 3 ],
[ -1, -5, 4 ], [ -1, 2, -1 ], [ 0, -7, 5 ], [ 0, 7, -5 ], [ 1, -2, 1 ],
[ 1, 5, -4 ], [ 2, 3, -3 ], [ 3, 1, -2 ], [ 4, -1, -1 ], [ 5, -3, 0 ],
[ 7, 0, -3 ] ]
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> MinimalPresentationOfAffineSemigroup(a);
[ [ [ 1, 0, 1 ], [ 0, 2, 0 ] ] ]
gap> GeneratorsOfAffineSemigroup(a);
[ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ] ]
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> BettiElementsOfAffineSemigroup(a);
[ [ 2, 2 ] ]
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> PrimitiveElementsOfAffineSemigroup(a);
[ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ], [ 2, 2 ] ]
gap> Set(FactorizationsVectorWRTList([5,5],[[2,0],[0,2],[1,1]]));
[ [ 0, 0, 5 ], [ 1, 1, 3 ], [ 2, 2, 1 ] ]
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> ElasticityOfAffineSemigroup(a);
1
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> DeltaSetOfAffineSemigroup(a);
[ ]
gap> s:=NumericalSemigroup(10,13,15,47);;
gap> a:=AsAffineSemigroup(s);;
gap> DeltaSetOfAffineSemigroup(a);
[ 1, 2, 3, 5 ]
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> CatenaryDegreeOfAffineSemigroup(a);
2
gap> a:=AffineSemigroup("inequalities",[[2,-1],[-1,3]]);
<Affine semigroup>
gap> GeneratorsOfAffineSemigroup(a);
[ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ], [ 3, 1 ] ]
gap> CatenaryDegreeOfAffineSemigroup(a);
3
gap> EqualCatenaryDegreeOfAffineSemigroup(a);
2
gap> HomogeneousCatenaryDegreeOfAffineSemigroup(a);
3
gap> MonotoneCatenaryDegreeOfAffineSemigroup(a);
3
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> TameDegreeOfAffineSemigroup(a);
2
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> OmegaPrimalityOfElementInAffineSemigroup([5,5],a);
6
gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
gap> OmegaPrimalityOfAffineSemigroup(a);
2
##good-semigroups.xml
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> l:=Cartesian([1..11],[1..11]);;
gap> Intersection(dup,l);
[ [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
[ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ],
[ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ],
[ 11, 9 ], [ 11, 11 ] ]
gap> [384938749837,349823749827] in dup;
true
gap> s:=NumericalSemigroup(2,3);;
gap> t:=NumericalSemigroup(3,4);;
gap> e:=3+t;;
gap> dup:=AmalgamationOfNumericalSemigroups(s,e,2);;
gap> [2,3] in dup;
true
gap> s:=NumericalSemigroup(2,3);;
gap> t:=NumericalSemigroup(3,4);;
gap> IsGoodSemigroup(CartesianProductOfNumericalSemigroups(s,t));
true
gap> G:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];
[ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],
[ 25, 12 ], [ 25, 16 ] ]
gap> C:=[25,27];
[ 25, 27 ]
gap> GoodSemigroup(G,C);
<Good semigroup>
gap> s:=NumericalSemigroup(2,3);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);;
gap> BelongsToGoodSemigroup([2,2],dup);
true
gap> [2,2] in dup;
true
gap> [3,2] in dup;
false
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> Conductor(dup);
[ 11, 11 ]
gap> ConductorOfGoodSemigroup(dup);
[ 11, 11 ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> SmallElementsOfGoodSemigroup(dup);
[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ],
[ 6, 10 ], [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ],
[ 9, 9 ], [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ],
[ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> SmallElements(dup);
[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ],
[ 6, 10 ], [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ],
[ 9, 9 ], [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ],
[ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> SmallElementsOfGoodSemigroup(dup);
[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
[ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],
[ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
gap> RepresentsSmallElementsOfGoodSemigroup(last);
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> SmallElementsOfGoodSemigroup(dup);
[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
[ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],
[ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
gap> G:=GoodSemigroupBySmallElements(last);
<Good semigroup>
gap> dup=G;
true
gap> G:=[[4,3],[7,13],[11,17]];;
gap> g:=GoodSemigroup(G,[11,17]);;
gap> mx:=MaximalElementsOfGoodSemigroup(g);
[ [ 0, 0 ], [ 4, 3 ], [ 7, 13 ], [ 8, 6 ] ]
gap> G:=[[4,3],[7,13],[11,17]];;
gap> g:=GoodSemigroup(G,[11,17]);;
gap> IrreducibleMaximalElementsOfGoodSemigroup(g);
[ [ 4, 3 ], [ 7, 13 ] ]
gap> G:=[[4,3],[7,13],[11,17]];;
gap> g:=GoodSemigroup(G,[11,17]);;
gap> sm:=SmallElements(g);;
gap> mx:=MaximalElementsOfGoodSemigroup(g);;
gap> s:=NumericalSemigroupBySmallElements(Set(sm,x->x[1]));;
gap> t:=NumericalSemigroupBySmallElements(Set(sm,x->x[2]));;
gap> Conductor(g);
[ 11, 15 ]
gap> gg:=GoodSemigroupByMaximalElements(s,t,mx,[11,15]);
<Good semigroup>
gap> gg=g;
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> MinimalGoodGeneratingSystemOfGoodSemigroup(dup);
[ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=6+s;;
gap> dup:=NumericalSemigroupDuplication(s,e);
<Good semigroup>
gap> MinimalGenerators(dup);
[ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=CanonicalIdealOfNumericalSemigroup(s);;
gap> e:=15+e;;
gap> dup:=NumericalSemigroupDuplication(s,e);;
gap> IsSymmetricGoodSemigroup(dup);
true
gap> G:=[[3,3],[4,4],[5,4],[4,6]];
[ [ 3, 3 ], [ 4, 4 ], [ 5, 4 ], [ 4, 6 ] ]
gap> C:=[6,6];
[ 6, 6 ]
gap> S:=GoodSemigroup(G,C);
<Good semigroup>
gap> SmallElements(S);
[ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ], [ 4, 6 ], [ 5, 4 ], [ 6, 6 ] ]
gap> A:=ArfGoodSemigroupClosure(S);
<Good semigroup>
gap> SmallElements(A);
[ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ] ]
#Good ideals
gap> G:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];
[ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],
[ 25, 12 ], [ 25, 16 ] ]
gap> C:=[25,27];
[ 25, 27 ]
gap> g := GoodSemigroup(G,C);
<Good semigroup>
gap> i:=GoodIdeal([[2,3]],g);
<Good ideal of good semigroup>
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> d:=NumericalSemigroupDuplication(s,e);;
gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],d);;
gap> GoodGeneratingSystemOfGoodIdeal(e);
[ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> a:=AmalgamationOfNumericalSemigroups(s,e,5);;
gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],a);;
gap> a=AmbientGoodSemigroupOfGoodIdeal(e);
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> d:=NumericalSemigroupDuplication(s,e);;
gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],d);;
gap> MinimalGoodGeneratingSystemOfGoodIdeal(e);
[ [ 2, 3 ], [ 3, 2 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> d:=NumericalSemigroupDuplication(s,e);;
gap> e:=GoodIdeal([[2,3],[3,2]],d);;
gap> [1,1] in e;
false
gap> [2,2] in e;
true
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> d:=NumericalSemigroupDuplication(s,e);;
gap> e:=GoodIdeal([[2,3],[3,2]],d);;
gap> SmallElements(e);
[ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ], [ 5, 5 ], [ 5, 6 ], [ 6, 5 ], [ 7, 7 ] ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> e:=10+s;;
gap> d:=NumericalSemigroupDuplication(s,e);;
gap> c:=CanonicalIdealOfGoodSemigroup(d);;
gap> MinimalGoodGeneratingSystemOfGoodIdeal(c);
[ [ 0, 0 ], [ 2, 2 ] ]
##generalstuff.xml
gap> BezoutSequence(4/5,53/27);
[ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/6, 13/7, 15/8, 17/9, 19/10, 21/11, 23/12,
25/13, 27/14, 29/15, 31/16, 33/17, 35/18, 37/19, 39/20, 41/21, 43/22,
45/23, 47/24, 49/25, 51/26, 53/27 ]
gap> IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/6]);
true
gap> IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/3]);
Take the 6 and the 7 elements of the sequence
false
gap> CeilingOfRational(3/5);
1
gap> RepresentsPeriodicSubAdditiveFunction([1,2,3,4,0]);
true
gap> IsListOfIntegersNS([1,-1,0]);
true
gap> IsListOfIntegersNS(2);
false
gap> IsListOfIntegersNS([[2],3]);
false
gap> IsListOfIntegersNS([]);
false
##random.xml
# RandomNumericalSemigroup(3,9,55);;
# RandomListForNS(13,1,79);;
# RandomModularNumericalSemigroup(9);;
# RandomModularNumericalSemigroup(10,25);;
# RandomProportionallyModularNumericalSemigroup(9);;
# RandomProportionallyModularNumericalSemigroup(10,25);;
# RandomListRepresentingSubAdditiveFunction(7,9);;
# RepresentsPeriodicSubAdditiveFunction(last);
# ns := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,50);;
# MinimalGeneratingSystem(ns);;
# SmallElements(ns);;
# ns2 := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,9);;
# ns3 := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,10);;
# MinimalGeneratingSystem(ns3);;
## get info level to the original state
gap> SetInfoLevel( InfoNumSgps, INFO_NSGPS);
gap> STOP_TEST( "testall.tst", 10000 );
## The first argument of STOP_TEST should be the name of the test file.
## The number is a proportionality factor that is used to output a
## "GAPstone" speed ranking after the file has been completely processed.
## For the files provided with the distribution this scaling is roughly
## equalized to yield the same numbers as produced by the test file
## tst/combinat.tst. For package tests, you may leave it unchnaged.
#############################################################################
##