Quelle  Almost_symmetric.xml   Sprache: XML

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


    A numerical semigroup is almost-symmetric (<Cite Key="BF97"></Cite>) if its genus is the arithmetic mean of its Frobenius number and type.
    We use a procedure presented in <Cite Key="MR3169635"></Cite> to determine the set of all almost-symmetric numerical semigroups with given Frobenius
    number. In order to do this, we first calculate the set of all almost-symmetric numerical semigroups that can be constructed from an irreducible
    numerical semigroup.
<P/>
                <ManSection>
                    <Func Arg="s" Name="AlmostSymmetricNumericalSemigroupsFromIrreducible"></Func>
                    <Description>
                        <A>s</A> is an irreducible numerical semigroup.  The output is the set of almost-symmetric numerical semigroups that can be constructed
      from <A>s</A> by removing some of its generators (as explained in <Cite Key="MR3169635"></Cite>).
                        <Example><![CDATA[
gap> ns := NumericalSemigroup(5,8,9,11);;
gap> AlmostSymmetricNumericalSemigroupsFromIrreducible(ns);
[ <Numerical semigroup with 4 generators>,
  <Numerical semigroup with 5 generators>,
  <Numerical semigroup with 5 generators> ]
gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
[ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                    <Func Arg="s, t" Name="AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType"></Func>
                    <Description>
                        <A>s</A> is an irreducible numerical semigroup and <A>t</A> is a positive integer.  The output is the set of almost-symmetric numerical semigroups with type <A>t</A> that can be constructed
      from <A>s</A> by removing some of its generators (as explained in <Cite Key="BOR18"></Cite>).
                        <Example><![CDATA[
gap> ns := NumericalSemigroup(5,8,9,11);;
gap> AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType(ns,4);
[ <Numerical semigroup with 5 generators>, 
  <Numerical semigroup with 5 generators> ]
gap> List(last,MinimalGenerators);
[ [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                  <Func Arg="s" Name="IsAlmostSymmetric"></Func>
                    <Func Arg="s" Name="IsAlmostSymmetricNumericalSemigroup"></Func>
                    <Description>
                        <A>s</A> is a numerical semigroup.  The output is <C>true</C> if the numerical semigroup is almost symmetric.
                        <Example><![CDATA[
gap> IsAlmostSymmetric(NumericalSemigroup(5,8,11,14,17));
true
gap> IsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17));
true
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Func Arg="f, [ts]" Name="AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber"></Func>
                    <Description>
                        <A>f</A> is an integer, and so is <A>ts</A>. The output is the set of all almost symmetric
                        numerical semigroups with Frobenius number <A>f</A>, and type greater than or equal to <A>ts</A>. If <A>ts</A> is not specified, then it is considered to be equal to one, and so the output is the set of all almost symmetric numerical semigroups with Frobenius number <A>f</A>.
                        <Example><![CDATA[
gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12));
15
gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(12));
2
gap> List(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12,4),Type);
[ 12, 10, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4 ]
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
                    <Func Arg="f, t" Name="AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType"></Func>
                    <Description>
                        <A>f</A> is an integer and so is <A>t</A>. The output is the set of all almost symmetric
                        numerical semigroups with Frobenius number <A>f</A> and type <A>t</A>.
                        <Example><![CDATA[
gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType(12,4)); 
5
]]></Example>
                    </Description>
                </ManSection>


</Section>