A Farey symbol is a compact and useful way to represent a subgroup
of finite index in <M>SL_2(&ZZ;)</M> from which one can deduce
independent generators for this subgroup. It consists of two
components, namely a so-called generalised Farey sequence
(<A>gfs</A>) and an ordered list of labels, giving additional
structure to the <A>gfs</A>.<P/>
A generalised Farey sequence (g.F.S.) is an ordered list of the form
<M>{ -infinity, x_0, x_1, ... , x_n, infinity }</M>, where<P/>
1. the <M>x_i = a_i/b_i</M> are rational
numbers in reduced form arranged in increasing order for
<M>i = 0, ... , n</M>;<P/>
2. <M>x_0, ... , x_n \in Z</M>, and some <M>x_i = 0</M>;<P/>
3. we define <M>x_{-1}=-infinity=-1/0</M> and <M>x_{n+1}=infinity=1/0</M>;<P/>
4. <M>a_{i+1}b_{i}-a_{i}b_{i+1}=1</M> for <M>i=-1, ... ,n</M>.<P/>
The ordered list of labels of a Farey symbol gives an additional
structure to the <A>gfs</A>. The labels correspond to each
consecutive pair of <M>x_i</M>'s and are of the following types:
1. even,<P/>
2. odd,<P/>
3. a natural number, which occurs in the list
of labels exactly twice or not at all.<P/>
Note that the actual values of numerical labels are not important;
it is the pairing of two intervals that matters.<P/>
The package &Congruence; provides functions to construct Farey symbols
by the given generalised Farey sequence and corresponding list of
labels. The returned Farey symbol will belong to the category
<C>IsFareySymbol</C> and will have the representation
<C>IsFareySymbolDefaultRep</C>.
<Section Label="FareyConstr">
<Heading>Construction of Farey symbols</Heading>
<ManSection>
<Func Name="FareySymbolByData"
Arg="gfs labels"/>
<Description>
This constructor creates the Farey symbol with the given generalized
Farey sequence and list of labels. It also checks conditions from
the definition of Farey symbol and returns an error if they are not
satisfied. The data used to create the Farey symbol are stored as
its attributes <Ref Attr="GeneralizedFareySequence"/> and <Ref
Attr="LabelsOfFareySymbol"/>.
</Description>
</ManSection>
<ManSection>
<Func Name="IsValidFareySymbol"
Arg="fs"/>
<Description>
This function is used in <Ref Func="FareySymbolByData"/> to validate its output.
</Description>
</ManSection>
<ManSection>
<Func Name="NumeratorOfGFSElement"
Arg="gfs i"/>
<Returns>
integer
</Returns>
<Description>
Returns the numerator of the i-th term of the generalised Farey
sequence <A>gfs</A>: for the 1st infinite entry returns -1, for the
last one returns 1, for all other entries returns the usual
numerator.
</Description>
</ManSection>
<ManSection>
<Func Name="DenominatorOfGFSElement"
Arg="gfs i"/>
<Returns>
integer
</Returns>
<Description>
Returns the denominator of the i-th term of the generalised Farey
sequence <A>gfs</A>: for both infinite entries returns 0, for the
other ones returns the usual denominator.
</Description>
</ManSection>
<ManSection>
<Attr Name="LabelsOfFareySymbol"
Arg="fs"/>
<Description>
Returns the list of labels of the Farey symbol. This list has "odd", "even" and paired integers as entries.
</Description>
</ManSection>
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