<Section>
                <Heading>Some basic tests</Heading>
                This section describes some basic tests on numerical semigroups.The first described tests refer to the way the semigroup
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="IsNumericalSemigroupByMinimalGenerators" 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 crieterium 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>
                    <Oper Name="BelongsToNumericalSemigroup" Arg="n,S"/>
                    <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>n in S</C> is the short for <C>BelongsToNumericalSemigroup(n,S)</C>.
                        <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>