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<Appendix><Heading>"Random" functions</Heading>

Here we describe some functions which allow to create
several "random" objects. We make use of the function <C>RandomList</C>.

<Section><Heading>Random functions for numerical semigroups</Heading>

<ManSection>
            <Func Name="RandomNumericalSemigroup" Arg="n,a[,b]"></Func>
            <Description>
                Returns a ``random" numerical semigroup with no more than n generators in [1..a] (or in [a..b], if b is present).
                <Example><![CDATA[
gap> RandomNumericalSemigroup(3,9);
<Numerical semigroup with 3 generators>
gap> RandomNumericalSemigroup(3,9,55);
<Numerical semigroup with 3 generators>
]]></Example>
            </Description>
        </ManSection>

        <ManSection>
            <Func Name="RandomListForNS" Arg="n,a,b"></Func>
            <Description>
Returns a set of length not greater than <A>n</A> of random integers in <A>[a..b]</A>
whose GCD is 1.
It is used to create "random" numerical semigroups.
                <Example><![CDATA[
gap> RandomListForNS(13,1,79);
[ 22, 26, 29, 31, 34, 46, 53, 61, 62, 73, 76 ]
]]></Example>
            </Description>
        </ManSection>

        <ManSection>
            <Func Name="RandomModularNumericalSemigroup" Arg="k[,m]"></Func>
            <Description>
                Returns a ``random" modular numerical semigroup S(a,b) with a \le k (see />) and multiplicity at least m, were m is the second argument, which may not be present..
                <Example><![CDATA[
gap> RandomModularNumericalSemigroup(9);
<Modular numerical semigroup satisfying 5x mod 6 <= x >
gap> RandomModularNumericalSemigroup(10,25);
<Modular numerical semigroup satisfying 4x mod 157 <= x >
]]></Example>
            </Description>
        </ManSection>

        <ManSection>
            <Func Name="RandomProportionallyModularNumericalSemigroup" Arg="k[,m]"></Func>
            <Description>
                Returns a ``random" proportionally modular numerical semigroup S(a,b,c) with a \le k (see ) and multiplicity at least m, were m is the second argument, which may not be present.
                <Example><![CDATA[
gap> RandomProportionallyModularNumericalSemigroup(9);
<Proportionally modular numerical semigroup satisfying 2x mod 3 <= 2x >
gap> RandomProportionallyModularNumericalSemigroup(10,25);
<Proportionally modular numerical semigroup satisfying 6x mod 681 <= 2x >
]]></Example>
            </Description>
        </ManSection>

        <ManSection>
            <Func Name="RandomListRepresentingSubAdditiveFunction" Arg="m, a"></Func>
            <Description>
                Produces a ``random" list representing a subadditive function (see ) which is periodic
with period <A>m</A> (or less). When possible, the images are in <A>[a..20*a]</A>.
(Otherwise, the list of possible images is enlarged.)

                <Example><![CDATA[
gap> RandomListRepresentingSubAdditiveFunction(7,9);
[ 173, 114, 67, 0 ]
gap> RepresentsPeriodicSubAdditiveFunction(last);
true
]]></Example>
            </Description>
        </ManSection>

        <ManSection>
            <Func Name="NumericalSemigroupWithRandomElementsAndFrobenius" Arg="n,mult,frob"></Func>
            <Description>
Produces a "random" semigroup containing (at least) <A>n</A> elements greater than or equal to <A>mult</A> and less than <A>frob</A>, chosen at random. The semigroup returned has multiplicity chosen at random but no smaller than <A>mult</A> and having Frobenius number chosen at random but not greater than <A>frob</A>. Returns <M>fail</M> if <A>frob</A> is greater than <A>mult</A>.
                <Example><![CDATA[
gap> ns := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,50);
<Numerical semigroup with 17 generators>
gap> MinimalGeneratingSystem(ns);
[ 12, 13, 19, 27, 47 ]
gap> SmallElements(ns);
[ 0, 12, 13, 19, 24, 25, 26, 27, 31, 32, 36, 37, 38, 39, 40, 43 ]
gap> ns2 := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,9);
#I  The third argument must not be smaller than the second
fail
gap> ns3 := NumericalSemigroupWithRandomElementsAndFrobenius(5,10,10);
<Proportionally modular numerical semigroup satisfying 20x mod 200 <= 10x >
gap> MinimalGeneratingSystem(ns3);
[ 10 .. 19 ]
gap> SmallElements(ns3);
[ 0, 10 ]
]]></Example>
            </Description>
        </ManSection>

<ManSection>
            <Func Name="RandomNumericalSemigroupWithGenus" Arg="g"></Func>
            <Description>
Produces a pseudo-random numerical semigroup with genus <A>g</A>.
                <Example><![CDATA[
gap> RandomNumericalSemigroupWithGenus(7);Gaps(last);
<Numerical semigroup with 7 generators>
[ 1, 2, 3, 4, 5, 6, 9 ]
]]></Example>
            </Description>
        </ManSection>

</Section>

<Section><Heading>Random functions for affine semigroups</Heading>

<ManSection>
            <Func Name="RandomAffineSemigroupWithGenusAndDimension" Arg="g,d"></Func>
            <Description>
Produces a pseudo-random affine semigroup with genus <A>g</A> and dimension <A>d</A>.
                <Example><![CDATA[
gap> RandomAffineSemigroupWithGenusAndDimension(10,3);Gaps(last);
<Affine semigroup in 3 dimensional space, with 66 generators>
[ [ 0, 1, 0 ], [ 0, 2, 0 ], [ 0, 3, 0 ], [ 0, 4, 0 ], [ 0, 5, 0 ],
  [ 0, 7, 0 ], [ 1, 0, 0 ], [ 1, 1, 0 ], [ 2, 0, 0 ], [ 3, 0, 0 ] ]
]]></Example>
            </Description>
        </ManSection>

<ManSection>
            <Func Name="RandomAffineSemigroup" Arg="n,d,m"></Func>
            <Description>
Returns an affine semigroup generated by a <A>n</A>*<A>d</A> matrix where <A>d</A> (the dimension) is randomly choosen from [1..<A>d</A>] and <A>n</A> (the number of generators) is randomly choosen from [1..<A>n</A>]. The entries of the matrix are randomly choosen from [0..<A>m</A>] (when the third argument is not present, m is taken as <A>n</A>*<A>d</A>)
                <Example><![CDATA[
gap> RandomAffineSemigroup(5,5);Generators(last);
<Affine semigroup in 5 dimensional space, with 4 generators>
[ [ 4, 10, 10, 8, 20 ], [ 9, 12, 16, 3, 16 ], [ 14, 19, 14, 3, 20 ],
  [ 16, 6, 0, 7, 13 ] ]
gap> RandomAffineSemigroup(5,5,3);Generators(last);
<Affine semigroup in 4 dimensional space, with 5 generators>
[ [ 0, 2, 1, 3 ], [ 1, 3, 3, 2 ], [ 2, 3, 3, 2 ], [ 3, 1, 2, 1 ],
  [ 3, 3, 1, 0 ] ]
]]></Example>
            </Description>
        </ManSection>

<ManSection>
            <Func Name="RandomFullAffineSemigroup" Arg="n,d,m"></Func>
            <Description>
Returns a full affine semigroup either given by equations or inequalities (when no string is given, one is choosen at random). The matrix is an <A>n</A>*<A>d</A> matrix where <A>d</A> (the dimension) is randomly choosen from [1..<A>d</A>] and <A>n</A> is randomly choosen from [1..<A>n</A>]. When it is given by equations, the moduli are choosen at random. The entries of the matrix (and moduli) are randomly choosen from [0..<A>m</A>] (when the third integer is not present, m is taken as <A>n</A>*<A>d</A>)                <Example><![CDATA[
gap> RandomFullAffineSemigroup(5,5,3);Generators(last);
<Affine semigroup>
#I  Using contejeanDevieAlgorithm for Hilbert Basis. Please, consider using
NormalizInterface, 4ti2Interface or 4ti2gap.
[ [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0 ],
  [ 1, 0, 0, 0, 0 ] ]
]]></Example>
            </Description>
        </ManSection>

</Section>

<Section><Heading>Random functions for good semigroups</Heading>

<ManSection>
      <Func Arg="m, cond" Name="RandomGoodSemigroupWithFixedMultiplicity"></Func>
      <Description>
        This function produces a "random" semigroup with multiplicity <A>m</A> and with conductor bounded by <A>cond</A>

        <Example><![CDATA[
gap> S:=RandomGoodSemigroupWithFixedMultiplicity([6,7],[30,30]);
<Good semigroup>
gap> SmallElements(S);
[ [ 0, 0 ], [ 6, 7 ], [ 9, 8 ], [ 9, 10 ], [ 9, 11 ], [ 9, 14 ], [ 9, 15 ],
  [ 9, 16 ], [ 10, 8 ], [ 11, 10 ], [ 11, 11 ], [ 12, 10 ], [ 12, 14 ],
  [ 13, 10 ], [ 13, 15 ], [ 13, 16 ], [ 15, 10 ], [ 15, 15 ], [ 15, 16 ],
  [ 16, 10 ], [ 16, 15 ], [ 17, 10 ], [ 17, 16 ] ]

]]></Example>
      </Description>
    </ManSection>

</Section>
</Appendix>

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