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<?xml version="1.0" encoding="UTF-8"?>
<Section>
                <Heading>
                    Complete intersection numerical semigroups
                </Heading>

    The cardinality of a minimal presentation of a numerical semigroup is always greater than or equal to its embedding
    dimension minus one. Complete intersection numerical semigroups are numerical semigroups reaching this bound, and they are
    irreducible.  It can be shown that every complete intersection (other that <M>\mathbb N</M>) is a complete intersection if
    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>,
    then we obtain a free semigroup in the sense of <Cite Key="BC77"></Cite>. Two special kinds of free semigroups are telescopic semigroups
    (<Cite Key="KP95"></Cite>) and those associated to an irreducible planar curve (<Cite Key="Z86"></Cite>). We use the algorithms presented
    in <Cite Key="AGS13"></Cite> to find the set of all complete intersections (also free, telescopic and associated to irreducible planar curves) numerical
    semigroups with given Frobenius number.
<P/>

                 <ManSection>
                    <Func Arg="s" Name="AsGluingOfNumericalSemigroups"></Func>
                    <Description>
                        <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
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>.
                        <Example><![CDATA[
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);
[  ]
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                  <Prop Arg="s" Name="IsCompleteIntersection"></Prop>
                    <Prop Arg="s" Name="IsACompleteIntersectionNumericalSemigroup"></Prop>
                    <Description>
                        <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.

                        <P/>
                        This filter implies <Ref Func="IsSymmetricNumericalSemigroup"/> and <Ref Func="IsCyclotomicNumericalSemigroup"/>.
                        <Example><![CDATA[
gap> s := NumericalSemigroup( 10, 15, 16 );
<Numerical semigroup with 3 generators>
gap> IsCompleteIntersection(s);
true
gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
<Numerical semigroup with 8 generators>
gap> IsACompleteIntersectionNumericalSemigroup(s);
false
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Func Arg="f" Name="CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber"></Func>
                    <Description>
                        <A>f</A> is an integer. The output is the set of all
                        complete intersection numerical semigroups with Frobenius number <A>f</A>.
                        <Example><![CDATA[
gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));
34
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Prop Arg="s" Name="IsFree"></Prop>
                    <Prop Arg="s" Name="IsFreeNumericalSemigroup"></Prop>
                    <Description>
                        <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>:
      it is either <M>\mathbb N</M> or the gluing of a copy of  <M>\mathbb N</M> with a free numerical semigroup.

                        <P/>
                        This filter implies <Ref Func="IsACompleteIntersectionNumericalSemigroup"/>.
                        <Example><![CDATA[
gap> IsFree(NumericalSemigroup(10,15,16));
true
gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));
false
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Func Arg="f" Name="FreeNumericalSemigroupsWithFrobeniusNumber"></Func>
                    <Description>
                        <A>f</A> is an integer. The output is the set of all
                        free numerical semigroups with Frobenius number <A>f</A>.
                        <Example><![CDATA[
gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57));
33
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Prop Arg="s" Name="IsTelescopic"></Prop>
                    <Prop Arg="s" Name="IsTelescopicNumericalSemigroup"></Prop>
                    <Description>
                        <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>:
      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,
      and <M>s' is again a telescopic numerical semigroup, where n_1 < \cdots < n_e are the minimal generators of s.

                        <P/>
                        This filter implies <Ref Func="IsAperySetBetaRectangular"/> and <Ref Func="IsFree"/>.
                        <Example><![CDATA[
gap> IsTelescopic(NumericalSemigroup(4,11,14));
false
gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));
false
gap> IsFree(NumericalSemigroup(4,11,14));
true
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Func Arg="f" Name="TelescopicNumericalSemigroupsWithFrobeniusNumber"></Func>
                    <Description>
                        <A>f</A> is an integer. The output is the set of all
                        telescopic numerical semigroups with Frobenius number <A>f</A>.
                        <Example><![CDATA[
gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));
20
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                    <Prop Arg="s" Name="IsUniversallyFree"></Prop>
                    <Prop Arg="s" Name="IsUniversallyFreeNumericalSemigroup"></Prop>
                    <Description>
                        <A>s</A> is a numerical semigroup.  The output is true if the numerical semigroup is free for all the arrangements of its minimal generators.

                        <P/>
                        This filter implies <Ref Func="IsTelescopic"/>.
                        <Example><![CDATA[
gap> s:=NumericalSemigroup(10,15,18);;
gap> IsUniversallyFree(s);
true
gap> s:=NumericalSemigroup(4,6,9);;
gap> IsFree(s);
true
gap> IsUniversallyFree(s);
false
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                    <Prop Arg="s" Name="IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity"></Prop>
                    <Description>
                        <A>s</A> is a numerical semigroup.  The output is true if the numerical semigroup is associated to an irreducible planar curve singularity
      (<Cite Key="Z86"></Cite>). These semigroups are telescopic.

                        <P/>
                        This filter implies <Ref Func="IsAperySetAlphaRectangular"/> and <Ref Func="IsTelescopicNumericalSemigroup"/>.
                        <Example><![CDATA[
gap> ns := NumericalSemigroup(4,11,14);;
gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
false
gap> ns := NumericalSemigroup(4,11,19);;
gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
true
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                  <Func Arg="f" Name="NumericalSemigroupsPlanarSingularityWithFrobeniusNumber"></Func>
                    <Description>
                        <A>f</A> is an integer. The output is the set of all
                        numerical semigroups associated to irreducible planar curves singularities with Frobenius number <A>f</A>.
                        <Example><![CDATA[
gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));
7
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                                    <Func Arg="S" Name="IsAperySetGammaRectangular"></Func>
                                    <Description>

                                        <A>S</A> is a numerical semigroup.
                                        <P/>

                                        Test for the <M>\gamma</M>-rectangularity of the Apéry Set of a numerical semigroup.
                                        This test is the implementation of the algorithm given in
                                        <Cite Key="DAMSClasses"></Cite>. Numerical Semigroups with this property are free and thus complete intersections.

                                        <P/>
                                        This filter implies <Ref Func="IsFreeNumericalSemigroup"/>.

                                        <Example><![CDATA[
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
                ]]></Example>
                                    </Description>
                                </ManSection>
                <ManSection>
                                    <Func Arg="S" Name="IsAperySetBetaRectangular"></Func>
                                    <Description>

                                        <A>S</A> is a numerical semigroup.
                                        <P/>

                                        Test for the <M>\beta</M>-rectangularity of the Apéry Set of a numerical semigroup.
                                        This test is the implementation of the algorithm given in
                                        <Cite Key="DAMSClasses"></Cite>; <M>\beta</M>-rectangularity implies <M>\gamma</M>-rectangularity.

                                        <P/>
                                        This filter implies <Ref Func="IsAperySetGammaRectangular"/>.
                                        <Example><![CDATA[
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
                ]]></Example>
                                    </Description>
                                </ManSection>
                <ManSection>
                                    <Func Arg="S" Name="IsAperySetAlphaRectangular"></Func>
                                    <Description>

                                        <A>S</A> is a numerical semigroup.
                                        <P/>

                                        Test for the <M>\alpha</M>-rectangularity of the Apéry Set of a numerical semigroup.
                                        This test is the implementation of the algorithm given in
                                        <Cite Key="DAMSClasses"></Cite>; <M>\alpha</M>-rectangularity implies <M>\beta</M>-rectangularity.
                                    <P/>
                                    This filter implies <Ref Func="IsAperySetBetaRectangular"/>.

                                        <Example><![CDATA[
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
]]></Example>
                                    </Description>
                </ManSection>

</Section>

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