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<div class="ChapSects"><a href="chap54.html#X860026B880BCB2A5">54 <span class="Heading">Transformations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X7EB5F6967BC9D340">54.1 <span class="Heading">Functions for Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7B6259467974FB70">54.1-1 IsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7E58AFA1832FF064">54.1-2 TransformationFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7B2E556E7BCF9CDF">54.1-3 Transformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X8268A58685BEFD6F">54.1-4 IdentityTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X8475448F87E8CB8A">54.1-5 RandomTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X78A209C87CF0E32B">54.1-6 DegreeOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7AEC9E6687B3505A">54.1-7 ImageListOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X839A6D6082A21D1F">54.1-8 ImageSetOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X85F22DDD84C28583">54.1-9 RankOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X80FCB5048789CF75">54.1-10 KernelOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X82F5DEEC837B60A3">54.1-11 PreimagesOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X846A6F6B7B715188">54.1-12 RestrictedTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7C5360B2799943F3">54.1-13 AsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X83DBA2A18719EFA8">54.1-14 PermLeftQuoTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7CCD27327CC84626">54.1-15 BinaryRelationTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap54.html#X7BF5FA5C8068EB3B">54.1-16 TransformationRelation</a></span>
</div></div>
</div>
<h3>54 <span class="Heading">Transformations</span></h3>
<p>This chapter describes functions for transformations.</p>
<p>A <em>transformation</em> in <strong class="pkg">GAP</strong> is an endomorphism of a set of integers of the form <span class="SimpleMath">{ 1, ..., n }</span>. Transformations are taken to act on the right, which defines the composition <span class="SimpleMath">i^(α β) = (i^α)^β</span> for <span class="SimpleMath">i</span> in <span class="SimpleMath">{ 1, ..., n }</span>.</p>
<p>For a transformation <span class="SimpleMath">α</span> on the set <span class="SimpleMath">{ 1, ..., n }</span>, we define its <em>degree</em> to be <span class="SimpleMath">n</span>, its <em>image list</em> to be the list <span class="SimpleMath">[1 α, ..., n α]</span>, its <em>image</em> to be the image list considered as a set, and its <em>rank</em> to be the size of the image. We also define the <em>kernel</em> of <span class="SimpleMath">α</span> to be the equivalence relation containing the pair <span class="SimpleMath">(i, j)</span> if and only if <span class="SimpleMath">i^α = j^α</span>.</p>
<p>Note that unlike permutations, we do not consider unspecified points to be fixed by a transformation. Therefore multiplication is only defined on two transformations of the same degree.</p>
<p><a id="X7EB5F6967BC9D340" name="X7EB5F6967BC9D340"></a></p>
<h4>54.1 <span class="Heading">Functions for Transformations</span></h4>
<p><a id="X7B6259467974FB70" name="X7B6259467974FB70"></a></p>
<h5>54.1-1 IsTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformation</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformationCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>We declare it as <code class="func">IsMultiplicativeElementWithOne</code> (<a href="chap31.html#X82BC294F7D388AE8"><span class="RefLink">31.14-11</span></a>) since the identity automorphism of <span class="SimpleMath">{ 1, ..., n }</span> is a multiplicative two sided identity for any transformation on the same set.</p>
<p><a id="X7E58AFA1832FF064" name="X7E58AFA1832FF064"></a></p>
<h5>54.1-2 TransformationFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationFamily</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationType</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationData</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For each <code class="code"><var class="Arg">n</var> > 0</code> there is a single family and type of transformations on <var class="Arg">n</var> points. To speed things up, we store these in a database of types. The three functions above a then access functions. If the <var class="Arg">n</var>th entry isn't yet created, they trigger creation as well.</p>
<p>For <code class="code"><var class="Arg">n</var> > 0</code>, element <var class="Arg">n</var> of the type database is <code class="code">[TransformationFamily(</code><var class="Arg">n</var><code class="code">), TransformationType(</code><var class="Arg">n</var><code class="code">)]</code></p>
<p><a id="X7B2E556E7BCF9CDF" name="X7B2E556E7BCF9CDF"></a></p>
<h5>54.1-3 Transformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transformation</code>( <var class="Arg">images</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationNC</code>( <var class="Arg">images</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>both return a transformation with the image list <var class="Arg">images</var>. The first version checks that the all the elements of the given list lie within the range <span class="SimpleMath">{ 1, ..., n }</span> where <span class="SimpleMath">n</span> is the length of <var class="Arg">images</var>, but for speed purposes, a non-checking version is also supplied.</p>
<p><a id="X8268A58685BEFD6F" name="X8268A58685BEFD6F"></a></p>
<h5>54.1-4 IdentityTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityTransformation</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the identity transformation of degree <var class="Arg">n</var>.</p>
<p><a id="X8475448F87E8CB8A" name="X8475448F87E8CB8A"></a></p>
<h5>54.1-5 RandomTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomTransformation</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a random transformation of degree <var class="Arg">n</var>.</p>
<p><a id="X78A209C87CF0E32B" name="X78A209C87CF0E32B"></a></p>
<h5>54.1-6 DegreeOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the degree of <var class="Arg">trans</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= Transformation([2, 3, 4, 2, 4]);</span>
Transformation( [ 2, 3, 4, 2, 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformation(t);</span>
5
</pre></div>
<p><a id="X7AEC9E6687B3505A" name="X7AEC9E6687B3505A"></a></p>
<h5>54.1-7 ImageListOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageListOfTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the image list of <var class="Arg">trans</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageListOfTransformation(t);</span>
[ 2, 3, 4, 2, 4 ]
</pre></div>
<p><a id="X839A6D6082A21D1F" name="X839A6D6082A21D1F"></a></p>
<h5>54.1-8 ImageSetOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageSetOfTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the image of <var class="Arg">trans</var> as a set.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageSetOfTransformation(t);</span>
[ 2, 3, 4 ]
</pre></div>
<p><a id="X85F22DDD84C28583" name="X85F22DDD84C28583"></a></p>
<h5>54.1-9 RankOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankOfTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the rank of <var class="Arg">trans</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RankOfTransformation(t);</span>
3
</pre></div>
<p><a id="X80FCB5048789CF75" name="X80FCB5048789CF75"></a></p>
<h5>54.1-10 KernelOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelOfTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the kernel of <var class="Arg">trans</var> as an equivalence relation, see <a href="chap33.html#X7DED7F1F78D31785"><span class="RefLink">33.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation(t); </span>
[ [ 1, 4 ], [ 2 ], [ 3, 5 ] ]
</pre></div>
<p><a id="X82F5DEEC837B60A3" name="X82F5DEEC837B60A3"></a></p>
<h5>54.1-11 PreimagesOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreimagesOfTransformation</code>( <var class="Arg">trans</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the subset of <span class="SimpleMath">{ 1, ..., n }</span> which maps to <var class="Arg">i</var> under <var class="Arg">trans</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PreimagesOfTransformation(t, 2);</span>
[ 1, 4 ]
</pre></div>
<p><a id="X846A6F6B7B715188" name="X846A6F6B7B715188"></a></p>
<h5>54.1-12 RestrictedTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedTransformation</code>( <var class="Arg">trans</var>, <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The transformation <var class="Arg">trans</var> is restricted to only those points of <var class="Arg">alpha</var>.</p>
<p><a id="X7C5360B2799943F3" name="X7C5360B2799943F3"></a></p>
<h5>54.1-13 AsTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsTransformation</code>( <var class="Arg">O</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsTransformationNC</code>( <var class="Arg">O</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the object <var class="Arg">O</var> as a transformation. Supported objects are permutations and binary relations on points. Called with two arguments, the operation returns a transformation of degree <var class="Arg">n</var>, signalling an error if such a representation is not possible. <code class="func">AsTransformationNC</code> does not perform this check.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation((1, 3)(2, 4));</span>
Transformation( [ 3, 4, 1, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation((1, 3)(2, 4), 10);</span>
Transformation( [ 3, 4, 1, 2, 5, 6, 7, 8, 9, 10 ] )
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation((1, 3)(2, 4), 3);</span>
Error, Permutation moves points over the degree specified called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
</pre></div>
<p><a id="X83DBA2A18719EFA8" name="X83DBA2A18719EFA8"></a></p>
<h5>54.1-14 PermLeftQuoTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermLeftQuoTransformation</code>( <var class="Arg">tr1</var>, <var class="Arg">tr2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Given transformations <var class="Arg">tr1</var> and <var class="Arg">tr2</var> with equal kernel and image, we compute the permutation induced by (<var class="Arg">tr1</var>)<span class="SimpleMath">^{-1} *</span> <var class="Arg">tr2</var> on the set of images of <var class="Arg">tr1</var>. If the kernels and images are not equal, an error is signaled.</p>
<p><a id="X7CCD27327CC84626" name="X7CCD27327CC84626"></a></p>
<h5>54.1-15 BinaryRelationTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BinaryRelationTransformation</code>( <var class="Arg">trans</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <var class="Arg">trans</var> when considered as a binary relation.</p>
<p><a id="X7BF5FA5C8068EB3B" name="X7BF5FA5C8068EB3B"></a></p>
<h5>54.1-16 TransformationRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the binary relation <var class="Arg">R</var> when considered as a transformation. Only makes sense for injective binary relations over <code class="code">[1..n]</code>. An error is signalled if the relation is not over <code class="code">[1..n]</code>, and <code class="keyw">fail</code> if it is not injective.</p>
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