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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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                    Adding and removing elements of a numerical semigroup

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                </Heading>

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                In this section we show how to construct new numerical semigroups from a given

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                numerical semigroup. Two dual operations are presented. The first one removes

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                a minimal generator from a numerical semigroup. The second adds a special gap

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                to a semigroup (see <Cite Key="RGGJ03"></Cite>).

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                <ManSection>

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                    <Func Arg="n, S" Name="RemoveMinimalGeneratorFromNumericalSemigroup"></Func>

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                    <Description>

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                        <A>S</A> is a numerical semigroup and <A>n</A> is one if its minimal

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                        generators.

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                        <P/>

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                        The output is the numerical semigroup <M> <A>S</A> \setminus\{<A>n</A>\} </M> 

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                        (see <Cite Key="RGGJ03"></Cite>; <M>S\setminus\{n\}</M> is a numerical

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                        semigroup if and only if <M>n</M> is a minimal generator of <M>S</M>).

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                        <Example><![CDATA[

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gap> s:=NumericalSemigroup(3,5,7);

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<Numerical semigroup with 3 generators>

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gap> RemoveMinimalGeneratorFromNumericalSemigroup(7,s);

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<Numerical semigroup with 3 generators>

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gap> MinimalGeneratingSystemOfNumericalSemigroup(last);

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[ 3, 5 ]

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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="g, S" Name="AddSpecialGapOfNumericalSemigroup"></Func>

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                    <Description>

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                        <A>S</A> is a numerical semigroup and <A>g</A> is a special gap of <A>S</A>

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                        <P/>

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                        The output is the numerical semigroup <M> <A>S</A> \cup\{<A>g</A>\} </M> 

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                        (see <Cite Key="RGGJ03"></Cite>, where it is explained why this set is a 

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                        numerical semigroup).            

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                        <Example><![CDATA[

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gap> s:=NumericalSemigroup(3,5,7);;

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gap> s2:=RemoveMinimalGeneratorFromNumericalSemigroup(5,s);

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<Numerical semigroup with 3 generators>

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gap> s3:=AddSpecialGapOfNumericalSemigroup(5,s2);

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<Numerical semigroup>

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gap> SmallElementsOfNumericalSemigroup(s) =

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> SmallElementsOfNumericalSemigroup(s3);

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true                

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gap> s=s3;

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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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