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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                    Irreducible numerical semigroups
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                </Heading>
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                An irreducible numerical semigroup is a semigroup that cannot be expressed as the
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                intersection of numerical semigroups properly containing it.
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                <P/>
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                It is not difficult to prove that a semigroup is irreducible if and only if it is
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                maximal (with respect to set inclusion) in the set of all numerical semigroups having
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                its same Frobenius number (see <Cite Key="RB03"></Cite>). Hence, according to
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                <Cite Key="FGH87"></Cite> (respectively <Cite Key="BDF97"></Cite>),
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                symmetric (respectively pseudo-symmetric) numerical semigroups are those irreducible
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                numerical semigroups with odd (respectively even) Frobenius number.
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                <P/>
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                In <Cite Key="RGGJ03"></Cite> it is shown that a nontrivial numerical semigroup is irreducible if and
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                only if it has only one special gap. We use this characterization.
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                <P/>
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                In this section we show how to construct the set of all numerical semigroups with a
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                given Frobenius number. In old versions of the package, we first constructed an irreducible numerical semigroup with
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                the given Frobenius number (as explained in <Cite Key="RGS04"></Cite>), and then
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                we constructed the rest from it. That is why we separated both functions.
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				The present version uses a faster procedure presented in <Cite Key="BR13"></Cite>.
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                <P/>
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                Every numerical semigroup can be expressed as an intersection of irreducible
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                numerical semigroups. If <M>S</M> can be expressed as <M>S=S_1\cap \cdots\cap S_n</M>,
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                with <M>S_i</M> irreducible numerical semigroups, and no factor can be removed, then
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                we say that this decomposition is minimal. Minimal decompositions can be computed by
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                using Algorithm 26 in <Cite Key="RGGJ03"></Cite>.
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                <P/>
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                <ManSection>
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                    <Prop Arg="s" Name="IsIrreducibleNumericalSemigroup"></Prop>
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                    <Prop Arg="s" Name="IsIrreducible" Label="for numerical semigroups"></Prop>
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                    <Description>
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                        <A>s</A> is a numerical semigroup. The output is true if <A>s</A> is irreducible,
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                        false otherwise.
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                        <Example><![CDATA[
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gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,9));
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true
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gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,7,9));
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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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                    <!--<Prop Arg="s" Name="IsSymmetric"></Prop>-->
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                    <Attr Arg="s" Name="IsSymmetricNumericalSemigroup"></Attr>
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                    <Attr Arg="s" Name="IsSymmetric" Label="for numerical semigroups"></Attr>
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                    <Description>
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                        <A>s</A> is a numerical semigroup. The output is true if <A>s</A> is symmetric,
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                        false otherwise.
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                        <Example><![CDATA[
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gap> IsSymmetric(NumericalSemigroup(10,23));
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true
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gap> IsSymmetricNumericalSemigroup(NumericalSemigroup(10,11,23));
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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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                  <Prop Arg="s" Name="IsPseudoSymmetric" Label="for numerical semigroups"></Prop>
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                    <Prop Arg="s" Name="IsPseudoSymmetricNumericalSemigroup"></Prop>
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                    <Description>
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                        <A>s</A> is a numerical semigroup. The output is true if <A>s</A> is pseudo-symmetric,
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                        false otherwise.
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                        <Example><![CDATA[
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gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(6,7,8,9,11));
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true
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gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(4,6,9));
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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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                    <Func Arg="f" Name="AnIrreducibleNumericalSemigroupWithFrobeniusNumber"></Func>
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                    <Description>
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                        <A>f</A> is an integer greater than or equal to -1. The output is an irreducible
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                        numerical semigroup with Frobenius number <A> f</A>. From the way the procedure is
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                        implemented, the resulting semigroup has at most four generators (see <Cite Key="RGS04"></Cite>).
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                        <Example><![CDATA[
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gap> s := AnIrreducibleNumericalSemigroupWithFrobeniusNumber(28);
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<Numerical semigroup with 3 generators>
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gap> MinimalGenerators(s);
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[ 3, 17, 31 ]
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gap> FrobeniusNumber(s);
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28
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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="f" Name="IrreducibleNumericalSemigroupsWithFrobeniusNumber"></Func>
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                    <Description>
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                        <A>f</A> is an integer greater than or equal to -1. The output is the set of all
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                        irreducible numerical semigroups with Frobenius number <A>f</A>.
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                        <Example><![CDATA[
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gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(19));
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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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                    <Func Arg="s" Name="DecomposeIntoIrreducibles" Label="for numerical semigroup"></Func>
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                    <Description>
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                        <A>s</A> is a numerical semigroup. The output is a set of irreducible
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                        numerical semigroups containing it. These elements appear in a minimal
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                        decomposition of <A>s</A> as intersection into irreducibles.
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                        <Example><![CDATA[
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gap> DecomposeIntoIrreducibles(NumericalSemigroup(5,6,8));
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[ <Numerical semigroup with 3 generators>,
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  <Numerical semigroup with 4 generators> ]
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]]></Example>
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                    </Description>
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                </ManSection>
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            </Section>
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