CoCalc -- Ideals_of_numerical

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Project: cocalc-sagemath-dev-slelievre
Path: gap4r8 / pkg / NumericalSgps-1.1.5 / doc / Ideals_of_numerical_semigroups.xml
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<?xml version="1.0" encoding="UTF-8"?>
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        Let <M>S</M> be a numerical semigroup. A set <M>I</M> of integers is an <E>ideal relative</E>
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        to a numerical semigroup <M>S</M> provided that <M>I+S\subseteq I</M> and that there
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        exists <M>d\in S</M> such that <M>d+I\subseteq S</M>.
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        <P/>
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        If <M>\{i_1,\ldots,i_k\}</M> is a subset of <M>{\mathbb Z}</M>, then the set
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        <M>I=\{i_1,\ldots,i_k\}+S=\bigcup_{n=1}^k i_n+S</M> is an ideal relative to <M>S</M>,
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        and <M>\{i_1,\ldots, i_k\}</M> is a system of generators of <M>I</M>. A system of generators
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        <M>M</M> is minimal if no proper subset of <M>M</M> generates the same ideal.
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        Usually, ideals are specified by means of its generators and the ambient numerical
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        semigroup to which they are ideals (for more information see for instance
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        <Cite Key="BDF97"></Cite>).
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<Section>
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                <Heading>
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                    Definitions and basic operations
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                </Heading>
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                <ManSection>
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                    <Func Arg="l,S" Name="IdealOfNumericalSemigroup"></Func>
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                    <Func Arg="l,S" Name="+" Label="for defining ideal of numerical semigroup"></Func>
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                    <Description>
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                        <A>S</A> is a numerical semigroup and <A>l</A> a list of integers.
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                        The output is the ideal of <A>S</A> generated by <A>l</A>.
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                        <P/>
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                        There are several shortcuts for this function, as shown in the example.
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                        <Example><![CDATA[
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gap> IdealOfNumericalSemigroup([3,5],NumericalSemigroup(9,11));
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<Ideal of numerical semigroup>
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gap> [3,5]+NumericalSemigroup(9,11);
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<Ideal of numerical semigroup>
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gap> last=last2;
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true
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gap> 3+NumericalSemigroup(5,9);
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<Ideal of numerical semigroup>
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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="Obj" Name="IsIdealOfNumericalSemigroup"></Func>
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                    <Description>
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                        Tests if the object <A>Obj</A> is an ideal of a
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                        numerical semigroup.
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                        <Example><![CDATA[
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gap> I:=[1..7]+NumericalSemigroup(7,19);;
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gap> IsIdealOfNumericalSemigroup(I);
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true
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gap> IsIdealOfNumericalSemigroup(2);
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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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<Attr Arg="I" Name="MinimalGenerators" Label="for ideal of numerical semigroup"></Attr>
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<Attr Arg="I" Name="MinimalGeneratingSystem" Label="for ideal of numerical semigroup"></Attr>
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                    <Attr Arg="I" Name="MinimalGeneratingSystemOfIdealOfNumericalSemigroup"></Attr>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup.
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                        The output is the minimal system of generators of <A>I</A>.
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                        <Example><![CDATA[
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gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
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[ 3 ]
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gap> MinimalGeneratingSystem(I);
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[ 3 ]
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gap> MinimalGenerators([3,5]+NumericalSemigroup(2,11));
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[ 3 ]
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]]></Example>
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                    </Description>
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                </ManSection>
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                <ManSection>
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                    <Attr Arg="I" Name="Generators" Label="for ideal of numerical semigroup"></Attr>
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                    <Attr Arg="I" Name="GeneratorsOfIdealOfNumericalSemigroup"></Attr>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup.
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                        The output is a system of generators of the ideal.
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			<P/>
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			Remark: from Version 1.0.1 on, this value does not change even when a set of minimal generators is computed.
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                        <Example><![CDATA[
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gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
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gap> GeneratorsOfIdealOfNumericalSemigroup(I);
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[ 3, 5, 9 ]
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gap> Generators(I);
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[ 3, 5, 9 ]
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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="I" Name="AmbientNumericalSemigroupOfIdeal"></Func>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup, say <M>S</M>.
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                        The output is <M>S</M>.
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                        <Example><![CDATA[
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gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
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gap> AmbientNumericalSemigroupOfIdeal(I);
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<Numerical semigroup with 2 generators>
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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="I" Name="IsIntegral"></Prop>
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                    <Prop Arg="I" Name="IsIntegralIdealOfNumericalSemigroup"></Prop>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup, say <M>S</M>.
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                        Detects if  <M>I\subseteq S</M>.
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                        <Example><![CDATA[
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gap> s:=NumericalSemigroup(3,7,5);;
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gap> IsIntegralIdealOfNumericalSemigroup(4+s);
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false
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gap> IsIntegralIdealOfNumericalSemigroup(10+s);
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true
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gap> IsIntegral(10+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="I" Name="SmallElements" Label="for ideal of numerical semigroup"></Func>
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                  <Func Arg="I" Name="SmallElementsOfIdealOfNumericalSemigroup"></Func>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup.
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                        The output is a list with the elements in <A>I</A> that are less than or equal
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                        to the greatest integer not belonging to the ideal plus one.
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                        <Example><![CDATA[
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gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
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gap> SmallElementsOfIdealOfNumericalSemigroup(I);
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[ 3, 5, 7, 9, 11, 13 ]
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gap> SmallElements(I) = SmallElementsOfIdealOfNumericalSemigroup(I);
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true
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gap> J:=[2,11]+NumericalSemigroup(2,11);;
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gap> SmallElementsOfIdealOfNumericalSemigroup(J);
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[ 2, 4, 6, 8, 10 ]
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]]></Example>
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                    </Description>
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                </ManSection>
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		<ManSection>
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                    <Attr Name="Conductor" Arg="NS" Label="for ideal of numerical semigroup"/>
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                    <Func Arg="I" Name="ConductorOfIdealOfNumericalSemigroup"></Func>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup.
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                        The output is the largest element in  <A>SmallElements(I)</A>.
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                        <Example><![CDATA[
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gap> s:=NumericalSemigroup(3,7,5);;
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gap> ConductorOfIdealOfNumericalSemigroup(10+s);
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15
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gap> Conductor(10+s);
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15
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]]></Example>
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                    </Description>
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                </ManSection>
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                <ManSection>
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                    <Oper Name="Minimum" Arg="I" Label="minimum of ideal of numerical semigroup"/>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup.
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                        The output is the minimum of <A>I</A>.
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                        <Example><![CDATA[
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gap> J:=[2,11]+NumericalSemigroup(2,11);;
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gap> Minimum(J);
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2
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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="n, I" Name="BelongsToIdealOfNumericalSemigroup"></Func>
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                    <Oper Name="\in" Arg="n, I" Label="membership test in ideal of numerical semigroup"/>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup, <A>n</A> is an integer.
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                        The output is true if <A>n</A> belongs to <A>I</A>.
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                        <P/>
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                        <A> n in I</A> can be used for short.
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                        <Example><![CDATA[
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gap> J:=[2,11]+NumericalSemigroup(2,11);;
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gap> BelongsToIdealOfNumericalSemigroup(9,J);
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false
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gap> 9 in J;
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false
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gap> BelongsToIdealOfNumericalSemigroup(10,J);
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true
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gap> 10 in J;
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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="I, J" Name="SumIdealsOfNumericalSemigroup"></Func>
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                    <Func Arg="I, J" Name="+" Label="for ideals of numerical semigroup"></Func>
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                    <Description>
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                        <A>I, J</A> are ideals of a numerical semigroup.
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                        The output is the sum of both ideals <M>\{ i+j \ |\  i\in <A>I</A>, j\in <A>J</A>\}</M>.
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                        <Example><![CDATA[
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gap> I:=[3,5,9]+NumericalSemigroup(2,11);;
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gap> J:=[2,11]+NumericalSemigroup(2,11);;
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gap> I+J;
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<Ideal of numerical semigroup>
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
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[ 5, 14 ]
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gap> SumIdealsOfNumericalSemigroup(I,J);
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<Ideal of numerical semigroup>
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
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[ 5, 14 ]
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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="n, I" Name="MultipleOfIdealOfNumericalSemigroup"></Func>
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                    <Func Arg="n, I" Name="*" Label="for multiple of ideal of numerical semigroup"></Func>
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                    <Description>
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                        <A>I</A> is an ideal of a numerical semigroup, <A>n</A> is a non negative integer.
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                        The output is the ideal <M><A>I</A>+\cdots+<A>I</A></M> (<A>n</A> times).
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                        <P/>
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                        <A> n * I</A> can be used for short.
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                        <Example><![CDATA[
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gap> I:=[0,1]+NumericalSemigroup(3,5,7);;
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(2*I);
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[ 0, 1, 2 ]
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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="I, J" Name="SubtractIdealsOfNumericalSemigroup"></Func>
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                    <Func Arg="I, J" Name="-" Label="for ideals of numerical semigroup"></Func>
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                    <Description>
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                        <A>I, J</A> are ideals of a numerical semigroup.
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                        The output is the ideal <M>\{ z\in {\mathbb Z}\ |\  z+<A>J</A>\subseteq <A>I</A>\}</M>.
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                        <P/>
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                        <M>I - J</M> is a synonym of <C>SubtractIdealsOfNumericalSemigroup</C>.
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                        <P/>
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                        <M>S-</M><A>J</A> is a synonym of <M>(0+S)-</M><A>J</A>, if <M>S</M> is the ambient semigroup of <A>I</A> and <A>J</A>.
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                        The following example appears in <Cite Key="HS04"></Cite>.
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                        <Example><![CDATA[
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gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);;
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gap> I:=[0,1]+S;;
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gap> II:=S-I;;
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
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[ 0, 1 ]
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(II);
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[ 14, 20 ]
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gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I+II);
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[ 14, 15, 20, 21 ]
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]]></Example>
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                    </Description>
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                </ManSection>
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                <ManSection>
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                    <Oper Name="Difference" Arg="I, J" Label="for ideals of numerical semigroups"/>
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                    <Func Arg="I, J" Name="DifferenceOfIdealsOfNumericalSemigroup"></Func>
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                    <Description>
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                        <A>I, J</A> are ideals of a numerical semigroup. <A>J</A> must be contained in <A>I</A>.
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                        The output is the set <M><A>I</A>\setminus <A>J</A></M>.
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                        <Example><![CDATA[
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gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);;
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gap> I:=[0,1]+S;
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<Ideal of numerical semigroup>
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gap> 2*I-2*I;
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<Ideal of numerical semigroup>
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gap> I-I;
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<Ideal of numerical semigroup>
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gap> ii := 2*I-2*I;
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<Ideal of numerical semigroup>
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gap> i := I-I;
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<Ideal of numerical semigroup>
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gap>  DifferenceOfIdealsOfNumericalSemigroup(last2,last);
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[ 26, 27, 37, 38 ]
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gap> Difference(ii,i);
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[ 26, 27, 37, 38 ]
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gap> Difference(i,ii);
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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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					<Func Arg="k, I" Name="TranslationOfIdealOfNumericalSemigroup"></Func>
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	                    <Func Arg="k, I" Name="+" Label="translation of ideal of numerical semigroup"></Func>
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				<Description>
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			Given an ideal <A>I</A> of a numerical semigroup S and an integer <A>k</A>
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			returns an ideal of the numerical semigroup S generated by
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			<M>\{i_1+k,\ldots,i_n+k\}</M> where <M>\{i_1,\ldots,i_n\}</M> is the system of generators of <A>I</A>.
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			<P/>
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			As a synonym to <C>TranslationOfIdealOfNumericalSemigroup(k, I)</C> the expression
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			<C>k + I</C> may be used.
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						<Example><![CDATA[
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gap> s:=NumericalSemigroup(13,23);;
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gap> l:=List([1..6], _ -> Random([8..34]));
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[ 22, 29, 34, 25, 10, 12 ]
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gap> I:=IdealOfNumericalSemigroup(l, s);;
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gap> It:=TranslationOfIdealOfNumericalSemigroup(7,I);
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<Ideal of numerical semigroup>
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gap> It2:=7+I;
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<Ideal of numerical semigroup>
336
gap> It2=It;
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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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                    <Oper Name="Intersection" Arg="I, J" Label="for ideals of numerical semigroups"/>
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                    <Func Arg="I, J" Name="IntersectionIdealsOfNumericalSemigroup"></Func>
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                    <Description>
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Given two ideals <A>I</A> and <A>J</A> of a numerical semigroup <A>S</A>
347
returns the ideal of the numerical semigroup <A>S</A> which is the
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intersection of the ideals <A>I</A> and <A>J</A>.
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                        <Example><![CDATA[
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gap> i:=IdealOfNumericalSemigroup([75,89],s);;
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gap> j:=IdealOfNumericalSemigroup([115,289],s);;
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gap> IntersectionIdealsOfNumericalSemigroup(i,j);
354
<Ideal of numerical semigroup>
355
]]></Example>
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                    </Description>
357
                </ManSection>
358

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<ManSection>
361
                    <Func Arg="S" Name="MaximalIdealOfNumericalSemigroup"></Func>
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                    <Description>
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                        Returns the maximal ideal of the numerical semigroup <A>S</A>.
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                        <Example><![CDATA[
366
gap> MaximalIdealOfNumericalSemigroup(NumericalSemigroup(3,7));
367
<Ideal of numerical semigroup>
368
]]></Example>
369
                    </Description>
370
                </ManSection>
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<ManSection>
374
                    <Func Arg="S" Name="CanonicalIdealOfNumericalSemigroup"></Func>
375
                    <Description>
376
		      <A>S</A> is a numerical semigroup.
377
                        Computes a canonical ideal of <A>S</A> (<Cite Key="BF06"></Cite>):
378
                        <M>\{ x \in \mathbb{Z} | g-x \not \in S\} </M>.
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                        <Example><![CDATA[
381
gap> s:=NumericalSemigroup(4,6,11);;
382
gap> m:=MaximalIdealOfNumericalSemigroup(s);;
383
gap> c:=CanonicalIdealOfNumericalSemigroup(s);
384
<Ideal of numerical semigroup>
385
gap> c-(c-m)=m;
386
true
387
gap> id:=3+s;
388
<Ideal of numerical semigroup>
389
gap> c-(c-id)=id;
390
true
391
]]></Example>
392
                    </Description>
393
                </ManSection>
394

395
<ManSection>
396
  <Prop Arg="E" Name="IsCanonicalIdeal"></Prop>
397
  <Prop Arg="E" Name="IsCanonicalIdealOfNumericalSemigroup"></Prop>
398
  <Description>
399
    <A>E</A> is an ideal of a numerical semigroup, say <M>S</M>. Determines if <A>E</A> is a translation of the canonical ideal of <M>S</M>, or equivalently, for every ideal <M>J</M>, <M>E-(E-J)=J</M>.
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401
<Example><![CDATA[
402
gap> s:=NumericalSemigroup(3,5,7);;
403
gap> c:=3+CanonicalIdealOfNumericalSemigroup(s);;
404
gap> c-(c-(3+s))=3+s;
405
true
406
gap> IsCanonicalIdealOfNumericalSemigroup(c);
407
true
408
]]></Example>
409
    </Description>
410
</ManSection>
411

412
<ManSection>
413
                    <Func Arg="S" Name="TypeSequenceOfNumericalSemigroup"></Func>
414
                    <Description>
415

416
                        <A>S</A> is a numerical semigroup.
417
                        <P/>
418

419
                         Computes the type sequence of a numerical semigroup. That is, the secuence <M>t_i(<A>S</A>)=\sharp(<A>S</A>(i)\setminus <A>S</A>(i-1))</M>, with <M><A>S</A>(i)=\{ s\in S\mid s\ge s_i\}</M> and <M>s_i</M> the <M>i</M>th element of <A>S</A>.
420
                         <P/>
421
                        This function is the implementation of the algorithm given in
422
                        <Cite Key="BDF97"></Cite>.
423

424

425
                        <Example><![CDATA[
426
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
427
gap> TypeSequenceOfNumericalSemigroup(s);
428
[ 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,
429
  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,
430
  1, 1, 1 ]
431
gap> s:=NumericalSemigroup(4,6,11);;
432
gap> TypeSequenceOfNumericalSemigroup(s);
433
[ 1, 1, 1, 1, 1, 1, 1 ]
434
]]></Example>
435
                    </Description>
436
</ManSection>
437

438
</Section>
439

440
<Section>
441
                <Heading>
442
                    Blow ups and closures
443
                </Heading>
444

445
                <ManSection>
446
                    <Func Arg="n, I" Name="HilbertFunctionOfIdealOfNumericalSemigroup"></Func>
447
                    <Description>
448
                        <A>I</A> is an ideal of a numerical semigroup, <A>n</A> is a non negative integer. <A>I</A> must be contained in its ambient semigroup.
449
                        The output is the cardinality of the set <M><A>n</A><A>I</A>\setminus (<A>n</A>+1)<A>I</A></M>.
450

451
                        <Example><![CDATA[
452
gap> I:=[6,9,11]+NumericalSemigroup(6,9,11);;
453
gap> List([1..7],n->HilbertFunctionOfIdealOfNumericalSemigroup(n,I));
454
[ 3, 5, 6, 6, 6, 6, 6 ]
455
]]></Example>
456
                    </Description>
457
                </ManSection>
458

459
                <ManSection>
460
                    <Func Arg="I" Name="BlowUpIdealOfNumericalSemigroup"></Func>
461
                    <Description>
462
                        <A>I</A> is an ideal of a numerical semigroup.
463
The output is the ideal <M>\bigcup_{n\geq 0} n<A>I</A>-n<A>I</A></M>.
464

465
                        <Example><![CDATA[
466
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
467
gap> BlowUpIdealOfNumericalSemigroup(I);;
468
gap> SmallElementsOfIdealOfNumericalSemigroup(last);
469
[ 0, 2, 4, 6, 8 ]
470
]]></Example>
471
                    </Description>
472
                </ManSection>
473

474

475

476
                <ManSection>
477
                    <Attr Arg="I" Name="ReductionNumber" Label="for ideals of numerical semigroups"></Attr>
478
                    <Attr Arg="I" Name="ReductionNumberIdealNumericalSemigroup"></Attr>
479
                    <Description>
480
                        <A>I</A> is an ideal of a numerical semigroup.
481
                        The output is the least integer
482
                        such that  <M>n <A>I</A> + i=(n+1)<A>I</A></M>, where <M>i=min(<A>I</A>)</M>.
483

484
                        <Example><![CDATA[
485
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
486
gap> ReductionNumberIdealNumericalSemigroup(I);
487
2
488
]]></Example>
489
                    </Description>
490
                </ManSection>
491

492

493

494

495

496
<ManSection>
497
                    <Func Arg="S" Name="BlowUpOfNumericalSemigroup"></Func>
498
                    <Description>
499
		      <A>S</A> is a numerical semigroup.
500
                        If <A>M</A> is the maximal ideal of the numerical semigroup, then
501
                        the output is the numerical semigroup
502
                        <M>\bigcup_{n\geq 0} n<A>M</A>-n<A>M</A></M>.
503

504

505
                        <Example><![CDATA[
506
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
507
gap> BlowUpOfNumericalSemigroup(s);
508
<Numerical semigroup with 10 generators>
509
gap> SmallElementsOfNumericalSemigroup(last);
510
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
511
  40, 41, 42, 44 ]
512
gap> m:=MaximalIdealOfNumericalSemigroup(s);
513
<Ideal of numerical semigroup>
514
gap> BlowUpIdealOfNumericalSemigroup(m);
515
<Ideal of numerical semigroup>
516
gap> SmallElementsOfIdealOfNumericalSemigroup(last);
517
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
518
  40, 41, 42, 44 ]
519
]]></Example>
520
                    </Description>
521
                </ManSection>
522
  <ManSection>
523
      <Func Name="LipmanSemigroup" Arg="S"></Func>
524
      <Description>
525
       This is just a synonym of <Ref Func="BlowUpOfNumericalSemigroup"/>.
526
      <Example><![CDATA[
527
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
528
gap> LipmanSemigroup(s);
529
<Numerical semigroup with 10 generators>
530
gap> SmallElementsOfNumericalSemigroup(last);
531
[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,
532
  40, 41, 42, 44 ]
533
]]></Example>
534

535
</Description>
536
</ManSection>
537

538

539
                <ManSection>
540
                    <Func Arg="I" Name="RatliffRushNumberOfIdealOfNumericalSemigroup"></Func>
541
                    <Description>
542
                        <A>I</A> is an ideal of a numerical semigroup.
543
                        The output is the least integer
544
                        such that  <M>(n+1)<A>I</A>-n<A>I</A></M> is the Ratliff-Rush closure of <A>I</A>.
545

546
                        <Example><![CDATA[
547
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
548
gap> RatliffRushNumberOfIdealOfNumericalSemigroup(I);
549
1
550
]]></Example>
551
                    </Description>
552
                </ManSection>
553

554

555
                <ManSection>
556
                    <Func Arg="I" Name="RatliffRushClosureOfIdealOfNumericalSemigroup"></Func>
557
                    <Description>
558
                        <A>I</A> is an ideal of a numerical semigroup.
559
                        The output is the Ratliff-Rush closure of <A>I</A>:  <M>\bigcup_{n\in \mathbb{N}}(n+1)<A>I</A>-n<A>I</A></M> (see <Cite Key="DA-G-H"></Cite>).
560

561
                        <Example><![CDATA[
562
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
563
gap> RatliffRushClosureOfIdealOfNumericalSemigroup(I);
564
<Ideal of numerical semigroup>
565
gap> MinimalGenerators(last);
566
[ 0, 2, 4 ]
567
]]></Example>
568
                    </Description>
569
                </ManSection>
570

571
                <ManSection>
572
                    <Func Arg="I" Name="AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup"></Func>
573
                    <Description>
574
                        <A>I</A> is an ideal of a numerical semigroup.
575
                        The output is the least <M>n</M> such that the Ratliff-Rush closure of <M>m<A>I</A></M> equals <M>m<A>I</A></M> for all <M>m\ge n</M>(see <Cite Key="DA-G-H"></Cite>).
576

577
                        <Example><![CDATA[
578
gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
579
gap> AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup(I);
580
2
581
]]></Example>
582
                    </Description>
583
                </ManSection>
584

585

586
<ManSection>
587
    <Func Name="MultiplicitySequenceOfNumericalSemigroup" Arg="S"/>
588
    <Description>
589
     <A>S</A> is a numerical semigroup. The output is a list with the multiplicities of the sequence <M>S\subseteq L(S)\subseteq \cdots \subseteq \mathbb{N}</M>, where <M>L(\cdot)</M> means <Ref Func="LipmanSemigroup"/>.
590
    <Example><![CDATA[
591
gap> s:=NumericalSemigroup(3,5);
592
<Numerical semigroup with 2 generators>
593
gap> MultiplicitySequenceOfNumericalSemigroup(s);
594
[ 3, 2, 1 ]
595
]]></Example>
596
</Description>
597
</ManSection>
598

599

600
<ManSection>
601
                    <Func Arg="S" Name="MicroInvariantsOfNumericalSemigroup"></Func>
602
                    <Description>
603
                        Returns the microinvariants of the numerical semigroup <A>S</A> defined
604
                        in <Cite Key="E01"></Cite>. For their computation we have used the formula
605
                        given in <Cite Key="BF06"></Cite>. The Apéry set of <A>S</A> and its
606
                        blow up are involved in this computation.
607

608
                        <Example><![CDATA[
609
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
610
gap> bu:=BlowUpOfNumericalSemigroup(s);;
611
gap> ap:=AperyListOfNumericalSemigroupWRTElement(s,30);;
612
gap> apbu:=AperyListOfNumericalSemigroupWRTElement(bu,30);;
613
gap> (ap-apbu)/30;
614
[ 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,
615
  5, 4, 3, 3, 2 ]
616
gap> MicroInvariantsOfNumericalSemigroup(s)=last;
617
true
618
]]></Example>
619
                    </Description>
620
                </ManSection>
621

622

623

624

625
                <Heading>
626
                    Apéry sets of ideals of numerical semigroups
627
                </Heading>
628

629

630
                <ManSection>
631
                    <Func Arg="I,n" Name="AperyListOfIdealOfNumericalSemigroupWRTElement"></Func>
632
                    <Description>
633
<A>I</A> is an ideal and <A>n</A> is an integer.
634
 					Computes the set of elements <M>x</M> of <A>I</A>  such that <M>x-</M><A>n</A> is not in the ideal <A>I</A>,
635
					where <A>n</A> is supposed to be in the ambient semigroup of <A>I</A>.
636
					The element in the <M>i</M>th position of the output list (starting in 0) is congruent with <M>i</M> modulo <A>n</A>.
637

638
                       <Example><![CDATA[
639
gap> s:=NumericalSemigroup(10,11,13);;
640
gap> i:=[12,14]+s;;
641
gap> AperyListOfIdealOfNumericalSemigroupWRTElement(i,10);
642
[ 40, 51, 12, 23, 14, 25, 36, 27, 38, 49 ]
643
]]></Example>
644
                     </Description>
645
                </ManSection>
646

647

648
                <ManSection>
649
                    <Func Arg="s" Name="AperyTableOfNumericalSemigroup"></Func>
650
                    <Description>
651

652
				 Computes the Apéry table associated to the numerical semigroup <A>s</A> as explained in <Cite Key="CJZ"></Cite>,
653
				 that is, a list containing the Apéry list of <A>s</A> with respect to its multiplicity and the Apéry lists of <M>kM</M>
654
				(with <M>M</M> the maximal ideal of <A>s</A>) with respect to the multiplicity of <A>s</A>, for <M>k\in\{1,\ldots,r\}</M>,
655
				where <M>r</M> is the reduction number of <M>M</M>  (see <Ref Func="ReductionNumberIdealNumericalSemigroup"/>).
656

657
                       <Example><![CDATA[
658
gap> s:=NumericalSemigroup(10,11,13);;
659
gap> AperyTableOfNumericalSemigroup(s);
660
[ [ 0, 11, 22, 13, 24, 35, 26, 37, 48, 39 ],
661
  [ 10, 11, 22, 13, 24, 35, 26, 37, 48, 39 ],
662
  [ 20, 21, 22, 23, 24, 35, 26, 37, 48, 39 ],
663
  [ 30, 31, 32, 33, 34, 35, 36, 37, 48, 39 ],
664
  [ 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 ] ]
665
]]></Example>
666
                     </Description>
667
                </ManSection>
668

669
                <ManSection>
670
                    <Func Arg="i, is" Name="StarClosureOfIdealOfNumericalSemigroup"></Func>
671
                    <Description>
672

673
						<A>i</A> is an ideal and <A>is</A> is a set of ideals (all from the same numerical semigroup<M>s</M>).
674
						The output is <M>i^{*_{is}}</M>, where <M>*_{is}</M> is the star operation generated by <A>is</A>: <M>(s-(s-i))\bigcap_{k\in is} (k-(k-i))</M>.
675
						The implementation uses Section 3 of <Cite Key="MR3354072"></Cite>.
676

677
                       <Example><![CDATA[
678
gap> s:=NumericalSemigroup(3,5,7);;
679
gap> StarClosureOfIdealOfNumericalSemigroup([0,2]+s,[[0,4]+s]);;
680
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
681
[ 0, 2, 4 ]
682
]]></Example>
683
                     </Description>
684
                </ManSection>
685

686

687
</Section>
688

689

690
<Section Label="sec:PatternsIdeals">
691
<Heading>
692
    Patterns for ideals
693
</Heading>
694

695
In this section we document the functions implemented by K. Stokes related to patterns of ideals in numerical semigroups. The correctness of the algorithms can be found in <Cite Key="Stokes"></Cite>.
696

697

698
<ManSection>
699
  <Func Arg="p" Name="IsAdmissiblePattern"></Func>
700
  <Description>
701

702
    <A>p</A> is the list of integers that are the coefficients of a pattern. <P/>
703

704
    Returns true or false depending if the pattern is admissible or not  (see <Cite Key="BA-GS"></Cite>).
705

706
    <Example><![CDATA[
707
gap> IsAdmissiblePattern([1,1,-1]);
708
true
709
gap> IsAdmissiblePattern([1,-2]);
710
false
711
]]></Example>
712
  </Description>
713
</ManSection>
714

715

716
<ManSection>
717
  <Func Arg="p" Name="IsStronglyAdmissiblePattern"></Func>
718
  <Description>
719

720
    <A>p</A> is the list of integers that are the coefficients of a pattern. <P/>
721

722
    Returns true or false depending if the pattern is strongly admissible or not  (see <Cite Key="BA-GS"></Cite>).
723

724
    <Example><![CDATA[
725
gap> IsAdmissiblePattern([1,-1]);
726
true
727
gap> IsStronglyAdmissiblePattern([1,-1]);
728
false
729
gap> IsStronglyAdmissiblePattern([1,1,-1]);
730
true
731
]]></Example>
732
  </Description>
733
</ManSection>
734

735

736
<ManSection>
737
  <Func Arg="I,T" Name="AsIdealOfNumericalSemigroup"></Func>
738
  <Description>
739

740
    <A>I</A> is an ideal of a numerical semigroup <M>S</M>, and <A>T</A> is a numerical semigroup. Detects if <A>I</A> is an ideal of <A>T</A> and contained in <A>T</A> (integral ideal), and if so, returns <A>I</A> as an ideal of <A>T</A>. It returns <C>fail</C> if <A>I</A> is an ideal of some semigroup but not an integral ideal of <A>t</A>.
741

742
    <Example><![CDATA[
743
gap> s:=NumericalSemigroup(3,7,5);;
744
gap> t:=NumericalSemigroup(10,11,14);;
745
gap> AsIdealOfNumericalSemigroup(10+s,t);
746
fail
747
gap> AsIdealOfNumericalSemigroup(100+s,t);
748
<Ideal of numerical semigroup>
749
]]></Example>
750
  </Description>
751
</ManSection>
752

753
<ManSection>
754
  <Func Arg="p,C" Name="BoundForConductorOfImageOfPattern"></Func>
755
  <Description>
756

757
     <A>p</A> is the list of integers that are the coefficients of an admissible pattern. <A>C</A> is a positive integer.
758

759
     Calculates an upper bound of the smallest element <M>K</M> in <M>p(I)</M> such that all integers larger than <M>K</M> are contained in <M>p(I)</M>, where <M>I</M> is an ideal of a
760
numerical semigroup. Instead of taking <M>I</M> as parameter, the function takes <M>C</M>, which is assumed to be the conductor of <M>I</M>.
761
    <Example><![CDATA[
762
gap> BoundForConductorOfImageOfPattern([1,1,-1],10);
763
10
764
]]></Example>
765
  </Description>
766
</ManSection>
767

768

769
<ManSection>
770
  <Func Arg="p,I" Name="ApplyPatternToIdeal"></Func>
771
  <Description>
772

773
    <A>p</A> is the list of integers that are the coefficients of a strongly admissible pattern. <A>I</A> is an ideal of a numerical semigroup.<P/>
774

775
    Calculates <M>p(I)</M>. Outputs <M>p(I)</M>, represented as <C>[d,p(I)/d]</C>, where <C>d</C > is the gcd of the coefficients of <A>p</A>. All elements of <M>p(I)</M> are divisible by <M>d</M>, and <M>p(I)/d</M> is an ideal of some numerical semigroup. It is returned as the maximal ideal of the numerical semigroup <M>p(I)/d \cup \{0\}</M>. The parent numerical semigroup can later be changed with the function <C>AsIdealOfNumericalSemigroup</C>.
776

777

778
     <Example><![CDATA[
779
gap> s:=NumericalSemigroup(3,7,5);;
780
gap> i:=10+s;;
781
gap> ApplyPatternToIdeal([1,1,-1],i);
782
[ 1, <Ideal of numerical semigroup> ]
783
]]></Example>
784
  </Description>
785
</ManSection>
786

787

788
<ManSection>
789
  <Func Arg="p,S" Name="ApplyPatternToNumericalSemigroup"></Func>
790
  <Description>
791

792
    <A>p</A> is the list of integers that are the coefficients of a strongly admissible pattern. <A>S</A> is  a numerical semigroup.<P/>
793

794
    Outputs <C>ApplyPatternToIdeal(p,0+S)</C>.
795

796
     <Example><![CDATA[
797
gap> s:=NumericalSemigroup(3,7,5);;
798
gap> ApplyPatternToNumericalSemigroup([1,1,-1],s);
799
[ 1, <Ideal of numerical semigroup> ]
800
gap> SmallElements(last[2]);
801
[ 0, 3, 5 ]
802
]]></Example>
803
  </Description>
804
</ManSection>
805

806
<ManSection>
807
  <Func Arg="p,I,J" Name="IsAdmittedPatternByIdeal"></Func>
808
  <Description>
809

810
    <A>p</A> is the list of integers that are the coefficients of a strongly admissible pattern. <A>I</A> and <A>J</A> are ideals of certain numerical semigroups.<P/>
811

812

813
    Tests whether or not <M>p(I)</M> is contained in <A>J</A>.
814

815
     <Example><![CDATA[
816
gap> s:=NumericalSemigroup(3,7,5);;
817
gap> i:=[3,5]+s;;
818
gap> IsAdmittedPatternByIdeal([1,1,-1],i,i);
819
false
820
gap> IsAdmittedPatternByIdeal([1,1,-1],i,0+s);
821
true
822
]]></Example>
823
  </Description>
824
</ManSection>
825

826
<ManSection>
827
  <Func Arg="p,S,T" Name="IsAdmittedPatternByNumericalSemigroup"></Func>
828
  <Description>
829

830
    <A>p</A> is the list of integers that are the coefficients of a strongly admissible pattern. <A>S</A> and <A>T</A> are numerical semigroups.<P/>
831

832

833
    Tests whether or not <M>p(S)</M> is contained in <A>T</A>.
834

835
     <Example><![CDATA[
836
gap> IsAdmittedPatternByNumericalSemigroup([1,1,-1],s,s);
837
true
838
gap> IsArfNumericalSemigroup(s);
839
true
840
]]></Example>
841
  </Description>
842
</ManSection>
843

844

845

846
</Section>
847

848
<Section>
849
  <Heading>Graded associated ring of numerical semigroup</Heading>
850

851
  This section contains several functions to test properties of the graded (with respect to the maximal ideal) of the semigroup ring <M>\mathbb{K}[\![S]\!]</M> (with <M>S</M> a numerical semigroup).
852

853
  <ManSection>
854
                      <Prop Arg="S" Name="IsGradedAssociatedRingNumericalSemigroupCM"></Prop>
855
                      <Description>
856
                          <A>S</A> is a numerical semigroup.
857
                          Returns true if the graded ring associated to <M>K[[<A>S</A>]]</M>
858
                          is Cohen-Macaulay, and false otherwise.
859
                          This test is the implementation of the algorithm given in
860
                          <Cite Key="BF06"></Cite>.
861

862
                          <Example><![CDATA[
863
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
864
gap> IsGradedAssociatedRingNumericalSemigroupCM(s);
865
false
866
gap> MicroInvariantsOfNumericalSemigroup(s);
867
[ 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,
868
  5, 4, 3, 3, 2 ]
869
gap> List(AperyListOfNumericalSemigroupWRTElement(s,30),
870
> w->MaximumDegreeOfElementWRTNumericalSemigroup (w,s));
871
[ 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,
872
  5, 4, 3, 1, 2 ]
873
gap> last=last2;
874
false
875
gap> s:=NumericalSemigroup(4,6,11);;
876
gap> IsGradedAssociatedRingNumericalSemigroupCM(s);
877
true
878
gap> MicroInvariantsOfNumericalSemigroup(s);
879
[ 0, 2, 1, 1 ]
880
gap> List(AperyListOfNumericalSemigroupWRTElement(s,4),
881
> w->MaximumDegreeOfElementWRTNumericalSemigroup(w,s));
882
[ 0, 2, 1, 1 ]
883
  ]]></Example>
884
                      </Description>
885
                  </ManSection>
886

887
<ManSection>
888
                    <Prop Arg="S" Name="IsGradedAssociatedRingNumericalSemigroupBuchsbaum"></Prop>
889
                    <Description>
890

891
                        <A>S</A> is a numerical semigroup.
892
                        <P/>
893

894
                        Returns true if the graded ring associated to <M>K[\![<A>S</A>]\!]</M>
895
                        is Buchsbaum, and false otherwise.
896
                        This test is the implementation of the algorithm given in <Cite Key="DA-M-M"></Cite>.
897

898

899
                        <Example><![CDATA[
900
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
901
gap> IsGradedAssociatedRingNumericalSemigroupBuchsbaum(s);
902
true
903
]]></Example>
904
                    </Description>
905
</ManSection>
906

907

908
<ManSection>
909
  <Func Arg="S" Name="TorsionOfAssociatedGradedRingNumericalSemigroup"></Func>
910
  <Description>
911

912
    <A>S</A> is a numerical semigroup.
913
    <P/>
914

915
    This function returns the set of elements in the numerical semigroup <A>S</A> corresponding to a <M>\mathbb{K}</M>-basis of the torsion submodule of the associated graded ring of the numerical semigroup ring <M>\mathbb{K}[\![S]\!]</M>. It uses the Apery table as explained in <Cite Key="CJZ"></Cite>.
916

917
<Example><![CDATA[
918
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
919
gap> TorsionOfAssociatedGradedRingNumericalSemigroup(s);
920
[ 181, 153, 157, 193, 169, 148 ]
921
]]></Example>
922
  </Description>
923
</ManSection>
924

925
<ManSection>
926
  <Func Arg="S" Name="BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup"></Func>
927
  <Description>
928

929
    <A>S</A> is a numerical semigroup.
930
    <P/>
931

932
    This function returns the smallest non-negative integer <M>k</M> for which the associated graded ring <M>G</M> of a given numerical semigroup ring is <M>k</M>-Buchsbaum, that is, the least <M>k</M> for which the torsion submodule of <M>G</M> is annihilated by the <M>k</M>-th power of the homogeneous maximal ideal of <M>G</M>.
933

934
<Example><![CDATA[
935
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
936
gap> BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup(s);
937
1
938
gap> IsGradedAssociatedRingNumericalSemigroupBuchsbaum(s);
939
true
940
]]></Example>
941
  </Description>
942
</ManSection>
943

944
<ManSection>
945
                    <Prop Arg="S" Name="IsMpure"></Prop>
946
                    <Prop Arg="S" Name="IsMpureNumericalSemigroup"></Prop>
947
                    <Description>
948

949
                        <A>S</A> is a numerical semigroup.
950
                        <P/>
951

952
                        Test for the M-Purity of the numerical semigroup S <A>S</A>.
953
			This test is based on
954
                        <Cite Key="Br"></Cite>.
955

956
                        <Example><![CDATA[
957
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
958
gap> IsMpureNumericalSemigroup(s);
959
false
960
gap> s:=NumericalSemigroup(4,6,11);;
961
gap> IsMpureNumericalSemigroup(s);
962
true
963
]]></Example>
964
                    </Description>
965
</ManSection>
966

967
<ManSection>
968
                    <Prop Arg="S" Name="IsPure"></Prop>
969
                    <Prop Arg="S" Name="IsPureNumericalSemigroup"></Prop>
970
                    <Description>
971

972
                        <A>S</A> is a numerical semigroup.
973
                        <P/>
974

975
                        Test for the purity of the numerical semigroup S <A>S</A>.
976
			This test is based on
977
                        <Cite Key="Br"></Cite>.
978

979
                        <Example><![CDATA[
980
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
981
gap> IsPureNumericalSemigroup(s);
982
false
983
gap> s:=NumericalSemigroup(4,6,11);;
984
gap> IsPureNumericalSemigroup(s);
985
true
986
]]></Example>
987
                    </Description>
988
</ManSection>
989

990
<ManSection>
991
                    <Func Arg="S" Name="IsGradedAssociatedRingNumericalSemigroupGorenstein"></Func>
992
                    <Description>
993

994
                        <A>S</A> is a numerical semigroup.
995
                        <P/>
996

997
                        Returns true if the graded ring associated to <M>K[[<A>S</A>]]</M>
998
                        is Gorenstein, and false otherwise.
999
                        This test is the implementation of the algorithm given in
1000
                        <Cite Key="DA-M-S"></Cite>.
1001

1002

1003
                        <Example><![CDATA[
1004
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
1005
gap> IsGradedAssociatedRingNumericalSemigroupGorenstein(s);
1006
false
1007
gap> s:=NumericalSemigroup(4,6,11);;
1008
gap> IsGradedAssociatedRingNumericalSemigroupGorenstein(s);
1009
true
1010
]]></Example>
1011
                    </Description>
1012
</ManSection>
1013

1014

1015
<!--the functions below first appeared in version 0.98-->
1016
<ManSection>
1017
                    <Func Arg="S" Name="IsGradedAssociatedRingNumericalSemigroupCI"></Func>
1018
                    <Description>
1019

1020
                        <A>S</A> is a numerical semigroup.
1021
                        <P/>
1022

1023
                        Returns true if the Complete Intersection property of the associated graded ring of a numerical semigroup ring associated to <M>K[[<A>S</A>]]</M>, and false otherwise.
1024
                        This test is the implementation of the algorithm given in
1025
                        <Cite Key="DAMSwhen"></Cite>.
1026

1027

1028
                        <Example><![CDATA[
1029
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
1030
gap> IsGradedAssociatedRingNumericalSemigroupCI(s);
1031
false
1032
gap> s:=NumericalSemigroup(4,6,11);;
1033
gap> IsGradedAssociatedRingNumericalSemigroupCI(s);
1034
true
1035
]]></Example>
1036
                    </Description>
1037
                </ManSection>
1038

1039

1040
</Section>
1041

1042
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