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<p><a id="X860026B880BCB2A5" name="X860026B880BCB2A5"></a></p>
<div class="ChapSects"><a href="chap2.html#X860026B880BCB2A5">2 <span class="Heading">Transformations</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X80F3086F87E93DF8">2.1 <span class="Heading">Creating Transformations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X82CB17E47CBCFA69">2.1-1 <span class="Heading">TransformationByKernelAndImage</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D1937FF85B76BB3">2.1-2 <span class="Heading">AllTransformationsWithKerAndImg</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X85D1071484CE004C">2.1-3 <span class="Heading">Idempotent</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C5BED247F770ECB">2.1-4 <span class="Heading">RandomIdempotent</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8475448F87E8CB8A">2.1-5 <span class="Heading">RandomTransformation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X814B3E6E7D3F1036">2.1-6 TransformationActionNC</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X84DD0B3981E5735B">2.2 <span class="Heading">Properties & Attributes</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79E9FFC97AFCEE61">2.2-1 IsTransversal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79A3FCED7E8A8B1B">2.2-2 IsKerImgOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E72E6117BFFC74E">2.2-3 KerImgOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7856A73E80DAF56F">2.2-4 IsRegularTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X863216CB7AF88BED">2.2-5 IndexPeriodOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84E1A41F84B70DBB">2.2-6 SmallestIdempotentPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X846B9EBB86A69BDC">2.2-7 <span class="Heading">InversesOfTransformation</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X80B8EF3D78E711BB">2.3 <span class="Heading">Changing Representation</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F867C337B18D84E">2.3-1 AsBooleanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A249D7781F31C22">2.3-2 AsPermOfRange</a></span>
</div>
</div>
<h3>2 <span class="Heading">Transformations</span></h3>
<p>The functions described in this section extend the functionality of <strong class="pkg">GAP</strong> relating to transformations.</p>
<p><a id="X80F3086F87E93DF8" name="X80F3086F87E93DF8"></a></p>
<h4>2.1 <span class="Heading">Creating Transformations</span></h4>
<p><a id="X82CB17E47CBCFA69" name="X82CB17E47CBCFA69"></a></p>
<h5>2.1-1 <span class="Heading">TransformationByKernelAndImage</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> TransformationByKernelAndImage</code>( <var class="Arg">ker, img</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">> TransformationByKernelAndImageNC</code>( <var class="Arg">ker, img</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the transformation <code class="code">f</code> with kernel <code class="code">ker</code> and image <code class="code">img</code> where <code class="code">(x)f=img[i]</code> for all <code class="code">x</code> in <code class="code">ker[i]</code>. The argument <code class="code">ker</code> should be a set of sets that partition the set <code class="code">1,...n</code> for some <code class="code">n</code> and <code class="code">img</code> should be a sublist of <code class="code">1,...n</code>.</p>
<p><code class="code">TransformationByKernelAndImage</code> first checks that <code class="code">ker</code> and <code class="code">img</code> describe the kernel and image of a transformation whereas <code class="code">TransformationByKernelAndImageNC</code> performs no such check.</p>
<table class="example">
<tr><td><pre>
gap> TransformationByKernelAndImageNC([[1,2,3,4],[5,6,7],[8]],[1,2,8]);
Transformation( [ 1, 1, 1, 1, 2, 2, 2, 8 ] )
gap> TransformationByKernelAndImageNC([[1,6],[2,5],[3,4]], [4,5,6]);
Transformation( [ 4, 5, 6, 6, 5, 4 ] )
</pre></td></tr></table>
<p><a id="X7D1937FF85B76BB3" name="X7D1937FF85B76BB3"></a></p>
<h5>2.1-2 <span class="Heading">AllTransformationsWithKerAndImg</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AllTransformationsWithKerAndImg</code>( <var class="Arg">ker, img</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">> AllTransformationsWithKerAndImgNC</code>( <var class="Arg">ker, img</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of all transformations with kernel <code class="code">ker</code> and image <code class="code">img</code>. The argument <code class="code">ker</code> should be a set of sets that partition the set <code class="code">1,...n</code> for some <code class="code">n</code> and <code class="code">img</code> should be a sublist of <code class="code">1,...n</code>.</p>
<p><code class="code">AllTransformationsWithKerAndImg</code> first checks that <code class="code">ker</code> and <code class="code">img</code> describe the kernel and image of a transformation whereas <code class="code">AllTransformationsWithKerAndImgNC</code> performs no such check.</p>
<table class="example">
<tr><td><pre>
gap> AllTransformationsWithKerAndImg([[1,6],[2,5],[3,4]], [4,5,6]);
[ Transformation( [ 4, 5, 6, 6, 5, 4 ] ),
Transformation( [ 6, 5, 4, 4, 5, 6 ] ),
Transformation( [ 6, 4, 5, 5, 4, 6 ] ),
Transformation( [ 4, 6, 5, 5, 6, 4 ] ),
Transformation( [ 5, 6, 4, 4, 6, 5 ] ),
Transformation( [ 5, 4, 6, 6, 4, 5 ] ) ]
</pre></td></tr></table>
<p><a id="X85D1071484CE004C" name="X85D1071484CE004C"></a></p>
<h5>2.1-3 <span class="Heading">Idempotent</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IdempotentNC</code>( <var class="Arg">ker, img</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">> Idempotent</code>( <var class="Arg">ker, img</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">IdempotentNC</code> returns an idempotent with kernel <code class="code">ker</code> and image <code class="code">img</code> without checking <code class="func">IsTransversal</code> (<a href="chap2.html#X79E9FFC97AFCEE61"><b>2.2-1</b></a>) with arguments <code class="code">ker</code> and <code class="code">im</code>.</p>
<p><code class="code">Idempotent</code> returns an idempotent with kernel <code class="code">ker</code> and image <code class="code">img</code> after checking that <code class="func">IsTransversal</code> (<a href="chap2.html#X79E9FFC97AFCEE61"><b>2.2-1</b></a>) with arguments <code class="code">ker</code> and <code class="code">im</code> returns <code class="code">true</code>.</p>
<table class="example">
<tr><td><pre>
gap> g1:=Transformation([2,2,4,4,5,6]);;
gap> g2:=Transformation([5,3,4,4,6,6]);;
gap> ker:=KernelOfTransformation(g2*g1);;
gap> im:=ImageListOfTransformation(g2);;
gap> Idempotent(ker, im);
Error, the image must be a transversal of the kernel
[ ... ]
gap> Idempotent([[1,2,3],[4,5],[6,7]], [1,5,6]);
Transformation( [ 1, 1, 1, 5, 5, 6, 6 ] )
gap> IdempotentNC([[1,2,3],[4,5],[6,7]], [1,5,6]);
Transformation( [ 1, 1, 1, 5, 5, 6, 6 ] )
</pre></td></tr></table>
<p><a id="X7C5BED247F770ECB" name="X7C5BED247F770ECB"></a></p>
<h5>2.1-4 <span class="Heading">RandomIdempotent</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RandomIdempotent</code>( <var class="Arg">arg</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">> RandomIdempotentNC</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If the argument is a kernel, then a random idempotent is return that has that kernel. A <em>kernel</em> is a set of sets that partition the set <code class="code">1,...n</code> for some <code class="code">n</code> and an <em>image</em> is a sublist of <code class="code">1,...n</code>.</p>
<p>If the first argument is an image <code class="code">img</code> and the second a positive integer <code class="code">n</code>, then a random idempotent of degree <code class="code">n</code> is returned with image <code class="code">img</code>.</p>
<p>The no check version does not check that the arguments can be the kernel and image of an idempotent.</p>
<table class="example">
<tr><td><pre>
gap> RandomIdempotent([[1,2,3], [4,5], [6,7,8]], [1,2,3]);;
fail
gap> RandomIdempotent([1,2,3],5);
Transformation( [ 1, 2, 3, 1, 3 ] )
gap> RandomIdempotent([[1,6], [2,4], [3,5]]);
Transformation( [ 1, 2, 5, 2, 5, 1 ] )
</pre></td></tr></table>
<p><a id="X8475448F87E8CB8A" name="X8475448F87E8CB8A"></a></p>
<h5>2.1-5 <span class="Heading">RandomTransformation</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RandomTransformation</code>( <var class="Arg">arg</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">> RandomTransformationNC</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These are new methods for the existing library function <code class="func">RandomTransformation</code>.</p>
<p>If the first argument is a kernel and the second an image, then a random transformation is returned with this kernel and image.A <em>kernel</em> is a set of sets that partition the set <code class="code">1,...n</code> for some <code class="code">n</code> and an <em>image</em> is a sublist of <code class="code">1,...n</code>.</p>
<p>If the argument is a kernel, then a random transformation is returned that has that kernel.</p>
<p>If the first argument is an image <code class="code">img</code> and the second a positive integer <code class="code">n</code>, then a random transformation of degree <code class="code">n</code> is returned with image <code class="code">img</code>.</p>
<p>The no check version does not check that the arguments can be the kernel and image of a transformation.</p>
<table class="example">
<tr><td><pre>
gap> RandomTransformation([[1,2,3], [4,5], [6,7,8]], [1,2,3]);;
Transformation( [ 2, 2, 2, 1, 1, 3, 3, 3 ] )
gap> RandomTransformation([[1,2,3],[5,7],[4,6]]);
Transformation( [ 3, 3, 3, 6, 1, 6, 1 ] )
gap> RandomTransformation([[1,2,3],[5,7],[4,6]]);
Transformation( [ 4, 4, 4, 7, 3, 7, 3 ] )
gap> RandomTransformationNC([[1,2,3],[5,7],[4,6]]);
Transformation( [ 1, 1, 1, 7, 5, 7, 5 ] )
gap> RandomTransformation([1,2,3], 6);
Transformation( [ 2, 1, 2, 1, 1, 2 ] )
gap> RandomTransformationNC([1,2,3], 6);
Transformation( [ 3, 1, 2, 2, 1, 2 ] )
</pre></td></tr></table>
<p><a id="X814B3E6E7D3F1036" name="X814B3E6E7D3F1036"></a></p>
<h5>2.1-6 TransformationActionNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> TransformationActionNC</code>( <var class="Arg">list, act, elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the list <code class="code">list</code> acted on by <code class="code">elm</code> via the action <code class="code">act</code>.</p>
<table class="example">
<tr><td><pre>
gap> mat:=OneMutable(GeneratorsOfGroup(GL(3,3))[1]);
[ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> mat[3][3]:=Z(3)*0;
0*Z(3)
gap> F:=BaseDomain(mat);
GF(3)
gap> TransformationActionNC(Elements(F^3), OnRight, mat);
Transformation( [ 1, 1, 1, 4, 4, 4, 7, 7, 7, 10, 10, 10, 13, 13, 13, 16, 16,
16, 19, 19, 19, 22, 22, 22, 25, 25, 25 ] )
</pre></td></tr></table>
<p><a id="X84DD0B3981E5735B" name="X84DD0B3981E5735B"></a></p>
<h4>2.2 <span class="Heading">Properties & Attributes</span></h4>
<p><a id="X79E9FFC97AFCEE61" name="X79E9FFC97AFCEE61"></a></p>
<h5>2.2-1 IsTransversal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsTransversal</code>( <var class="Arg">list1, list2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code">true</code> if the list <code class="code">list2</code> is a transversal of the list of lists <code class="code">list1</code>. That is, if every list in <code class="code">list1</code> contains exactly one element in <code class="code">list2</code>.</p>
<table class="example">
<tr><td><pre>
gap> g1:=Transformation([2,2,4,4,5,6]);;
gap> g2:=Transformation([5,3,4,4,6,6]);;
gap> ker:=KernelOfTransformation(g2*g1);
[ [ 1 ], [ 2, 3, 4 ], [ 5, 6 ] ]
gap> im:=ImageListOfTransformation(g2);
[ 5, 3, 4, 4, 6, 6 ]
gap> IsTransversal(ker, im);
false
gap> IsTransversal([[1,2,3],[4,5],[6,7]], [1,5,6]);
true
</pre></td></tr></table>
<p><a id="X79A3FCED7E8A8B1B" name="X79A3FCED7E8A8B1B"></a></p>
<h5>2.2-2 IsKerImgOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsKerImgOfTransformation</code>( <var class="Arg">ker, img</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code">true</code> if the arguments <code class="code">ker</code> and <code class="code">img</code> can be the kernel and image of a single transformation, respectively. The argument <code class="code">ker</code> should be a set of sets that partition the set <code class="code">1,...n</code> for some <code class="code">n</code> and <code class="code">img</code> should be a sublist of <code class="code">1,...n</code>.</p>
<table class="example">
<tr><td><pre>
gap> ker:=[[1,2,3],[5,6],[8]];
[ [ 1, 2, 3 ], [ 5, 6 ], [ 8 ] ]
gap> img:=[1,2,9];
[ 1, 2, 9 ]
gap> IsKerImgOfTransformation(ker,img);
false
gap> ker:=[[1,2,3,4],[5,6,7],[8]];
[ [ 1, 2, 3, 4 ], [ 5, 6, 7 ], [ 8 ] ]
gap> IsKerImgOfTransformation(ker,img);
false
gap> img:=[1,2,8];
[ 1, 2, 8 ]
gap> IsKerImgOfTransformation(ker,img);
true
</pre></td></tr></table>
<p><a id="X7E72E6117BFFC74E" name="X7E72E6117BFFC74E"></a></p>
<h5>2.2-3 KerImgOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> KerImgOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the kernel and image set of the transformation <code class="code">f</code>. These attributes of <code class="code">f</code> can be obtain separately using <code class="func">KernelOfTransformation</code> (<a href="/Users/jdm/Maths/Computation/GAP/gapdev/doc/ref/chap52.html#X80FCB5048789CF75"><b>Reference: KernelOfTransformation</b></a>) and <code class="func">ImageSetOfTransformation</code> (<a href="/Users/jdm/Maths/Computation/GAP/gapdev/doc/ref/chap52.html#X839A6D6082A21D1F"><b>Reference: ImageSetOfTransformation</b></a>), respectively.</p>
<table class="example">
<tr><td><pre>
gap> t:=Transformation( [ 10, 8, 7, 2, 8, 2, 2, 6, 4, 1 ] );;
gap> KerImgOfTransformation(t);
[ [ [ 1 ], [ 2, 5 ], [ 3 ], [ 4, 6, 7 ], [ 8 ], [ 9 ], [ 10 ] ],
[ 1, 2, 4, 6, 7, 8, 10 ] ]
</pre></td></tr></table>
<p><a id="X7856A73E80DAF56F" name="X7856A73E80DAF56F"></a></p>
<h5>2.2-4 IsRegularTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsRegularTransformation</code>( <var class="Arg">S, f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>if <code class="code">f</code> is a regular element of the transformation semigroup <code class="code">S</code>, then <code class="code">true</code> is returned. Otherwise <code class="code">false</code> is returned.</p>
<p>A transformation <code class="code">f</code> is regular inside a transformation semigroup <code class="code">S</code> if it lies inside a regular D-class. This is equivalent to the orbit of the image of <code class="code">f</code> containing a transversal of the kernel of <code class="code">f</code>.</p>
<table class="example">
<tr><td><pre>
gap> g1:=Transformation([2,2,4,4,5,6]);;
gap> g2:=Transformation([5,3,4,4,6,6]);;
gap> m1:=Monoid(g1,g2);;
gap> IsRegularTransformation(m1, g1);
true
gap> img:=ImageSetOfTransformation(g1);
[ 2, 4, 5, 6 ]
gap> ker:=KernelOfTransformation(g1);
[ [ 1, 2 ], [ 3, 4 ], [ 5 ], [ 6 ] ]
gap> ForAny(MonoidOrbit(m1, img), x-> IsTransversal(ker, x));
true
gap> IsRegularTransformation(m1, g2);
false
gap> IsRegularTransformation(FullTransformationSemigroup(6), g2);
true
</pre></td></tr></table>
<p><a id="X863216CB7AF88BED" name="X863216CB7AF88BED"></a></p>
<h5>2.2-5 IndexPeriodOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IndexPeriodOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the minimum numbers <code class="code">m, r</code> such that <code class="code">f^(m+r)=f^m</code>; known as the <em>index</em> and <em>period</em> of the transformation.</p>
<table class="example">
<tr><td><pre>
gap> f:=Transformation( [ 3, 4, 4, 6, 1, 3, 3, 7, 1 ] );;
gap> IndexPeriodOfTransformation(f);
[ 2, 3 ]
gap> f^2=f^5;
true
</pre></td></tr></table>
<p><a id="X84E1A41F84B70DBB" name="X84E1A41F84B70DBB"></a></p>
<h5>2.2-6 SmallestIdempotentPower</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SmallestIdempotentPower</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the least natural number <code class="code">n</code> such that the transformation <code class="code">f^n</code> is an idempotent.</p>
<table class="example">
<tr><td><pre>
gap> t:=Transformation( [ 6, 7, 4, 1, 7, 4, 6, 1, 3, 4 ] );;
gap> SmallestIdempotentPower(t);
6
gap> t:=Transformation( [ 6, 6, 6, 2, 7, 1, 5, 3, 10, 6 ] );;
gap> SmallestIdempotentPower(t);
4
</pre></td></tr></table>
<p><a id="X846B9EBB86A69BDC" name="X846B9EBB86A69BDC"></a></p>
<h5>2.2-7 <span class="Heading">InversesOfTransformation</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> InversesOfTransformation</code>( <var class="Arg">S, f</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">> InversesOfTransformationNC</code>( <var class="Arg">S, f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of the inverses of the transformation <code class="code">f</code> in the transformation semigroup <code class="code">S</code>. The function <code class="code">InversesOfTransformationNC</code> does not check that <code class="code">f</code> is an element of <code class="code">S</code>.</p>
<table class="example">
<tr><td><pre>
gap> S:=Semigroup([ Transformation( [ 3, 1, 4, 2, 5, 2, 1, 6, 1 ] ),
Transformation( [ 5, 7, 8, 8, 7, 5, 9, 1, 9 ] ),
Transformation( [ 7, 6, 2, 8, 4, 7, 5, 8, 3 ] ) ]);;
gap> f:=Transformation( [ 3, 1, 4, 2, 5, 2, 1, 6, 1 ] );;
gap> InversesOfTransformationNC(S, f);
[ ]
gap> IsRegularTransformation(S, f);
false
gap> f:=Transformation( [ 1, 9, 7, 5, 5, 1, 9, 5, 1 ] );;
gap> inv:=InversesOfTransformation(S, f);
[ Transformation( [ 1, 5, 1, 1, 5, 1, 3, 1, 2 ] ),
Transformation( [ 1, 5, 1, 2, 5, 1, 3, 2, 2 ] ),
Transformation( [ 1, 2, 3, 5, 5, 1, 3, 5, 2 ] ) ]
gap> IsRegularTransformation(S, f);
true
</pre></td></tr></table>
<p><a id="X80B8EF3D78E711BB" name="X80B8EF3D78E711BB"></a></p>
<h4>2.3 <span class="Heading">Changing Representation</span></h4>
<p><a id="X7F867C337B18D84E" name="X7F867C337B18D84E"></a></p>
<h5>2.3-1 AsBooleanMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AsBooleanMatrix</code>( <var class="Arg">f[, n]</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the transformation or permutation <code class="code">f</code> represented as an <code class="code">n</code> by <code class="code">n</code> Boolean matrix where <code class="code">i,f(i)</code>th entries equal <code class="code">1</code> and all other entries are <code class="code">0</code>.</p>
<p>If <code class="code">f</code> is a transformation, then <code class="code">n</code> is the size of the domain of <code class="code">f</code>.</p>
<p>If <code class="code">f</code> is a permutation, then <code class="code">n</code> is the number of points moved by <code class="code">f</code>.</p>
<table class="example">
<tr><td><pre>
gap> t:=Transformation( [ 4, 2, 2, 1 ] );;
gap> AsBooleanMatrix(t);
[ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ] ]
gap> t:=(1,4,5);;
gap> AsBooleanMatrix(t);
[ [ 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 0 ] ]
gap> AsBooleanMatrix(t,3);
fail
gap> AsBooleanMatrix(t,5);
[ [ 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 0 ] ]
gap> AsBooleanMatrix(t,6);
[ [ 0, 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ] ]
</pre></td></tr></table>
<p><a id="X7A249D7781F31C22" name="X7A249D7781F31C22"></a></p>
<h5>2.3-2 AsPermOfRange</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AsPermOfRange</code>( <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>converts a transformation <code class="code">x</code> that is a permutation of its image into that permutation.</p>
<table class="example">
<tr><td><pre>
gap> t:=Transformation([1,2,9,9,9,8,8,8,4]);
Transformation( [ 1, 2, 9, 9, 9, 8, 8, 8, 4 ] )
gap> AsPermOfRange(t);
(4,9)
gap> t*last;
Transformation( [ 1, 2, 4, 4, 4, 8, 8, 8, 9 ] )
gap> AsPermOfRange(last);
()
</pre></td></tr></table>
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