In this section we present formulas to compute invariants and notable elements of a numerical semigroup. Some tests depending on these invariants are provided here, like being an acute or an ordinary numerical semigroup. We also present procedures to construct iterators from a numerical semigroup, or to retrieve several elements from a numerical semigroup as if it were a list (with infinitely many elements).
<ManSection>
<Attr Name="Multiplicity" Arg="NS" Label="for numerical semigroup"/>
<Attr Name="MultiplicityOfNumericalSemigroup" Arg="NS"/>
<Description>
<A>NS</A> is a numerical semigroup. Returns the multiplicity of <A>NS</A>, which is the smallest positive integer belonging to <A>NS</A>. Depending on the information known about <A>NS</A>, different methods are implemented. There are methods for the following cases: generators are known, Apéry set is known, it is a modular numerical semigroup, or it is proportionally modular (and thus is defined by a closed interval <Cite Key="RosalesVasco2008MIA"/>).
<Example><![CDATA[
gap> NumericalSemigroup(3,5);
<Numerical semigroup with 2 generators>
gap> Multiplicity(last);
3
gap> S := NumericalSemigroup("modular", 7,53);
<Modular numerical semigroup satisfying 7x mod 53 <= x >
gap> MultiplicityOfNumericalSemigroup(S);
8
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Generators" Arg="S" Label="for numerical semigroup"/>
<Attr Name="GeneratorsOfNumericalSemigroup" Arg="S"/>
<Attr Name="MinimalGenerators" Arg="S" Label="for numerical semigroup"/>
<Attr Name="MinimalGeneratingSystemOfNumericalSemigroup" Arg="S"/>
<Attr Name="MinimalGeneratingSystem" Arg="S" Label="for numerical semigroup"/>
<Description>
<A>S</A>
is a numerical semigroup.
<C>GeneratorsOfNumericalSemigroup</C>
returns a set of generators of <C>S</C>, which may
not be minimal. The shorter name <C>Generators</C> may be used.
<!-- <C>GeneratorsOfNumericalSemigroupNC</C> returns the set of generators recorded in <C>S!.generators</C>, which may not be minimal <C>ReducedSetOfGeneratorsOfNumericalSemigroupNC</C> Returns a set with possibly fewer generators than those recorded in <C>S!.generators</C>. It changes <C>S!.generators</C> to the set returned.
The function has 1 to 3 arguments. One of them a numerical semigroup. Then an argument is a boolean (<E>true</E> means that all the elements not belonging to the Apery set with respect to the multiplicity are removed; the default is "false") and another argument is a positive integer <M>n</M> (meaning that generators that can be written as the sum of <M>n</M> or less generators are removed; the default is "2"). The boolean or the integer may not be present. If a minimal generating set for <M>S</M> is known or no generating set is known, then the minimal generating system is returned.-->
<C>MinimalGeneratingSystemOfNumericalSemigroup</C>
returns the minimal set of generators of <C>S</C>.
The shorter names <C>MinimalGenerators</C> or <C>MinimalGeneratingSystem</C> may be used.
<P/> <!--
From Version 0.980, <C>ReducedSetOfGeneratorsOfNumericalSemigroup</C> is a synonym of <C>MinimalGeneratingSystemOfNumericalSemigroup</C>; <C>GeneratorsOfNumericalSemigroupNC</C> is a synonym of <C>GeneratorsOfNumericalSemigroup</C>. The names are kept for compatibility with code produced for previous versions, but will be removed in the future. -->
<ManSection>
<Attr Name="EmbeddingDimension" Arg="NS" Label="for numerical semigroup"/>
<Attr Name="EmbeddingDimensionOfNumericalSemigroup" Arg="NS"/>
<Description>
<C>NS</C>
is a numerical semigroup. It returns the cardinality of its minimal generating system.
<Example><![CDATA[
gap> s := NumericalSemigroup(3,5,7,15);
<Numerical semigroup with 4 generators>
gap> EmbeddingDimension(s);
3
gap> EmbeddingDimensionOfNumericalSemigroup(s);
3
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="SmallElements" Arg="NS" Label="for numerical semigroup"/>
<Attr Name="SmallElementsOfNumericalSemigroup" Arg="NS"/>
<Description>
<C>NS</C>
is a numerical semigroup. It returns the list of small elements of <C>NS</C>. Of course, the time consumed to return a result may depend on the way the semigroup is given.
<Example><![CDATA[
gap> SmallElements(NumericalSemigroup(3,5,7));
[ 0, 3, 5 ]
gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 0, 3, 5 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Length" Arg="NS" Label="for numerical semigroup"/>
<Description>
<C>NS</C>
is a numerical semigroup. It returns the number of small elements of <C>NS</C> below the conductor. This corresponds with the length of the semigroup ring modulo the conductor ideal. See also <Ref Func="LengthOfGoodSemigroup"/>.
<Example><![CDATA[
gap> Length(NumericalSemigroup(3,5,7));
2
]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="FirstElementsOfNumericalSemigroup" Arg="n, NS"></Func>
<Description>
<C>NS</C> is a numerical semigroup. It returns the list with the
first <A>n</A> elements of <C>NS</C>.
<Example><![CDATA[
gap> FirstElementsOfNumericalSemigroup(2,NumericalSemigroup(3,5,7));
[ 0, 3 ]
gap> FirstElementsOfNumericalSemigroup(10,NumericalSemigroup(3,5,7));
[ 0, 3, 5, 6, 7, 8, 9, 10, 11, 12 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="ElementsUpTo" Arg="NS, b"></Func>
<Description>
<A>NS</A> is a numerical semigroup, <A>b</A> a positve integer. It returns the set of
elements of <A>NS</A> up to <A>b</A>.
<Example><![CDATA[
gap> ns := NumericalSemigroup(5,7);;
gap> SmallElements(ns);
[ 0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24 ]
gap> ElementsUpTo(ns,18);
[ 0, 5, 7, 10, 12, 14, 15, 17 ]
gap> ElementsUpTo(ns,27);
[ 0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 26, 27 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Arg="S,r" Name="\[ \]" Label="for numerical semigroups"/>
<Description>
<A>S</A> is a numerical semigroup and <A>r</A> is an integer. It returns the <A>r</A>-th element of <A>S</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> S[53];
68
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Arg="S,ls" Name="\{ \}" Label="for numerical semigroups"/>
<Description>
<A>S</A> is a numerical semigroup and <A>ls</A> is a list of integers. It returns the list <A>[S[r] : r in ls]</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> S{[1..5]};
[ 0, 7, 8, 14, 15 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Arg="S,r" Name="NextElementOfNumericalSemigroup"/>
<Description>
<A>S</A> is a numerical semigroup and <A>r</A> is an integer. It returns the returns the
least integer greater than <A>r</A> belonging to <A>S</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> NextElementOfNumericalSemigroup(S,9);
14
gap> NextElementOfNumericalSemigroup(16,S);
17
gap> NextElementOfNumericalSemigroup(S,FrobeniusNumber(S))=Conductor(S);
true
]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Arg="S,r" Name="ElementNumber_NumericalSemigroup"/>
<Oper Arg="S,r" Name="RthElementOfNumericalSemigroup"/>
<Description>
<A>S</A> is a numerical semigroup and <A>r</A> is an integer. Both functions (which are like synonyms) return the <A>r</A>-th element of <A>S</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> ElementNumber_NumericalSemigroup(S,53);
68
gap> RthElementOfNumericalSemigroup(S,53);
68
]]></Example>
</Description>
</ManSection> <!-- <ManSection> <Oper Arg="S,r" Name="RthElementOfNumericalSemigroup"/> <Description> This operation works as a synonym of <Ref Func="ElementNumber_NumericalSemigroup"/>. <Example><![CDATA[ gap> S := NumericalSemigroup(7,8,17);; gap> RthElementOfNumericalSemigroup(S,53); 68 ]]></Example> </Description> </ManSection>
-->
<ManSection>
<Func Arg="S,r" Name="NumberElement_NumericalSemigroup"/>
<Description>
<A>S</A> is a numerical semigroup and <A>r</A> is an integer. It returns the position of <A>r</A> in <A>S</A> (and <C>fail</C> if the integer is not in the semigroup).
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> NumberElement_NumericalSemigroup(S,68);
53
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Arg="S" Name="Iterator" Label="for numerical semigroups"/>
<Description>
<A>S</A> is a numerical semigroup. It returns an iterator over <A>S</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup(7,8,17);;
gap> iter:=Iterator(S);
<iterator>
gap> NextIterator(iter);
0
gap> NextIterator(iter);
7
gap> NextIterator(iter);
8
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="Difference" Arg="S, T" Label="for numerical semigroups"/>
<Func Arg="S, T" Name="DifferenceOfNumericalSemigroups"></Func>
<Description>
<A>S, T</A> are numerical semigroups.
The output is the set <M><A>S</A>\setminus <A>T</A></M>.
<ManSection>
<Attr Name="AperyList" Arg="S, n" Label="for numerical semigroup with respect to element"/>
<Oper Name="AperyListOfNumericalSemigroupWRTElement" Arg="S, n"/>
<Description>
<A>S</A> is a numerical semigroup and <A>n</A> is a positive element of
<A>S</A>. Computes the Apéry list of <A>S</A> with respect to <A>n</A>. It contains for
every <M>i\in \{0,\ldots,<A>n</A>-1\}</M>, in the <M>i+1</M>th position,
the smallest element in the semigroup congruent with <M>i</M> modulo <A>n</A>.
<ManSection>
<Attr Name="AperyList" Arg="S" Label="for numerical semigroup with respect to multiplicity"/>
<Attr Name="AperyListOfNumericalSemigroup" Arg="S"/>
<Description>
<A>S</A> is a numerical semigroup. It computes the Apéry list of <A>S</A> with respect to the multiplicity of <A>S</A>.
<Example><![CDATA[
gap> AperyList(NumericalSemigroup(5,7,11));
[ 0, 11, 7, 18, 14 ]
gap> S := NumericalSemigroup("modular", 5,53);;
gap> AperyListOfNumericalSemigroup(S);
[ 0, 12, 13, 25, 26, 38, 39, 51, 52, 53, 32 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="AperyList" Arg="S, m" Label="for numerical semigroup with respect to integer"/>
<Func Name="AperyListOfNumericalSemigroupWRTInteger" Arg="S, m"/>
<Description>
<A>S</A> is a numerical semigroup and <A>m</A> is an integer.
Computes the Apéry list of <A>S</A> with respect to <A>m</A>, that is, the
set of elements <M>x</M> in <A>S</A> such that <M>x-</M><A>m</A> is not in <A>S</A>.
If <A>m</A> is an element in <A>S</A>, then the output of <C>AperyListOfNumericalSemigroupWRTInteger</C>, as sets, is the same as <C>AperyListOfNumericalSemigroupWRTElement</C>,
though without side effects,
in the sense that this information is no longer used by the package. The output of <C>AperyList</C> is the same as <C>AperyListOfNumericalSemigroupWRTInteger</C>.
<ManSection>
<Func Name="AperyListOfNumericalSemigroupAsGraph" Arg="ap"/>
<Description>
<A>ap</A> is the Apéry list of a numerical semigroup.
This function returns the adjacency list of the graph <M>(ap, E)</M> where the edge <M>u -> v</M> is
in <M>E</M> iff <M>v - u</M> is in <M>ap</M>.
The 0 is ignored.
<ManSection>
<Oper Name="KunzCoordinates" Arg="S[,m]" Label="for a numerical semigroup and (optionally) an integer"/>
<Func Name="KunzCoordinatesOfNumericalSemigroup" Arg="S[,m]"/>
<Description>
<A>S</A> is a numerical semigroup, and <A>m</A> is a nonzero element of <A>S</A>. The second argument is optional, and if missing it is assumed to be the multiplicity of <A>S</A>.
<P/>
Then the Apéry set of <A>m</A> in <A>S</A> has the form <M>[0,k_1m+1,...,k_{m-1}m+m-1]</M>, and the output is the <M>(m-1)</M>-uple <M>[k_1,k_2,...,k_{m-1}]</M>
<ManSection>
<Func Name="KunzPolytope" Arg="m"/>
<Description>
<A>m</A> is a positive integer.
<P/>
The Kunz coordinates of the semigroups with multiplicity <A>m</A> are solutions of a system of inequalities <M>Ax\ge b</M> (see <Cite Key="R-GS-GG-B"></Cite>). The output is the matrix <M>(A|-b)</M>.
<Example><![CDATA[
gap> KunzPolytope(3);
[ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 2, -1, 0 ], [ -1, 2, 1 ] ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="CocycleOfNumericalSemigroupWRTElement" Arg="S,m"/>
<Description>
<A>S</A> is a numerical semigroup, and <A>m</A> is a nonzero element of <A>S</A>. The output is the matrix <M>h(i,j)=(w(i)+w(j)-w((i+j)\bmod m))/m</M>,
where <M>w(i)</M> is the smallest element in <A>S</A> congruent with <M>i</M> modulo <M>m</M> (and thus it is in the Apéry set of <M>m</M>), <Cite Key="arf-frob-gen"></Cite>.
<ManSection>
<Attr Name="FrobeniusNumber" Arg="NS" Label="for numerical semigroup"/>
<Attr Name="FrobeniusNumberOfNumericalSemigroup" Arg="NS"/>
<Description>
The largest nonnegative integer not belonging to a numerical semigroup
<M>S</M> is the <E>Frobenius number</E> of <M>S</M>. If <M>S</M> is the set of
nonnegative integers, then clearly its Frobenius number is <M>-1</M>,
otherwise its Frobenius number coincides with the maximum of the gaps (or
fundamental gaps) of <M>S</M>.
<P/>
<C>NS</C>
is a numerical semigroup. It returns the Frobenius number of <C>NS</C>.
Of
course, the time consumed to return a result may depend on the way the
semigroup is given or on the knowledge already produced on the semigroup.
<Example><![CDATA[
gap> FrobeniusNumber(NumericalSemigroup(3,5,7));
4
gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
4
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Conductor" Arg="NS" Label="for numerical Semigroup"/>
<Attr Name="ConductorOfNumericalSemigroup" Arg="NS"/>
<Description>
This is just a synonym of <C> FrobeniusNumberOfNumericalSemigroup</C> (<C>NS</C>)<M>+1</M>.
<Example><![CDATA[
gap> Conductor(NumericalSemigroup(3,5,7));
5
gap> ConductorOfNumericalSemigroup(NumericalSemigroup(3,5,7));
5
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="PseudoFrobenius" Arg="S"/>
<Attr Name="PseudoFrobeniusOfNumericalSemigroup" Arg="S"/>
<Description>
An integer <M>z</M> is a <E>pseudo-Frobenius
number</E> of <M>S</M> if <M>z+S\setminus\{0\}\subseteq S</M>.
<P/>
<C>S</C>
is a numerical semigroup. It returns the set of pseudo-Frobenius numbers of <A>S</A>.
<Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> PseudoFrobenius(S);
[ 21, 40, 41, 42 ]
gap> PseudoFrobeniusOfNumericalSemigroup(S);
[ 21, 40, 41, 42 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="Type" Arg="NS" Label="of a numerical semigroup"/>
<Attr Name="TypeOfNumericalSemigroup" Arg="NS"/>
<Description>
Stands for <C>Length(PseudoFrobeniusOfNumericalSemigroup (NS))</C>.
<Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> Type(S);
4
gap> TypeOfNumericalSemigroup(S);
4
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Gaps" Arg="NS" Label="for numerical semigroup" />
<Attr Name="GapsOfNumericalSemigroup" Arg="NS"/>
<Description>
A <E>gap</E> of a numerical semigroup <M>S</M> is a nonnegative integer not belonging to
<M>S</M>.
<C>NS</C>
is a numerical semigroup. Both return the set of gaps of <C>NS</C>.
<Example><![CDATA[
gap> Gaps(NumericalSemigroup(5,7,11));
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
[ 1, 2, 4 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Weight" Arg="NS" Label="for numerical semigroup" />
<Description>
If <M>l_1<\cdots <l_g</M> are the gaps of <C>NS</C>, then its (Weierstrass) weight is <M>\sum_{i=1}^g (l_i-i)</M>.
<ManSection>
<Oper Arg="NS" Name="Deserts"/>
<Func Arg="NS" Name="DesertsOfNumericalSemigroup"></Func>
<Description>
<A>NS</A> is a numerical semigroup.
The output is the list with the runs of gaps of <A>NS</A>.
<P/>
<ManSection>
<Prop Arg="NS" Name="IsOrdinary" Label="for numerical semigroups"></Prop>
<Prop Arg="NS" Name="IsOrdinaryNumericalSemigroup"></Prop>
<Description>
<A>NS</A> is a numerical semigroup.
Dectects if the semigroup is ordinary, that is, with less than two deserts.
<P/>
This filter implies <Ref Func="IsAcuteNumericalSemigroup"/>.
<Example><![CDATA[
gap> s:=NumericalSemigroup(3,5,7);;
gap> IsOrdinary(s);
false
]]></Example>
</Description>
</ManSection>
<ManSection>
<Prop Arg="NS" Name="IsAcute" Label="for numerical semigroups"></Prop>
<Prop Arg="NS" Name="IsAcuteNumericalSemigroup"></Prop>
<Description>
<A>NS</A> is a numerical semigroup.
Dectects if the semigroup is acute, that is, it is either ordinary or its last desert (the one with the Frobenius number) has less elements than the preceding one (<Cite Key="Br-acute"></Cite>).
<P/>
<ManSection>
<Attr Name="Holes" Arg="NS" Label="for numerical semigroup" />
<Attr Name="HolesOfNumericalSemigroup" Arg="S"/>
<Description>
<C>S</C>
is a numerical semigroup. Returns the set of gaps <M>x</M> of <C>S</C> such that <M>F(S)-x</M> is also a gap, where <M>F(S)</M> stands for the Frobenius number of <C>S</C>.
<Example><![CDATA[
gap> s:=NumericalSemigroup(3,5);;
gap> Holes(s);
[ ]
gap> s:=NumericalSemigroup(3,5,7);;
gap> HolesOfNumericalSemigroup(s);
[ 2 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="LatticePathAssociatedToNumericalSemigroup" Arg="S,p,q"/>
<Description>
<C>S</C> is a numerical semigroup and <C>p,q</C> are two coprime elements in <C>S</C>.
<P/>In this setting <C>S</C> is an oversemigroup of <M>\langle p,q\rangle</M>, and consequently every gap of <C>S</C> is a gap of <M>\langle p,q\rangle</M>. If <M>c</M> is the conductor of <M>\langle p,q\rangle</M>, then every gap <M>g</M> of <M>\langle p,q\rangle</M> can be written uniquely as <M>g=c-1-(ap+bp)</M> for some nonnegative integers <M>a,b</M>. We say that <M>(a,b)</M> are the coordinates associated to <M>g</M>.
<P/> The output is a path in <M>\mathbb{N}^2</M> such that the coordinates of the gaps of <M>S</M> correspond exactly with the points in <M>\mathbb{N}^2</M> that are between the path and the line <M>ax+by=c-1</M>. See <Cite Key="K-W"></Cite>.
<Example><![CDATA[
gap> s:=NumericalSemigroup(16,17,71,72);;
gap> LatticePathAssociatedToNumericalSemigroup(s,16,17);
[ [ 0, 14 ], [ 1, 13 ], [ 2, 12 ], [ 3, 11 ], [ 4, 10 ], [ 5, 9 ], [ 6, 8 ],
[ 7, 7 ], [ 8, 6 ], [ 9, 5 ], [ 10, 4 ], [ 11, 3 ], [ 12, 2 ], [ 13, 1 ],
[ 14, 0 ] ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="Genus" Arg="NS" Label="for numerical semigroup"/>
<Attr Name="GenusOfNumericalSemigroup" Arg="NS"/>
<Description>
<C>NS</C>
is a numerical semigroup. It returns the number of gaps of <C>NS</C>.
<Example><![CDATA[
gap> s:=NumericalSemigroup(16,17,71,72);;
gap> Genus(s);
80
gap> GenusOfNumericalSemigroup(s);
80
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> Genus(S);
26
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="FundamentalGaps" Arg="S" Label="for numerical semigroup"/>
<Attr Name="FundamentalGapsOfNumericalSemigroup" Arg="S"/>
<Description>
<C>S</C>
The <E>fundamental gaps</E> of <M>S</M> are those gaps that are maximal with respect to the partial order induced by
division in <M>{\mathbb N}</M>.
It returns the set of fundamental gaps of <A>S</A>.
<Example><![CDATA[
gap> FundamentalGaps(NumericalSemigroup(5,7,11));
[ 6, 8, 9, 13 ]
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> FundamentalGapsOfNumericalSemigroup(S);
[ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ]
gap> GapsOfNumericalSemigroup(S);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29,
30, 31, 40, 41, 42 ]
gap> Gaps(NumericalSemigroup(5,7,11));
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="SpecialGaps" Arg="S" Label="for numerical semigroup"/>
<Attr Name="SpecialGapsOfNumericalSemigroup" Arg="S"/>
<Description>
The <E>special gaps</E> of a numerical semigroup <M>S</M> are
those fundamental gaps such that if they are added to the given numerical semigroup,
then the resulting set is again a numerical semigroup.
<C>S</C>
is a numerical semigroup. It returns the special gaps of <A>S</A>.
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
