<h3>3 <span class="Heading">Farey symbols and their properties</span></h3>
<p>A Farey symbol is a compact and useful way to represent a subgroup of finite index in <span class="SimpleMath">SL_2(ℤ)</span> from which one can deduce independent generators for this subgroup. It consists of two components, namely a so-called generalised Farey sequence (<var class="Arg">gfs</var>) and an ordered list of labels, giving additional structure to the <var class="Arg">gfs</var>.</p>
<p>A generalised Farey sequence (g.F.S.) is an ordered list of the form <span class="SimpleMath">-infinity, x_0, x_1, ... , x_n, infinity</span>, where</p>
<p>1. the <span class="SimpleMath">x_i = a_i/b_i</span> are rational numbers in reduced form arranged in increasing order for <span class="SimpleMath">i = 0, ... , n</span>;</p>
<p>2. <span class="SimpleMath">x_0, ... , x_n ∈ Z</span>, and some <span class="SimpleMath">x_i = 0</span>;</p>
<p>3. we define <span class="SimpleMath">x_-1=-infinity=-1/0</span> and <span class="SimpleMath">x_n+1=infinity=1/0</span>;</p>
<p>4. <span class="SimpleMath">a_i+1b_i-a_ib_i+1=1</span> for <span class="SimpleMath">i=-1, ... ,n</span>.</p>
<p>The ordered list of labels of a Farey symbol gives an additional structure to the <var class="Arg">gfs</var>. The labels correspond to each consecutive pair of <span class="SimpleMath">x_i</span>'s and are of the following types:
<p>1. even,</p>
<p>2. odd,</p>
<p>3. a natural number, which occurs in the list of labels exactly twice or not at all.</p>
<p>Note that the actual values of numerical labels are not important; it is the pairing of two intervals that matters.</p>
<p>The package <strong class="pkg">Congruence</strong> 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 <code class="code">IsFareySymbol</code> and will have the representation <code class="code">IsFareySymbolDefaultRep</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FareySymbolByData</code>( <var class="Arg">gfs</var>, <var class="Arg">labels</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>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 <code class="func">GeneralizedFareySequence</code> (<a href="chap3.html#X8245766978F02751"><spanclass="RefLink">3.2-1</span></a>) and <code class="func">LabelsOfFareySymbol</code> (<a href="chap3.html#X83C941047D486000"><span class="RefLink">3.2-4</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsValidFareySymbol</code>( <var class="Arg">fs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function is used in <code class="func">FareySymbolByData</code> (<a href="chap3.html#X7F8F5919870A46FE"><span class="RefLink">3.1-1</span></a>) to validate its output.</p>
<p>Returns the numerator of the i-th term of the generalised Farey sequence <var class="Arg">gfs</var>: for the 1st infinite entry returns -1, for the last one returns 1, for all other entries returns the usual numerator.</p>
<p>Returns the denominator of the i-th term of the generalised Farey sequence <var class="Arg">gfs</var>: for both infinite entries returns 0, for the other ones returns the usual denominator.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LabelsOfFareySymbol</code>( <var class="Arg">fs</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the list of labels of the Farey symbol. This list has "odd", "even" and paired integers as entries.</p>
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