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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>
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<Section>
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<Heading>
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Complete intersection numerical semigroups
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</Heading>
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The cardinality of a minimal presentation of a numerical semigroup is alwas greater than or equal to its embedding
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dimension minus one. Complete intersection numerical semigroups are numerical semigroups reching this bound, and they are
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irreducible. It can be shown that every complete intersection (other that <M>\mathbb N</M>) is a complete intersection if
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and 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>,
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then we obtain a free semigroup in the sense of <Cite Key="BC77"></Cite>. Two special kinds of free semigroups are telescopic semigroups
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(<Cite Key="KP95"></Cite>) and those associated to an irreducible planar curve (<Cite Key="Z86"></Cite>). We use the algorithms presented
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in <Cite Key="AGS13"></Cite> to find the set of all complete intersections (also free, telescopic and associated to irreducible planar curves) numerical
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semigroups with given Frobenius number.
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<P/>
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<ManSection>
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<Func Arg="s" Name="AsGluingOfNumericalSemigroups"></Func>
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<Description>
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<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> such
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that <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>
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<Example><![CDATA[
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gap> s := NumericalSemigroup( 10, 15, 16 );
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<Numerical semigroup with 3 generators>
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gap> AsGluingOfNumericalSemigroups(s);
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[ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ]
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gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
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<Numerical semigroup with 8 generators>
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gap> AsGluingOfNumericalSemigroups(s);
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[ ]
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Prop Arg="s" Name="IsCompleteIntersection"></Prop>
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<Prop Arg="s" Name="IsACompleteIntersectionNumericalSemigroup"></Prop>
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<Description>
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<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.
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<Example><![CDATA[
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gap> s := NumericalSemigroup( 10, 15, 16 );
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<Numerical semigroup with 3 generators>
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gap> IsACompleteIntersectionNumericalSemigroup(s);
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true
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gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
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<Numerical semigroup with 8 generators>
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gap> IsACompleteIntersectionNumericalSemigroup(s);
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false
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="f" Name="CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber"></Func>
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<Description>
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<A>f</A> is an integer greater than or equal to -1. The output is the set of all
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complete intersection numerical semigroups with frobenius number <A>f</A>.
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<Example><![CDATA[
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gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));
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34
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Prop Arg="s" Name="IsFree"></Prop>
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<Prop Arg="s" Name="IsFreeNumericalSemigroup"></Prop>
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<Description>
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<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>:
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it is either <M>\mathbb N</M> or the gluing of a copy of <M>\mathbb N</M> with a free numerical semigroup.
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<Example><![CDATA[
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gap> IsFreeNumericalSemigroup(NumericalSemigroup(10,15,16));
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true
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gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));
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false
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="f" Name="FreeNumericalSemigroupsWithFrobeniusNumber"></Func>
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<Description>
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<A>f</A> is an integer greater than or equal to -1. The output is the set of all
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free numerical semigroups with frobenius number <A>f</A>.
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<Example><![CDATA[
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gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57));
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33
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Prop Arg="s" Name="IsTelescopic"></Prop>
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<Prop Arg="s" Name="IsTelescopicNumericalSemigroup"></Prop>
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<Description>
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<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>:
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it 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>,
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and <M>s'</M> is again a telescopic numerical semigroup, where <M>n_1 &lt; \cdots &lt; n_e </M> are the minimal generators of <A>s</A>.
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<Example><![CDATA[
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gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));
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false
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gap> IsFreeNumericalSemigroup(NumericalSemigroup(4,11,14));
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true
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="f" Name="TelescopicNumericalSemigroupsWithFrobeniusNumber"></Func>
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<Description>
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<A>f</A> is an integer greater than or equal to -1. The output is the set of all
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telescopic numerical semigroups with frobenius number <A>f</A>.
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<Example><![CDATA[
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gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Prop Arg="s" Name="IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity"></Prop>
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<Description>
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<A>s</A> is a numerical semigroup. The output is true if the numerical semigroup is associated to an irreducible planar curve singularity
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(<Cite Key="Z86"></Cite>). These semigroups are telescopic.
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<Example><![CDATA[
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gap> ns := NumericalSemigroup(4,11,14);;
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gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
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false
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gap> ns := NumericalSemigroup(4,11,19);;
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gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
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true
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="f" Name="NumericalSemigroupsPlanarSingularityWithFrobeniusNumber"></Func>
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<Description>
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<A>f</A> is an integer greater than or equal to -1. The output is the set of all
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numerical semigroups associated to irreducible planar curves singularities with frobenius number <A>f</A>.
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<Example><![CDATA[
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gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));
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7
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="S" Name="IsAperySetGammaRectangular"></Func>
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<Description>
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<A>S</A> is a numerical semigroup.
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<P/>
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Test for the <M>\gamma</M>-rectangularity of the Apéry Set of a numerical semigroup.
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This test is the implementation of the algorithm given in
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<Cite Key="DAMSClasses"></Cite>. Numerical Semigroups with this property are free and thus complete intersections.
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<Example><![CDATA[
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gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
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gap> IsAperySetGammaRectangular(s);
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false
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gap> s:=NumericalSemigroup(4,6,11);;
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gap> IsAperySetGammaRectangular(s);
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true
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="S" Name="IsAperySetBetaRectangular"></Func>
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<Description>
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<A>S</A> is a numerical semigroup.
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<P/>
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Test for the <M>\beta</M>-rectangularity of the Apéry Set of a numerical semigroup.
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This test is the implementation of the algorithm given in
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<Cite Key="DAMSClasses"></Cite>; <M>\beta</M>-rectangularity implies <M>\gamma</M>-rectangularity.
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<Example><![CDATA[
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gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
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gap> IsAperySetBetaRectangular(s);
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false
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gap> s:=NumericalSemigroup(4,6,11);;
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gap> IsAperySetBetaRectangular(s);
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true
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]]></Example>
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</Description>
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</ManSection>
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<ManSection>
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<Func Arg="S" Name="IsAperySetAlphaRectangular"></Func>
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<Description>
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<A>S</A> is a numerical semigroup.
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<P/>
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Test for the <M>\alpha</M>-rectangularity of the Apéry Set of a numerical semigroup.
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This test is the implementation of the algorithm given in
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<Cite Key="DAMSClasses"></Cite>; <M>\alpha</M>-rectangularity implies <M>\beta</M>-rectangularity.
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<Example><![CDATA[
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gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
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gap> IsAperySetAlphaRectangular(s);
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false
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gap> s:=NumericalSemigroup(4,6,11);;
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gap> IsAperySetAlphaRectangular(s);
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true
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]]></Example>
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</Description>
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</ManSection>
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</Section>
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