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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<?xml version="1.0" encoding="UTF-8"?>1<Section>2<Heading>3Complete intersection numerical semigroups4</Heading>56The cardinality of a minimal presentation of a numerical semigroup is alwas greater than or equal to its embedding7dimension minus one. Complete intersection numerical semigroups are numerical semigroups reching this bound, and they are8irreducible. It can be shown that every complete intersection (other that <M>\mathbb N</M>) is a complete intersection if9and only if it is the gluing of two complete intersections. When in this gluing, one of the copies is isomorphic to <M>\mathbb N</M>,10then we obtain a free semigroup in the sense of <Cite Key="BC77"></Cite>. Two special kinds of free semigroups are telescopic semigroups11(<Cite Key="KP95"></Cite>) and those associated to an irreducible planar curve (<Cite Key="Z86"></Cite>). We use the algorithms presented12in <Cite Key="AGS13"></Cite> to find the set of all complete intersections (also free, telescopic and associated to irreducible planar curves) numerical13semigroups with given Frobenius number.14<P/>1516<ManSection>17<Func Arg="s" Name="AsGluingOfNumericalSemigroups"></Func>18<Description>19<A>s</A> is a numerical semigroup. Returns all partitions <M>\{A_1,A_2\}</M> of the minimal generating set of <A>s</A> such20that <A>s</A> is a gluing of <M>\langle A_1\rangle</M> and <M>\langle A_2\rangle</M> by <M>gcd(A_1)gcd(A_2)</M>21<Example><![CDATA[22gap> s := NumericalSemigroup( 10, 15, 16 );23<Numerical semigroup with 3 generators>24gap> AsGluingOfNumericalSemigroups(s);25[ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ]26gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );27<Numerical semigroup with 8 generators>28gap> AsGluingOfNumericalSemigroups(s);29[ ]30]]></Example>31</Description>32</ManSection>3334<ManSection>35<Prop Arg="s" Name="IsCompleteIntersection"></Prop>36<Prop Arg="s" Name="IsACompleteIntersectionNumericalSemigroup"></Prop>37<Description>38<A>s</A> is a numerical semigroup. The output is true if the numerical semigroup is a complete intersection, that is, the cardinality of a (any) minimal presentation equals its embedding dimension minus one.39<Example><![CDATA[40gap> s := NumericalSemigroup( 10, 15, 16 );41<Numerical semigroup with 3 generators>42gap> IsACompleteIntersectionNumericalSemigroup(s);43true44gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );45<Numerical semigroup with 8 generators>46gap> IsACompleteIntersectionNumericalSemigroup(s);47false48]]></Example>49</Description>50</ManSection>5152<ManSection>53<Func Arg="f" Name="CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber"></Func>54<Description>55<A>f</A> is an integer greater than or equal to -1. The output is the set of all56complete intersection numerical semigroups with frobenius number <A>f</A>.57<Example><![CDATA[58gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));593460]]></Example>61</Description>62</ManSection>6364<ManSection>65<Prop Arg="s" Name="IsFree"></Prop>66<Prop Arg="s" Name="IsFreeNumericalSemigroup"></Prop>67<Description>68<A>s</A> is a numerical semigroup. The output is true if the numerical semigroup is free in the sense of <Cite Key="BC77"></Cite>:69it is either <M>\mathbb N</M> or the gluing of a copy of <M>\mathbb N</M> with a free numerical semigroup.70<Example><![CDATA[71gap> IsFreeNumericalSemigroup(NumericalSemigroup(10,15,16));72true73gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));74false75]]></Example>76</Description>77</ManSection>7879<ManSection>80<Func Arg="f" Name="FreeNumericalSemigroupsWithFrobeniusNumber"></Func>81<Description>82<A>f</A> is an integer greater than or equal to -1. The output is the set of all83free numerical semigroups with frobenius number <A>f</A>.84<Example><![CDATA[85gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57));863387]]></Example>88</Description>89</ManSection>9091<ManSection>92<Prop Arg="s" Name="IsTelescopic"></Prop>93<Prop Arg="s" Name="IsTelescopicNumericalSemigroup"></Prop>94<Description>95<A>s</A> is a numerical semigroup. The output is true if the numerical semigroup is telescopic in the sense of <Cite Key="KP95"></Cite>:96it is either <M>\mathbb N</M> or the gluing of <M>\langle n_e\rangle</M> and <M>s'=\langle n_1/d,\ldots, n_{e-1}/d\rangle</M>,97and <M>s'</M> is again a telescopic numerical semigroup, where <M>n_1 < \cdots < n_e </M> are the minimal generators of <A>s</A>.98<Example><![CDATA[99gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));100false101gap> IsFreeNumericalSemigroup(NumericalSemigroup(4,11,14));102true103]]></Example>104</Description>105</ManSection>106107<ManSection>108<Func Arg="f" Name="TelescopicNumericalSemigroupsWithFrobeniusNumber"></Func>109<Description>110<A>f</A> is an integer greater than or equal to -1. The output is the set of all111telescopic numerical semigroups with frobenius number <A>f</A>.112<Example><![CDATA[113gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));11420115]]></Example>116</Description>117</ManSection>118119<ManSection>120<Prop Arg="s" Name="IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity"></Prop>121<Description>122<A>s</A> is a numerical semigroup. The output is true if the numerical semigroup is associated to an irreducible planar curve singularity123(<Cite Key="Z86"></Cite>). These semigroups are telescopic.124<Example><![CDATA[125gap> ns := NumericalSemigroup(4,11,14);;126gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);127false128gap> ns := NumericalSemigroup(4,11,19);;129gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);130true131]]></Example>132</Description>133</ManSection>134135<ManSection>136<Func Arg="f" Name="NumericalSemigroupsPlanarSingularityWithFrobeniusNumber"></Func>137<Description>138<A>f</A> is an integer greater than or equal to -1. The output is the set of all139numerical semigroups associated to irreducible planar curves singularities with frobenius number <A>f</A>.140<Example><![CDATA[141gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));1427143]]></Example>144</Description>145</ManSection>146147148<ManSection>149<Func Arg="S" Name="IsAperySetGammaRectangular"></Func>150<Description>151152<A>S</A> is a numerical semigroup.153<P/>154155Test for the <M>\gamma</M>-rectangularity of the Apéry Set of a numerical semigroup.156This test is the implementation of the algorithm given in157<Cite Key="DAMSClasses"></Cite>. Numerical Semigroups with this property are free and thus complete intersections.158159160<Example><![CDATA[161gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;162gap> IsAperySetGammaRectangular(s);163false164gap> s:=NumericalSemigroup(4,6,11);;165gap> IsAperySetGammaRectangular(s);166true167]]></Example>168</Description>169</ManSection>170<ManSection>171<Func Arg="S" Name="IsAperySetBetaRectangular"></Func>172<Description>173174<A>S</A> is a numerical semigroup.175<P/>176177Test for the <M>\beta</M>-rectangularity of the Apéry Set of a numerical semigroup.178This test is the implementation of the algorithm given in179<Cite Key="DAMSClasses"></Cite>; <M>\beta</M>-rectangularity implies <M>\gamma</M>-rectangularity.180181182<Example><![CDATA[183gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;184gap> IsAperySetBetaRectangular(s);185false186gap> s:=NumericalSemigroup(4,6,11);;187gap> IsAperySetBetaRectangular(s);188true189]]></Example>190</Description>191</ManSection>192<ManSection>193<Func Arg="S" Name="IsAperySetAlphaRectangular"></Func>194<Description>195196<A>S</A> is a numerical semigroup.197<P/>198199Test for the <M>\alpha</M>-rectangularity of the Apéry Set of a numerical semigroup.200This test is the implementation of the algorithm given in201<Cite Key="DAMSClasses"></Cite>; <M>\alpha</M>-rectangularity implies <M>\beta</M>-rectangularity.202203204<Example><![CDATA[205gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;206gap> IsAperySetAlphaRectangular(s);207false208gap> s:=NumericalSemigroup(4,6,11);;209gap> IsAperySetAlphaRectangular(s);210true211]]></Example>212</Description>213</ManSection>214215</Section>216217218