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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>Intersections, and quotients and multiples by integers</Heading>

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  <ManSection>

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    <Oper Name="Intersection" Arg="S, T" Label="for numerical semigroups"/>

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    <Func Name="IntersectionOfNumericalSemigroups" Arg="S, T"/>

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    <Description>

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      <A>S</A> and <A>T</A>

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      are numerical semigroups. Computes the intersection of <A>S</A> and <A>T</A>

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      (which is a numerical semigroup).

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      <Example><![CDATA[

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gap> S := NumericalSemigroup("modular", 5,53);

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<Modular numerical semigroup satisfying 5x mod 53 <= x >

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gap> T := NumericalSemigroup(2,17);

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<Numerical semigroup with 2 generators>

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gap> SmallElements(S);

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[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]

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gap> SmallElements(T);

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[ 0, 2, 4, 6, 8, 10, 12, 14, 16 ]

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gap> IntersectionOfNumericalSemigroups(S,T);

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<Numerical semigroup>

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gap> SmallElements(last);

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[ 0, 12, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]

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

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    <Oper Name="\/" Arg="S, n" Label="quotient of numerical semigroup"/>

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    <Description>

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      <A>S</A> is a numerical semigroup and <A>n</A> is an integer.

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      Computes the quotient of <A>S</A> by <A>n</A>, that is, the set <M>\{ x\in {\mathbb N}\ |\  nx \in S\}</M>, which is again a numerical semigroup.

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      <C>S / n</C> may be used as a short for <C>QuotientOfNumericalSemigroup(S, n)</C>.

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      <Example><![CDATA[

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gap> s:=NumericalSemigroup(3,29);

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<Numerical semigroup with 2 generators>

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gap> SmallElements(s);

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[ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 30, 32, 33, 35, 36, 38,

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39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56 ]

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gap> t:=QuotientOfNumericalSemigroup(s,7);

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<Numerical semigroup>

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gap> SmallElements(t);

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[ 0, 3, 5, 6, 8 ]

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gap> u := s / 7;

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<Numerical semigroup>

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gap> SmallElements(u);

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[ 0, 3, 5, 6, 8 ]

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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 Name="MultipleOfNumericalSemigroup" Arg="S, a, b"/>

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    <Description>

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      <A>S</A> is a numerical semigroup, and <A>a</A> and <A>b</A> are positive integers.

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      Computes  <M>a S\cup \{b,b+1,\to\}</M>. If <A>b</A> is smaller than <M>a c</M>, with <M>c</M> the conductor of <M>S</M>, then a warning is displayed.

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      <Example><![CDATA[

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gap> N:=NumericalSemigroup(1);;

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gap> s:=MultipleOfNumericalSemigroup(N,4,20);;

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gap> SmallElements(s);

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[ 0, 4, 8, 12, 16, 20 ]

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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="S, T" Label="for numerical semigroups"/>

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                    <Func Arg="S, T" Name="DifferenceOfNumericalSemigroups"></Func>

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                    <Description>

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                        <A>S, T</A> are numerical semigroups.

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                        The output is the set <M><A>S</A>\setminus <A>T</A></M>.

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                        <Example><![CDATA[

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gap> ns1 := NumericalSemigroup(5,7);;

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gap> ns2 := NumericalSemigroup(7,11,12);;

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gap> Difference(ns1,ns2);

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[ 5, 10, 15, 17, 20, 27 ]

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gap> Difference(ns2,ns1);

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[ 11, 18, 23 ]

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gap> DifferenceOfNumericalSemigroups(ns2,ns1);

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[ 11, 18, 23 ]

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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 Name="NumericalDuplication" Arg="S, E, b"/>

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  <Description>

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    <A>S</A> is a numerical semigroup, and <A>E</A> and ideal of <A>S</A>, and <A>b</A> is a positive odd integer, so that <M>2S\cup (2E+b)</M> is a numerical semigroup

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    (this extends slightly the original definition where <A>b</A> was imposed to be in <A>S</A>, <Cite Key="duplication"></Cite>; now the condition imposed is <M>E+E+b\subseteq S</M>).

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    Computes  <M>2S\cup (2E+b)</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> e:=6+s;

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<Ideal of numerical semigroup>

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gap> ndup:=NumericalDuplication(s,e,3);

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<Numerical semigroup with 4 generators>

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gap> SmallElements(ndup);

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[ 0, 6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 ]

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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 Name="InductiveNumericalSemigroup" Arg="S, a, b"/>

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    <Description>

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      <A>S</A> is a numerical semigroup, and <A>a</A> and <A>b</A> are lists of positive integers, such that <M>b[i+1]\ge a[i]b[i]</M>.

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      Computes inductively <M>S_0=\mathbb N</M> and <M>S_{i+1}=a[i]S_i\cup \{a[i]b[i],a[i]b[i]+1,\to\}</M>, and returns <M>S_{k}</M>, with <M>k</M> the length of <A>a</A> and <A>b</A>.

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      <Example><![CDATA[

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gap> s:=InductiveNumericalSemigroup([4,2],[5,23]);;

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gap> SmallElements(s);

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[ 0, 8, 16, 24, 32, 40, 42, 44, 46 ]

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]]></Example>

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    </Description>

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  </ManSection>

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</Section>

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