Quelle Some_basic_tests.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>
<Section>
                <Heading>Some basic tests</Heading>
                This section describes some basic tests on numerical semigroups. The first described tests refer to what the semigroup is currently known to be (not necessarily the way it was created).
Then are presented functions to test if a given list represents
the small elements, gaps or the Apéry set (see <Ref Label="zlab1" />) of a numerical semigroup; to test if an integer belongs to a numerical semigroup
and if a numerical semigroup is a subsemigroup of another one.

                <ManSection>
                    <Attr Name="IsNumericalSemigroup" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByGenerators" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByInterval" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByOpenInterval" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupBySubAdditiveFunction" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByAperyList" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupBySmallElements" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByGaps" Arg="NS"/>
                    <Attr Name="IsNumericalSemigroupByFundamentalGaps" Arg="NS"/>
                    <Attr Name="IsProportionallyModularNumericalSemigroup" Arg="NS"/>
                    <Attr Name="IsModularNumericalSemigroup" Arg="NS"/>
                    <Description>

                        <A>NS</A> is a numerical semigroup and these attributes are available
                        (their names should be self explanatory).

                        <Example><![CDATA[
gap> s:=NumericalSemigroup(3,7);
<Numerical semigroup with 2 generators>
gap> AperyListOfNumericalSemigroupWRTElement(s,30);;
gap> t:=NumericalSemigroupByAperyList(last);
<Numerical semigroup>
gap> IsNumericalSemigroupByGenerators(s);
true
gap> IsNumericalSemigroupByGenerators(t);
false
gap> IsNumericalSemigroupByAperyList(s);
false
gap> IsNumericalSemigroupByAperyList(t);
true
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                    <Attr Name="RepresentsSmallElementsOfNumericalSemigroup" Arg="L"/>
                    <Description>
                        Tests if the list <A>L</A> (which has to be a set) may represent the ``small" elements of a numerical semigroup.
                        <Example><![CDATA[
gap> L:=[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ];
[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ]
gap> RepresentsSmallElementsOfNumericalSemigroup(L);
true
gap> L:=[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ];
[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ]
gap> RepresentsSmallElementsOfNumericalSemigroup(L);
false
]]></Example>
                    </Description>
                </ManSection>



                <ManSection>
                    <Attr Name="RepresentsGapsOfNumericalSemigroup" Arg="L"/>
                    <Description>
                        Tests if the list <A>L</A>  may represent the gaps
                        (see <Ref Label="xx1" />)   of a numerical semigroup.
                        <Example><![CDATA[
gap> s:=NumericalSemigroup(3,7);
<Numerical semigroup with 2 generators>
gap> L:=GapsOfNumericalSemigroup(s);
[ 1, 2, 4, 5, 8, 11 ]
gap> RepresentsGapsOfNumericalSemigroup(L);
true
gap> L:=Set(List([1..21],i->RandomList([1..50])));
[ 2, 6, 7, 8, 10, 12, 14, 19, 24, 28, 31, 35, 42, 50 ]
gap> RepresentsGapsOfNumericalSemigroup(L);
false
]]></Example>
                    </Description>
                </ManSection>

                <ManSection>
<Func Name="IsAperyListOfNumericalSemigroup" Arg="L"/>
<Description>
Tests whether a list <A>L</A> of integers may represent the Apéry list of a
numerical semigroup. It returns <K>true</K> when the periodic function
represented by <A>L</A> is subadditive
(see <Ref Func="RepresentsPeriodicSubAdditiveFunction" />)
and the remainder of the division of
<C>L[i]</C> by the length of <A>L</A> is <C>i</C> and returns <K>false</K>
otherwise (the criterium used is the one explained in <Cite Key="R96"></Cite>).
<Example><![CDATA[
gap> IsAperyListOfNumericalSemigroup([0,21,7,28,14]);
true
]]></Example>
</Description>
</ManSection>

                <ManSection>
                    <Func Name="IsSubsemigroupOfNumericalSemigroup" Arg="S, T"/>
                    <Description>
                        <A>S</A> and <A>T</A>
                        are numerical semigroups. Tests whether <A>T</A> is contained in <A>S</A>.
                        <Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> T:=NumericalSemigroup(2,3);
<Numerical semigroup with 2 generators>
gap> IsSubsemigroupOfNumericalSemigroup(T,S);
true
gap> IsSubsemigroupOfNumericalSemigroup(S,T);
false
]]></Example>
                    </Description>
                </ManSection>
                <ManSection>
                    <Attr Name="IsSubset" Arg="S, T"/>
                    <Description>
<A>S</A> is a numerical semigroup. <A>T</A> can be a numerical semigroup, in which case the function is just a synonym of <Ref Func="IsSubsemigroupOfNumericalSemigroup" />, or a list of integers, in which case tests whether all elements of the list belong to <A>S</A>.
<Example><![CDATA[
gap> ns1 := NumericalSemigroup(5,7);;
gap> ns2 := NumericalSemigroup(5,7,11);;
gap> IsSubset(ns1,ns2);
false
gap> IsSubset(ns2,[5,15]);
true
gap> IsSubset(ns1,[5,11]);
false
gap> IsSubset(ns2,ns1);
true
]]></Example>
                    </Description>
                </ManSection>


                <ManSection>
                    <Oper Name="BelongsToNumericalSemigroup" Arg="n,S"/>
                    <Oper Name="\in" Arg="n,S" Label="membership test for numerical semigroup"/>
                    <Description>
                        <A>n</A> is an integer and <A>S</A> is a numerical semigroup. Tests whether
                        <A>n</A> belongs to <A>S</A>. <C>\in(n,S)</C> calls the infix variant <C>n in S</C>, and both can be seen as a short for <C>BelongsToNumericalSemigroup(n,S)</C>. Several methods are implemented for membership, depending on the properties of <A>S</A> known. For instance, there are methods if any of the following information is known: Apéry set, small elements, defining (proportionally) modular Diophantine equation, fundamental gaps, gaps, generators.
                        <Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> BelongsToNumericalSemigroup(15,S);
false
gap> 15 in S;
false
gap> SmallElementsOfNumericalSemigroup(S);
[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
gap> BelongsToNumericalSemigroup(13,S);
true
gap> 13 in S;
true
]]></Example>
                    </Description>
                </ManSection>
            </Section>

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