Quelle Operations_Numerical_Semigroups.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>
<Section>
  <Heading>Intersections, sums, quotients, dilatations, numerical duplications and multiples by integers</Heading>

We provide functions to build numerical semigroups from others by means of intersections, quotients, multiples and related constructions.

  <ManSection>
    <Oper Name="Intersection" Arg="S, T" Label="for numerical semigroups"/>
    <Func Name="IntersectionOfNumericalSemigroups" Arg="S, T"/>
    <Description>
      <A>S</A> and <A>T</A>
      are numerical semigroups. Computes the intersection of <A>S</A> and <A>T</A>
      (which is a numerical semigroup).
      <Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> T := NumericalSemigroup(2,17);
<Numerical semigroup with 2 generators>
gap> SmallElements(S);
[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> SmallElements(T);
[ 0, 2, 4, 6, 8, 10, 12, 14, 16 ]
gap> Intersection(S,T);
<Numerical semigroup>
gap> SmallElements(last);
[ 0, 12, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> IntersectionOfNumericalSemigroups(S,T) = Intersection(S,T);
true
]]></Example>
    </Description>
  </ManSection>

  <ManSection>
    <Oper Name="\+" Arg="S, T" Label="for numerical semigroups"/>
    <Description>
      <A>S</A> and <A>T</A>
      are numerical semigroups. Computes the sum of <A>S</A> and <A>T</A>
      (which is a numerical semigroup).
      <Example><![CDATA[
gap> s:=NumericalSemigroup(4,9);;
gap> t:=NumericalSemigroup(6,7);;
gap> MinimalGenerators(s+t);
[ 4, 6, 7, 9 ]
]]></Example>
    </Description>
  </ManSection>

  <ManSection>
    <Func Name="QuotientOfNumericalSemigroup" Arg="S, n"></Func>
    <Oper Name="\/" Arg="S, n" Label="quotient of numerical semigroup"/>
    <Description>
      <A>S</A> is a numerical semigroup and <A>n</A> is an integer.
      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.
      <C>S / n</C> may be used as a short for <C>QuotientOfNumericalSemigroup(S, n)</C>.
      <Example><![CDATA[
gap> s:=NumericalSemigroup(3,29);
<Numerical semigroup with 2 generators>
gap> SmallElements(s);
[ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 30, 32, 33, 35, 36, 38,
39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56 ]
gap> t:=QuotientOfNumericalSemigroup(s,7);
<Numerical semigroup>
gap> SmallElements(t);
[ 0, 3, 5, 6, 8 ]
gap> u := s / 7;
<Numerical semigroup>
gap> SmallElements(u);
[ 0, 3, 5, 6, 8 ]
]]></Example>
    </Description>
  </ManSection>

    <ManSection>
    <Func Name="MultipleOfNumericalSemigroup" Arg="S, a, b"/>
    <Description>
      <A>S</A> is a numerical semigroup, and <A>a</A> and <A>b</A> are positive integers.
      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.
      <Example><![CDATA[
gap> N:=NumericalSemigroup(1);;
gap> s:=MultipleOfNumericalSemigroup(N,4,20);;
gap> SmallElements(s);
[ 0, 4, 8, 12, 16, 20 ]
]]></Example>
    </Description>
</ManSection>

<ManSection>
  <Func Name="NumericalDuplication" Arg="S, E, b"/>
  <Description>
    <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
    (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>).

    Computes  <M>2S\cup (2E+b)</M>.
    <Example><![CDATA[
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> e:=6+s;
<Ideal of numerical semigroup>
gap> ndup:=NumericalDuplication(s,e,3);
<Numerical semigroup with 4 generators>
gap> SmallElements(ndup);
[ 0, 6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 ]
]]></Example>
  </Description>
</ManSection>

<ManSection>
  <Func Name="AsNumericalDuplication" Arg="T"/>
  <Description>
    <A>T</A> is a numerical semigroup. Detects whether or not <A>T</A> can be expressed as <C>NumericalDuplication</C>(S,E,b), with E a proper ideal of S. Returns the list [S,E,b] if this is possible, and <C>fail</C> otherwise.
    <P/>
    Notice that a numerical semigroup can be represented in different ways as a numerical duplication.

    <Example><![CDATA[
gap> s:=NumericalSemigroup(3,5,7);;
gap> ndup:=NumericalDuplication(s,6+s,11);;
gap> asdup:=AsNumericalDuplication(ndup);
[ <Numerical semigroup with 3 generators>, <Ideal of numerical semigroup>, 3 ]
gap> ndup = CallFuncList(NumericalDuplication,asdup);
true
]]></Example>
  </Description>
</ManSection>

  <ManSection>
    <Func Name="InductiveNumericalSemigroup" Arg="a, b"/>
    <Description>
      <A>a</A> and <A>b</A> are lists of positive integers, with <M>k</M> the length of <A>a</A> and <A>b</A>, and such that <M>b[i+1]\ge a[i]b[i]</M> (<M>0\le i\le k-1</M>).
      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>.
      <Example><![CDATA[
gap> s:=InductiveNumericalSemigroup([4,2],[5,23]);;
gap> SmallElements(s);
[ 0, 8, 16, 24, 32, 40, 42, 44, 46 ]
]]></Example>
    </Description>
  </ManSection>

    <ManSection>
    <Func Name="DilatationOfNumericalSemigroup" Arg="S, a"/>
    <Description>
      <A>S</A> is a numerical semigroup, and <A>a</A> is a positive integer. If <M>M</M> is the maximal ideal of <A>S</A>, then <A>a</A> must be in <M>M-2M</M>.
      Computes the numerical semigroup <M>\{0\} \cup \{a+s \mid s\in M\}</M>, see <Cite Key="dilatation"/>.
      <Example><![CDATA[
gap> s:=NumericalSemigroup(3,4,5);;
gap> m:=MaximalIdeal(s);;
gap> SmallElements(m-2*m);
[ -3 ]
gap> d:=DilatationOfNumericalSemigroup(s,3);
<Numerical semigroup>
gap> SmallElements(d);
[ 0, 6 ]
]]></Example>
    </Description>
</ManSection>

</Section>

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