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<div class="ChapSects"><a href="chap53.html#X860026B880BCB2A5">53 <span class="Heading">Transformations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7CF9291C7CC42340">53.1 <span class="Heading">The family and categories of transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7B6259467974FB70">53.1-1 IsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7A6747CE85F2E6EA">53.1-2 IsTransformationCollection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7E58AFA1832FF064">53.1-3 TransformationFamily</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X80F3086F87E93DF8">53.2 <span class="Heading">Creating transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X86ADBDE57A20E323">53.2-1 Transformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8040642687531E7F">53.2-2 Transformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7E82EBD68455EE4A">53.2-3 TransformationByImageAndKernel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X85D1071484CE004C">53.2-4 Idempotent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7C2A3FC9782F2099">53.2-5 TransformationOp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7D6FCC417DE86CD1">53.2-6 TransformationNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8475448F87E8CB8A">53.2-7 <span class="Heading">RandomTransformation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8268A58685BEFD6F">53.2-8 IdentityTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7F1E4B5184210D2B">53.2-9 ConstantTransformation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7F81A18B813C9DF0">53.3 <span class="Heading">Changing the representation of a transformation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7C5360B2799943F3">53.3-1 AsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X846A6F6B7B715188">53.3-2 RestrictedTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8708AE247F5B129B">53.3-3 PermutationOfImage</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X812CEC008609A8A2">53.4 <span class="Heading">Operators for transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X83DBA2A18719EFA8">53.4-1 PermLeftQuoTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8275DFAA8270BB59">53.4-2 IsInjectiveListTrans</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X834A313B7DAF06D5">53.4-3 ComponentTransformationInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X82F5DEEC837B60A3">53.4-4 PreImagesOfTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X86DE4F7A7C535820">53.5 <span class="Heading">Attributes for transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X78A209C87CF0E32B">53.5-1 DegreeOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7AEC9E6687B3505A">53.5-2 ImageListOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X839A6D6082A21D1F">53.5-3 ImageSetOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X818EBB167C7EA37B">53.5-4 RankOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X844F00F982D5BD3C">53.5-5 MovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7FA6A4B57FDA003D">53.5-6 NrMovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X86C0DDDC7881273A">53.5-7 SmallestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8383A7727AC97724">53.5-8 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7CCFE27E83676572">53.5-9 SmallestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7E7172567C3A3E63">53.5-10 LargestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8083794579274E87">53.5-11 FlatKernelOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X80FCB5048789CF75">53.5-12 KernelOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X860306EB7FAAD2D4">53.5-13 InverseOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7BB9DB6E8558356D">53.5-14 Inverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X863216CB7AF88BED">53.5-15 IndexPeriodOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X85FE9F20810BCC70">53.5-16 SmallestIdempotentPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X858E944481F6B591">53.5-17 ComponentsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X8640AE1C79201470">53.5-18 NrComponentsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X784650B583CEAF7D">53.5-19 ComponentRepsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7EAA15557D55D93B">53.5-20 CyclesOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X786EB02A829260DB">53.5-21 CycleTransformationInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X845869E0815A6AA6">53.5-22 LeftOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7F19C9C77F9F8981">53.5-23 TrimTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X810D23017A5527B7">53.6 <span class="Heading">Displaying transformations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7B51CE257B814B09">53.7 <span class="Heading">Semigroups of transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7EAF835D7FE4026F">53.7-1 IsTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7EA699C687952544">53.7-2 DegreeOfTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X7D2B0685815B4053">53.7-3 FullTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X85C58E1E818C838C">53.7-4 IsFullTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X78F29C817CF6827F">53.7-5 IsomorphismTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap53.html#X820ECE00846E480F">53.7-6 AntiIsomorphismTransformationSemigroup</a></span>
</div></div>
</div>
<h3>53 <span class="Heading">Transformations</span></h3>
<p>This chapter describes the functions in <strong class="pkg">GAP</strong> for transformations.</p>
<p>A <em>transformation</em> in <strong class="pkg">GAP</strong> is simply a function from the positive integers to the positive integers. Transformations are to semigroup theory what permutations are to group theory, in the sense that every semigroup can be realised as a semigroup of transformations. In <strong class="pkg">GAP</strong> transformation semigroups are always finite, and so only finite semigroups can be realised in this way.</p>
<p>A transformation in <strong class="pkg">GAP</strong> acts on the positive integers (up to some architecture dependent limit) on the right. The image of a point <code class="code">i</code> under a transformation <code class="code">f</code> is expressed as <code class="code">i ^ f</code> in <strong class="pkg">GAP</strong>. This action is also implemented by the function <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>). If <code class="code">i ^ f</code> is different from <code class="code">i</code>, then <code class="code">i</code> is <em>moved</em> by <em>f</em> and otherwise it is <em>fixed</em> by <code class="code">f</code>. Transformations in <strong class="pkg">GAP</strong> are created using the operations described in Section <a href="chap53.html#X80F3086F87E93DF8"><span class="RefLink">53.2</span></a>.</p>
<p>The <em>degree</em> of a transformation <code class="code">f</code> is usually defined as the largest positive integer where <code class="code">f</code> is defined. In previous versions of <strong class="pkg">GAP</strong>, transformations were only defined on positive integers less than their degree, it was only possible to multiply transformations of equal degree, and a transformation did not act on any point exceeding its degree. Starting with version 4.7 of <strong class="pkg">GAP</strong>, transformations behave more like permutations, in that they fix unspecified points and it is possible to multiply arbitrary transformations; see Chapter <a href="chap42.html#X80F808307A2D5AB8"><span class="RefLink">42</span></a>. The definition of the degree of a transformation <code class="code">f</code> in the current version of <strong class="pkg">GAP</strong> is the largest value <code class="code">n</code> such that <code class="code">n ^ f <> n</code> or <code class="code">i ^ f = n</code> for some <code class="code">i <> n</code>. Equivalently, the degree of a transformation is the least value <code class="code">n</code> such that <code class="code">[ n + 1, n + 2, ... ]</code> is fixed pointwise by <code class="code">f</code>.</p>
<p>The transformations of a given degree belong to the full transformation semigroup of that degree; see <code class="func">FullTransformationSemigroup</code> (<a href="chap53.html#X7D2B0685815B4053"><span class="RefLink">53.7-3</span></a>). Transformation semigroups are hence subsemigroups of the full transformation semigroup.</p>
<p>It is possible to use transformations in <strong class="pkg">GAP</strong> without reference to the degree, much as it is possible to use permutations in this way. However, for backwards compatibility, and because it is sometimes useful, it is possible to access the degree of a transformation using <code class="func">DegreeOfTransformation</code> (<a href="chap53.html#X78A209C87CF0E32B"><span class="RefLink">53.5-1</span></a>). Certain attributes of transformations are also calculated with respect to the degree, such as the rank, image set, or kernel (these values can also be calculated with respect to any positive integer). So, it is possible to ignore the degree of a transformation if you prefer to think of transformations as acting on the positive integers in a similar way to permutations. For example, this approach is used in the <strong class="pkg">FR</strong> package. It is also possible to think of transformations as only acting on the positive integers not exceeding their degree. For example, this was the approach formerly used in <strong class="pkg">GAP</strong> and it is also useful in the <strong class="pkg">Semigroups</strong> package.</p>
<p>Transformations are displayed, by default, using the list <code class="code">[ 1 ^ f .. n ^ f ]</code> where <code class="code">n</code> is the degree of <code class="code">f</code>. This behaviour differs from that of versions of <strong class="pkg">GAP</strong> earlier than 4.7. See Section <a href="chap53.html#X810D23017A5527B7"><span class="RefLink">53.6</span></a> for more information.</p>
<p>The <em>rank</em> of a transformation on the positive integers up to <code class="code">n</code> is the number of distinct points in <code class="code">[ 1 ^ f .. n ^ f ]</code>. The <em>kernel</em> of a transformation <code class="code">f</code> on <code class="code">[ 1 .. n ]</code> is the equivalence relation on <code class="code">[ 1 .. n ]</code> consisting of those pairs <code class="code">(i, j)</code> of positive integers such that <code class="code">i ^ f = j ^ f</code>. The kernel of a transformation is represented in two ways: as a partition of <code class="code">[ 1 .. n ]</code> or as the image list of a transformation <code class="code">g</code> such that the kernel of <code class="code">g</code> on <code class="code">[ 1 .. n ]</code> equals the kernel of <code class="code">f</code> and <code class="code">j ^ g = i</code> for all <code class="code">j</code> in <code class="code">i</code>th class. The latter is referred to as the <em>flat kernel</em> of <code class="code">f</code>. For any given transformation <code class="code">f</code> and value <code class="code">n</code>, there is a unique transformation <code class="code">g</code> with this property.</p>
<p>A <em>functional digraph</em> is a directed graph where every vertex has out-degree <span class="SimpleMath">1</span>. A transformation <var class="Arg">f</var> can be thought of as a functional digraph with vertices the positive integers and edges from <code class="code">i</code> to <code class="code">i ^ f</code> for every <code class="code">i</code>. A <em>component</em> of a transformation is defined as a component of the corresponding functional digraph. More specifically, <code class="code">i</code> and <code class="code">j</code> are in the same component if and only if there are <span class="SimpleMath">i = v_0, v_1, ..., v_n = j</span> such that either <span class="SimpleMath">v_k+1=v_k^f</span> or <span class="SimpleMath">v_k=v_k+1^f</span> for all <span class="SimpleMath">k</span>. A <em>cycle</em> of a transformation is defined as a cycle (or strongly connected component) of the corresponding functional digraph. More specifically, <code class="code">i</code> belongs to a cycle of <var class="Arg">f</var> if there are <span class="SimpleMath">i=v_0, v_1, ..., v_n=i</span> such that either <span class="SimpleMath">v_k+1=v_k^f</span> or <span class="SimpleMath">v_k=v_k+1^f</span> for all <span class="SimpleMath">k</span>.</p>
<p>Internally, <strong class="pkg">GAP</strong> stores a transformation <code class="code">f</code> as a list consisting of the images <code class="code">i ^ f</code> for all values of <code class="code">i</code> less than a value which is at least the degree of <code class="code">f</code> and which is determined at the time of the creation of <code class="code">f</code>. When the degree of a transformation <code class="code">f</code> is at most 65536, the images of points under <code class="code">f</code> are stored as 16-bit integers, the kernel and image set are subobjects of <code class="code">f</code> which are plain lists of <strong class="pkg">GAP</strong> integers. When the degree of <code class="code">f</code> is greater than 65536, the images of points under <code class="code">f</code> are stored as 32-bit integers; the kernel and image set are stored in the same way as before. A transformation belongs to <code class="code">IsTrans2Rep</code> if it is stored using 16-bit integers and to <code class="code">IsTrans4Rep</code> if it is stored using 32-bit integers.</p>
<p><a id="X7CF9291C7CC42340" name="X7CF9291C7CC42340"></a></p>
<h4>53.1 <span class="Heading">The family and categories of transformations</span></h4>
<p><a id="X7B6259467974FB70" name="X7B6259467974FB70"></a></p>
<h5>53.1-1 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>
<p>Every transformation in <strong class="pkg">GAP</strong> belongs to the category <code class="code">IsTransformation</code>. Basic operations for transformations are <code class="func">ImageListOfTransformation</code> (<a href="chap53.html#X7AEC9E6687B3505A"><span class="RefLink">53.5-2</span></a>), <code class="func">ImageSetOfTransformation</code> (<a href="chap53.html#X839A6D6082A21D1F"><span class="RefLink">53.5-3</span></a>), <code class="func">KernelOfTransformation</code> (<a href="chap53.html#X80FCB5048789CF75"><span class="RefLink">53.5-12</span></a>), <code class="func">FlatKernelOfTransformation</code> (<a href="chap53.html#X8083794579274E87"><span class="RefLink">53.5-11</span></a>), <code class="func">RankOfTransformation</code> (<a href="chap53.html#X818EBB167C7EA37B"><span class="RefLink">53.5-4</span></a>), <code class="func">DegreeOfTransformation</code> (<a href="chap53.html#X78A209C87CF0E32B"><span class="RefLink">53.5-1</span></a>), multiplication of two transformations via <code class="keyw">*</code>, and exponentiation with the first argument a positive integer <code class="code">i</code> and second argument a transformation <code class="code">f</code> where the result is the image <code class="code">i ^ f</code> of the point <code class="code">i</code> under <code class="code">f</code>.</p>
<p><a id="X7A6747CE85F2E6EA" name="X7A6747CE85F2E6EA"></a></p>
<h5>53.1-2 IsTransformationCollection</h5>
<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>Every collection of transformations belongs to the category <code class="code">IsTransformationCollection</code>. For example, transformation semigroups belong to <code class="code">IsTransformationCollection</code>.</p>
<p><a id="X7E58AFA1832FF064" name="X7E58AFA1832FF064"></a></p>
<h5>53.1-3 TransformationFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationFamily</code></td><td class="tdright">( family )</td></tr></table></div>
<p>The family of all transformations is <code class="code">TransformationFamily</code>.</p>
<p><a id="X80F3086F87E93DF8" name="X80F3086F87E93DF8"></a></p>
<h4>53.2 <span class="Heading">Creating transformations</span></h4>
<p>There are several ways of creating transformations in <strong class="pkg">GAP</strong>, which are described in this section.</p>
<p><a id="X86ADBDE57A20E323" name="X86ADBDE57A20E323"></a></p>
<h5>53.2-1 Transformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transformation</code>( <var class="Arg">list</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">‣ Transformation</code>( <var class="Arg">list</var>, <var class="Arg">func</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">‣ TransformationList</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p><code class="code">TransformationList</code> returns the transformation <code class="code">f</code> such that <code class="code">i ^ <var class="Arg">f</var> = <var class="Arg">list</var>[i]</code> if <code class="code">i</code> is between <code class="code">1</code> and the length of <var class="Arg">list</var> and <code class="code">i ^ <var class="Arg">f</var> = i</code> if <code class="code">i</code> is larger than the length of <var class="Arg">list</var>. An error will occur in <code class="code">TransformationList</code> if <var class="Arg">list</var> is not dense, if <var class="Arg">list</var> contains an element which is not a positive integer, or if <var class="Arg">list</var> contains an integer not in <code class="code">[ 1 .. Length( <var class="Arg">list</var> ) ]</code>.</p>
<p><code class="code">TransformationList</code> is the analogue in the context of transformations of <code class="func">PermList</code> (<a href="chap42.html#X78D611D17EA6E3BC"><span class="RefLink">42.5-2</span></a>). <code class="code">Transformation</code> is a synonym of <code class="code">TransformationList</code> when the argument is a list.</p>
<p>When the arguments are a list of positive integers <var class="Arg">list</var> and a function <var class="Arg">func</var>, <code class="code">Transformation</code> returns the transformation <code class="code">f</code> such that <code class="code"><var class="Arg">list</var>[i] ^ f = <var class="Arg">func</var>( <var class="Arg">list</var>[i] )</code> if <code class="code">i</code> is in the range <code class="code">[ 1 .. Length( <var class="Arg">list</var> ) ]</code> and <code class="code">f</code> fixes all other points.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "NotationForTransformations", "input" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 11, 10, 2, 11, 4, 4, 7, 6, 9, 10, 1, 11 ] );</span>
Transformation( [ 11, 10, 2, 11, 4, 4, 7, 6, 9, 10, 1, 11 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">f := TransformationList( [ 2, 3, 3, 1 ] );</span>
Transformation( [ 2, 3, 3, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "NotationForTransformations", "fr" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 10, 11 ], x -> x ^ 2 );</span>
<transformation: 1,2,3,4,5,6,7,8,9,100,121>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "NotationForTransformations", "input" );</span>
</pre></div>
<p><a id="X8040642687531E7F" name="X8040642687531E7F"></a></p>
<h5>53.2-2 Transformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transformation</code>( <var class="Arg">src</var>, <var class="Arg">dst</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">‣ TransformationListList</code>( <var class="Arg">src</var>, <var class="Arg">dst</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p>If <var class="Arg">src</var> and <var class="Arg">dst</var> are lists of positive integers of the same length, such that <var class="Arg">src</var> contains no element twice, then <code class="code">TransformationListList( <var class="Arg">src</var>, <var class="Arg">dst</var> )</code> returns a transformation <code class="code">f</code> such that <code class="code">src[i] ^ <var class="Arg">f</var> = dst[i]</code>. The transformation <var class="Arg">f</var> fixes all points larger than the maximum of the entries in <var class="Arg">src</var> and <var class="Arg">dst</var>.</p>
<p><code class="code">TransformationListList</code> is the analogue in the context of transformations of <code class="func">MappingPermListList</code> (<a href="chap42.html#X8087DCC780B9656A"><span class="RefLink">42.5-3</span></a>). <code class="code">Transformation</code> is a synonym of <code class="code">TransformationListList</code> when its arguments are two lists of positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Transformation( [ 10, 11 ],[ 11, 12 ] );</span>
Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 12 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationListList( [ 1, 2, 3 ], [ 4, 5, 6 ] );</span>
Transformation( [ 4, 5, 6, 4, 5, 6 ] )
</pre></div>
<p><a id="X7E82EBD68455EE4A" name="X7E82EBD68455EE4A"></a></p>
<h5>53.2-3 TransformationByImageAndKernel</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationByImageAndKernel</code>( <var class="Arg">im</var>, <var class="Arg">ker</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation or <code class="keyw">fail</code>.</p>
<p>This operation returns the transformation <code class="code">f</code> where <code class="code">i ^ f = <var class="Arg">im</var>[<var class="Arg">ker</var>[i]]</code> for <code class="code">i</code> in the range <code class="code">[ 1 .. Length( <var class="Arg">ker</var> ) ]</code>. This transformation has flat kernel equal to <var class="Arg">ker</var> and image set equal to <code class="code">Set( <var class="Arg">im</var> )</code>.</p>
<p>The argument <var class="Arg">im</var> should be a duplicate free list of positive integers and <var class="Arg">ker</var> should be the flat kernel of a transformation with rank equal to the length of <var class="Arg">im</var>. If the arguments do not fulfil these conditions, then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationByImageAndKernel( [ 8, 1, 3, 4 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 3, 1, 2, 1, 2, 4 ] );</span>
Transformation( [ 8, 1, 3, 8, 1, 8, 1, 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationByImageAndKernel( [ 1, 3, 8, 4 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 3, 1, 2, 1, 2, 4 ] );</span>
Transformation( [ 1, 3, 8, 1, 3, 1, 3, 4 ] )
</pre></div>
<p><a id="X85D1071484CE004C" name="X85D1071484CE004C"></a></p>
<h5>53.2-4 Idempotent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Idempotent</code>( <var class="Arg">im</var>, <var class="Arg">ker</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation or <code class="keyw">fail</code>.</p>
<p><code class="func">Idempotent</code> returns the idempotent transformation with image set <var class="Arg">im</var> and flat kernel <var class="Arg">ker</var> if such a transformation exists and <code class="keyw">fail</code> if it does not. More specifically, a transformation is returned when the argument <var class="Arg">im</var> is a set of positive integers and <var class="Arg">ker</var> is the flat kernel of a transformation with rank equal to the length of <var class="Arg">im</var> and where <var class="Arg">im</var> has one element in every class of the kernel corresponding to <var class="Arg">ker</var>.</p>
<p>Note that this is function does not always return the same transformation as <code class="code">TransformationByImageAndKernel</code> with the same arguments.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Idempotent( [ 2, 4, 6, 7, 8, 10, 11 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 1, 3, 3, 4, 5, 1, 6, 6, 7, 5 ] );</span>
Transformation( [ 8, 2, 8, 4, 4, 6, 7, 8, 10, 10, 11, 7 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationByImageAndKernel( [ 2, 4, 6, 7, 8, 10, 11 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 1, 3, 3, 4, 5, 1, 6, 6, 7, 5 ] );</span>
Transformation( [ 2, 4, 2, 6, 6, 7, 8, 2, 10, 10, 11, 8 ] )
</pre></div>
<p><a id="X7C2A3FC9782F2099" name="X7C2A3FC9782F2099"></a></p>
<h5>53.2-5 TransformationOp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationOp</code>( <var class="Arg">obj</var>, <var class="Arg">list</var>[, <var class="Arg">func</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">‣ TransformationOpNC</code>( <var class="Arg">obj</var>, <var class="Arg">list</var>[, <var class="Arg">func</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation or <code class="keyw">fail</code>.</p>
<p><code class="func">TransformationOp</code> returns the transformation that corresponds to the action of the object <var class="Arg">obj</var> on the domain or list <var class="Arg">list</var> via the function <var class="Arg">func</var>. If the optional third argument <var class="Arg">func</var> is not specified, then the action <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) is used by default. Note that the returned transformation refers to the positions in <var class="Arg">list</var> even if <var class="Arg">list</var> itself consists of integers.</p>
<p>This function is the analogue in the context of transformations of <code class="func">Permutation</code> (<span class="RefLink">Reference: Permutation for a group, an action domain, etc.</span>).</p>
<p>If <var class="Arg">obj</var> does not map elements of <var class="Arg">list</var> into <var class="Arg">list</var>, then <code class="keyw">fail</code> is returned.</p>
<p><code class="func">TransformationOpNC</code> does not check that <var class="Arg">obj</var> maps elements of <var class="Arg">list</var> to elements of <var class="Arg">list</var> or that a transformation is defined by the action of <var class="Arg">obj</var> on <var class="Arg">list</var> via <var class="Arg">func</var>. This function should be used only with caution, and in situations where it is guaranteed that the arguments have the required properties.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 10, 2, 3, 10, 5, 10, 7, 2, 5, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationOp( f, [ 2, 3 ] );</span>
IdentityTransformation
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationOp( f, [ 1, 2, 3 ] );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">S := SemigroupByMultiplicationTable( [ [ 1, 1, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 2 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationOp( Elements( S )[1], S, OnRight );</span>
Transformation( [ 1, 1, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationOp( Elements( S )[3], S, OnRight );</span>
Transformation( [ 1, 1, 2 ] )
</pre></div>
<p><a id="X7D6FCC417DE86CD1" name="X7D6FCC417DE86CD1"></a></p>
<h5>53.2-6 TransformationNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransformationNumber</code>( <var class="Arg">m</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">‣ NumberTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation or a number.</p>
<p>These functions implement a bijection from the transformations with degree at most <var class="Arg">n</var> to the numbers <code class="code">[ 1 .. <var class="Arg">n</var> ^ <var class="Arg">n</var> ]</code>.</p>
<p>More precisely, if <var class="Arg">m</var> and <var class="Arg">n</var> are positive integers such that <var class="Arg">m</var> is at most <code class="code"><var class="Arg">n</var> ^ <var class="Arg">n</var></code>, then <code class="code">TransformationNumber</code> returns the <var class="Arg">m</var>th transformation with degree at most <var class="Arg">n</var>.</p>
<p>If <var class="Arg">f</var> is a transformation and <var class="Arg">n</var> is a positive integer, which is greater than or equal to the degree of <var class="Arg">f</var>, then <code class="code">NumberTransformation</code> returns the number in <code class="code">[ 1 .. <var class="Arg">n</var> ^ <var class="Arg">n</var> ]</code> that corresponds to <var class="Arg">f</var>. If the optional second argument <var class="Arg">n</var> is not specified, then the degree of <var class="Arg">f</var> is used by default.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 3, 3, 5, 3, 3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberTransformation( f, 5 );</span>
1613
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberTransformation( f, 10 );</span>
2242256790
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationNumber( 2242256790, 10 );</span>
Transformation( [ 3, 3, 5, 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformationNumber( 1613, 5 ); </span>
Transformation( [ 3, 3, 5, 3, 3 ] )</pre></div>
<p><a id="X8475448F87E8CB8A" name="X8475448F87E8CB8A"></a></p>
<h5>53.2-7 <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">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A random transformation.</p>
<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">RandomTransformation</code> returns a random transformation with degree at most <var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomTransformation( 6 ); </span>
Transformation( [ 2, 1, 2, 1, 1, 2 ] )</pre></div>
<p><a id="X8268A58685BEFD6F" name="X8268A58685BEFD6F"></a></p>
<h5>53.2-8 IdentityTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityTransformation</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Returns: The identity transformation.</p>
<p>Returns the identity transformation, which has degree <code class="code">0</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentityTransformation;</span>
IdentityTransformation
</pre></div>
<p><a id="X7F1E4B5184210D2B" name="X7F1E4B5184210D2B"></a></p>
<h5>53.2-9 ConstantTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstantTransformation</code>( <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p>This function returns a constant transformation <code class="code">f</code> such that <code class="code">i ^ f = <var class="Arg">n</var></code> for all <code class="code">i</code> less than or equal to <var class="Arg">m</var>, when <var class="Arg">n</var> and <var class="Arg">m</var> are positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConstantTransformation( 5, 1 );</span>
Transformation( [ 1, 1, 1, 1, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ConstantTransformation( 6, 4 );</span>
Transformation( [ 4, 4, 4, 4, 4, 4 ] )
</pre></div>
<p><a id="X7F81A18B813C9DF0" name="X7F81A18B813C9DF0"></a></p>
<h4>53.3 <span class="Heading">Changing the representation of a transformation</span></h4>
<p>It is possible that a transformation in <strong class="pkg">GAP</strong> can be represented as another type of object, or that another type of <strong class="pkg">GAP</strong> object can be represented as a transformation.</p>
<p>The operations <code class="func">AsPermutation</code> (<a href="chap42.html#X8353AB8987E35DF3"><span class="RefLink">42.5-6</span></a>) and <code class="func">AsPartialPerm</code> (<a href="chap54.html#X87EC67747B260E98"><span class="RefLink">54.4-2</span></a>) can be used to convert transformations into permutations or partial permutations, where appropriate. In this section we describe functions for converting other types of objects into transformations.</p>
<p><a id="X7C5360B2799943F3" name="X7C5360B2799943F3"></a></p>
<h5>53.3-1 AsTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p><code class="code">AsTransformation</code> returns the permutation, transformation, partial permutation or binary relation <var class="Arg">f</var> as a transformation.</p>
<dl>
<dt><strong class="Mark">for permutations</strong></dt>
<dd><p>If <var class="Arg">f</var> is a permutation and <var class="Arg">n</var> is a non-negative integer, then <code class="code">AsTransformation( <var class="Arg">f</var>, <var class="Arg">n</var> )</code> returns the transformation <code class="code">g</code> such that <code class="code">i ^ g = i ^ f</code> for all <code class="code">i</code> in the range <code class="code">[ 1 .. <var class="Arg">n</var> ]</code>.</p>
<p>If no non-negative integer <var class="Arg">n</var> is specified, then the largest moved point of <var class="Arg">f</var> is used as the value for <var class="Arg">n</var>; see <code class="func">LargestMovedPoint</code> (<a href="chap42.html#X84AA603987C94AC0"><span class="RefLink">42.3-2</span></a>).</p>
</dd>
<dt><strong class="Mark">for transformations</strong></dt>
<dd><p>If <var class="Arg">f</var> is a transformation and <var class="Arg">n</var> is a non-negative integer less than the degree of <var class="Arg">f</var> such that <var class="Arg">f</var> is a transformation of <code class="code">[ 1 .. <var class="Arg">n</var> ]</code>, then <code class="code">AsTransformation</code> returns the restriction of <var class="Arg">f</var> to <code class="code">[ 1 .. <var class="Arg">n</var> ]</code>.</p>
<p>If <var class="Arg">f</var> is a transformation and <var class="Arg">n</var> is not specified or is greater than or equal to the degree of <var class="Arg">f</var>, then <var class="Arg">f</var> is returned.</p>
</dd>
<dt><strong class="Mark">for partial permutations</strong></dt>
<dd><p>A partial permutation <var class="Arg">f</var> can be converted into a transformation <code class="code">g</code> as follows. The degree <code class="code">m</code> of <code class="code">g</code> is equal to the maximum of <var class="Arg">n</var>, the largest moved point of <var class="Arg">f</var> plus <code class="code">1</code>, and the largest image of a moved point plus <code class="code">1</code>. The transformation <code class="code">g</code> agrees with <var class="Arg">f</var> on the domain of <var class="Arg">f</var> and maps the points in <code class="code">[ 1 .. m ]</code>, which are not in the domain of <var class="Arg">f</var> to <code class="code">n</code>, i.e. <code class="code">i ^ g = i ^ <var class="Arg">f</var></code> for all <code class="code">i</code> in the domain of <var class="Arg">f</var>, <code class="code">i ^ g = n</code> for all <code class="code">i</code> in <code class="code">[ 1 .. n ]</code>, and <code class="code">i ^ g = i</code> for all <code class="code">i</code> greater than <var class="Arg">n</var>. <code class="code">AsTransformation( <var class="Arg">f</var> )</code> returns the transformation <code class="code">g</code> defined in the previous sentences.</p>
<p>If the optional argument <var class="Arg">n</var> is not present, then the default value of the maximum of the largest moved point and the largest image of a moved point of <var class="Arg">f</var> plus <code class="code">1</code> is used.</p>
</dd>
<dt><strong class="Mark">for binary relations</strong></dt>
<dd><p>In the case that <var class="Arg">f</var> is a binary relation, which defines a transformation, <code class="code">AsTransformation</code> returns that transformation.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 3, 5, 3, 4, 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( f, 5 );</span>
Transformation( [ 3, 5, 3, 4, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( f, 10 );</span>
Transformation( [ 3, 5, 3, 4, 1, 2 ] )
<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 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">f := PartialPerm( [ 1, 2, 3, 4, 5, 6 ], [ 6, 7, 1, 4, 3, 2 ] );</span>
[5,3,1,6,2,7](4)
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( f, 11 );</span>
Transformation( [ 6, 7, 1, 4, 3, 2, 11, 11, 11, 11, 11 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPartialPerm( last, DomainOfPartialPerm( f ) );</span>
[5,3,1,6,2,7](4)
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( f, 14 );</span>
Transformation( [ 6, 7, 1, 4, 3, 2, 14, 14, 14, 14, 14, 14, 14, 14 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPartialPerm( last, DomainOfPartialPerm( f ) );</span>
[5,3,1,6,2,7](4)
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( f );</span>
Transformation( [ 6, 7, 1, 4, 3, 2, 8, 8 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation( Transformation( [ 1, 1, 2 ] ), 0 );</span>
IdentityTransformation
</pre></div>
<p><a id="X846A6F6B7B715188" name="X846A6F6B7B715188"></a></p>
<h5>53.3-2 RestrictedTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedTransformation</code>( <var class="Arg">f</var>, <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p><code class="code">RestrictedTransformation</code> returns the new transformation <code class="code">g</code> such that <code class="code"> i ^ g = i ^ <var class="Arg">f</var></code> for all <code class="code">i</code> in <var class="Arg">list</var> and such that <code class="code">i ^ g = i</code> for all <code class="code">i</code> not in <var class="Arg">list</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 10, 5, 9, 10, 9, 6, 3, 8, 4, 6, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RestrictedTransformation( f, [ 1, 2, 3, 10, 11, 12 ] );</span>
Transformation( [ 2, 10, 5, 4, 5, 6, 7, 8, 9, 4, 6, 5 ] )</pre></div>
<p><a id="X8708AE247F5B129B" name="X8708AE247F5B129B"></a></p>
<h5>53.3-3 PermutationOfImage</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermutationOfImage</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A permutation or <code class="keyw">fail</code>.</p>
<p>If the transformation <var class="Arg">f</var> is a permutation of the points in its image, then <code class="code">PermutationOfImage</code> returns this permutation. If <var class="Arg">f</var> does not permute its image, then <code class="keyw">fail</code> is returned.</p>
<p>If <var class="Arg">f</var> happens to be a permutation, then <code class="code">PermutationOfImage</code> with argument <var class="Arg">f</var> returns the same value as <code class="code">AsPermutation</code> with argument <var class="Arg">f</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 5, 8, 3, 5, 8, 6, 2, 2, 7, 8 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermutationOfImage( f );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 8, 2, 10, 2, 4, 4, 7, 6, 9, 10 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermutationOfImage( f );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1, 3, 6, 6, 2, 10, 2, 3, 10, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermutationOfImage( f );</span>
(2,3,6,10,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 5, 2, 8, 4, 1, 8, 10, 3, 5, 7 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermutationOfImage( f );</span>
(1,5)(3,8)(7,10)
</pre></div>
<p><a id="X812CEC008609A8A2" name="X812CEC008609A8A2"></a></p>
<h4>53.4 <span class="Heading">Operators for transformations</span></h4>
<dl>
<dt><strong class="Mark"><code class="code"><var class="Arg">i</var> ^ <var class="Arg">f</var></code></strong></dt>
<dd><p>returns the image of the positive integer <var class="Arg">i</var> under the transformation <var class="Arg">f</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">f</var> ^ <var class="Arg">g</var></code></strong></dt>
<dd><p>returns <code class="code"><var class="Arg">g</var> ^ -1 * <var class="Arg">f</var> * <var class="Arg">g</var></code> when <var class="Arg">f</var> is a transformation and <var class="Arg">g</var> is a permutation <code class="func">\^</code> (<span class="RefLink">Reference: ^</span>). This operation requires essentially the same number of steps as multiplying a transformation by a permutation, which is approximately one third of the number required to first invert <var class="Arg">g</var>, take the product with <var class="Arg">f</var>, and then the product with <var class="Arg">g</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">f</var> * <var class="Arg">g</var></code></strong></dt>
<dd><p>returns the composition of <var class="Arg">f</var> and <var class="Arg">g</var> when <var class="Arg">f</var> and <var class="Arg">g</var> are transformations or permutations. The product of a permutation and a transformation is returned as a transformation.</p>
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">f</var> / <var class="Arg">g</var></code></strong></dt>
<dd><p>returns <code class="code"><var class="Arg">f</var> * <var class="Arg">g</var> ^ -1</code> when <var class="Arg">f</var> is a transformation and <var class="Arg">g</var> is a permutation. This operation requires essentially the same number of steps as multiplying a transformation by a permutation, which is approximately half the number required to first invert <var class="Arg">g</var> and then take the product with <var class="Arg">f</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">LQUO( <var class="Arg">g</var>, <var class="Arg">f</var> )</code></strong></dt>
<dd><p>returns <code class="code"><var class="Arg">g</var> ^ -1 * <var class="Arg">f</var></code> when <var class="Arg">f</var> is a transformation and <var class="Arg">g</var> is a permutation. This operation uses essentially the same number of steps as multiplying a transformation by a permutation, which is approximately half the number required to first invert <var class="Arg">g</var> and then take the product with <var class="Arg">f</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">f</var> < <var class="Arg">g</var></code></strong></dt>
<dd><p>returns <code class="keyw">true</code> if the image list of <var class="Arg">f</var> is lexicographically less than the image list of <var class="Arg">g</var> and <code class="keyw">false</code> if it is not.</p>
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">f</var> = <var class="Arg">g</var></code></strong></dt>
<dd><p>returns <code class="keyw">true</code> if the transformation <var class="Arg">f</var> equals the transformation <var class="Arg">g</var> and returns <code class="keyw">false</code> if it does not.</p>
</dd>
</dl>
<p><a id="X83DBA2A18719EFA8" name="X83DBA2A18719EFA8"></a></p>
<h5>53.4-1 PermLeftQuoTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermLeftQuoTransformation</code>( <var class="Arg">f</var>, <var class="Arg">g</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">‣ PermLeftQuoTransformationNC</code>( <var class="Arg">f</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A permutation.</p>
<p>Returns the permutation on the image set of <var class="Arg">f</var> induced by <code class="code"><var class="Arg">f</var> ^ -1 * <var class="Arg">g</var></code> when the transformations <var class="Arg">f</var> and <var class="Arg">g</var> have equal kernel and image set.</p>
<p><code class="code">PermLeftQuoTransformation</code> verifies that <var class="Arg">f</var> and <var class="Arg">g</var> have equal kernels and image sets, and returns an error if they do not. <code class="code">PermLeftQuoTransformationNC</code> does no checks.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 5, 6, 7, 1, 4, 3, 2, 7 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Transformation( [ 5, 7, 1, 6, 4, 3, 2, 1 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermLeftQuoTransformation( f, g );</span>
(1,6,7)
<span class="GAPprompt">gap></span> <span class="GAPinput">PermLeftQuoTransformation( g, f );</span>
(1,7,6)
</pre></div>
<p><a id="X8275DFAA8270BB59" name="X8275DFAA8270BB59"></a></p>
<h5>53.4-2 IsInjectiveListTrans</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInjectiveListTrans</code>( <var class="Arg">list</var>, <var class="Arg">obj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>The argument <var class="Arg">obj</var> should be a transformation or the list of images of a transformation and <var class="Arg">list</var> should be a list of positive integers. <code class="code">IsInjectiveListTrans</code> checks if <var class="Arg">obj</var> is injective on <var class="Arg">list</var>.</p>
<p>More precisely, if <var class="Arg">obj</var> is a transformation, then we define <code class="code">f := <var class="Arg">obj</var></code> and if <var class="Arg">obj</var> is the image list of a transformation we define <code class="code">f := Transformation( <var class="Arg">obj</var> )</code>. <code class="code">IsInjectiveListTrans</code> returns <code class="keyw">true</code> if <code class="code">f</code> is injective on <var class="Arg">list</var> and <code class="keyw">false</code> if it is not. If <var class="Arg">list</var> is not duplicate free, then <code class="keyw">false</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveListTrans( [ 1, 5 ], f );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveListTrans( [ 5, 1 ], f );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveListTrans( [ 5, 1, 5, 1, 1, ], f );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveListTrans( [ 5, 1, 2, 3 ], [ 1, 2, 3, 4, 5 ] );</span>
true
</pre></div>
<p><a id="X834A313B7DAF06D5" name="X834A313B7DAF06D5"></a></p>
<h5>53.4-3 ComponentTransformationInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComponentTransformationInt</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of positive integers.</p>
<p>If <var class="Arg">f</var> is a transformation and <var class="Arg">n</var> is a positive integer, then <code class="code">ComponentTransformationInt</code> returns those elements <code class="code">i</code> such that <code class="code"><var class="Arg">n</var> ^ <var class="Arg">f</var> ^ j = i</code> for some positive integer <code class="code">j</code>, i.e. the elements of the component of <var class="Arg">f</var> containing <var class="Arg">n</var> that can be obtained by applying powers of <var class="Arg">f</var> to <var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 2, 8, 4, 7, 5, 8, 3, 5, 8 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentTransformationInt( f, 1 );</span>
[ 1, 6, 5, 7, 8, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentTransformationInt( f, 12 );</span>
[ 12 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentTransformationInt( f, 5 ); </span>
[ 5, 7, 8, 3 ]
</pre></div>
<p><a id="X82F5DEEC837B60A3" name="X82F5DEEC837B60A3"></a></p>
<h5>53.4-4 PreImagesOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreImagesOfTransformation</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A set of positive integers.</p>
<p>Returns the preimages of the positive integer <var class="Arg">n</var> under the transformation <var class="Arg">f</var>, i.e. the positive integers <code class="code">i</code> such that <code class="code">i ^ <var class="Arg">f</var> = n</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesOfTransformation( f, 1 );</span>
[ 8, 9 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesOfTransformation( f, 3 );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesOfTransformation( f, 100 );</span>
[ 100 ]
</pre></div>
<p><a id="X86DE4F7A7C535820" name="X86DE4F7A7C535820"></a></p>
<h4>53.5 <span class="Heading">Attributes for transformations</span></h4>
<p>In this section we describe the functions available in <strong class="pkg">GAP</strong> for finding various properties and attributes of transformations.</p>
<p><a id="X78A209C87CF0E32B" name="X78A209C87CF0E32B"></a></p>
<h5>53.5-1 DegreeOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfTransformation</code>( <var class="Arg">f</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">‣ DegreeOfTransformationCollection</code>( <var class="Arg">coll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A positive integer.</p>
<p>The <em>degree</em> of a transformation <var class="Arg">f</var> is the largest value such that <code class="code">n ^ <var class="Arg">f</var> <> n</code> or <code class="code">i ^ <var class="Arg">f</var> = n</code> for some <code class="code">i <> n</code>. Equivalently, the degree of a transformation is the least value <code class="code">n</code> such that <code class="code">[ n + 1, n + 2, ... ]</code> is fixed pointwise by <var class="Arg">f</var>.</p>
<p>The degree of a collection of transformations <var class="Arg">coll</var> is the maximum degree of any transformation in <var class="Arg">coll</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformation( IdentityTransformation );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformationCollection(</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ Transformation( [ 1, 3, 4, 1 ] ), </span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation( [ 3, 1, 1, 3, 4 ] ), </span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation( [ 2, 4, 1, 2 ] ) ] );</span>
5
</pre></div>
<p><a id="X7AEC9E6687B3505A" name="X7AEC9E6687B3505A"></a></p>
<h5>53.5-2 ImageListOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageListOfTransformation</code>( <var class="Arg">f</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">‣ ListTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The list of images of a transformation.</p>
<p>Returns the list of images of <code class="code">[ 1 .. <var class="Arg">n</var> ]</code> under the transformation <var class="Arg">f</var>, which is <code class="code">[ 1 ^ <var class="Arg">f</var> .. <var class="Arg">n</var> ^ <var class="Arg">f</var> ]</code>. If the optional second argument <var class="Arg">n</var> is not present, then the degree of <var class="Arg">f</var> is used by default.</p>
<p>This is the analogue for transformations of <code class="func">ListPerm</code> (<a href="chap42.html#X7A9DCFD986958C1E"><span class="RefLink">42.5-1</span></a>) for permutations.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2 ,3, 4, 2, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageListOfTransformation( f );</span>
[ 2, 3, 4, 2, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageListOfTransformation( f, 10 );</span>
[ 2, 3, 4, 2, 4, 6, 7, 8, 9, 10 ]
</pre></div>
<p><a id="X839A6D6082A21D1F" name="X839A6D6082A21D1F"></a></p>
<h5>53.5-3 ImageSetOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageSetOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The set of images of the transformation.</p>
<p>Returns the set of points in the list of images of <code class="code">[ 1 .. <var class="Arg">n</var> ]</code> under <var class="Arg">f</var>, i.e. the sorted list of images with duplicates removed. If the optional second argument <var class="Arg">n</var> is not given, then the degree of <var class="Arg">f</var> is used.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 5, 6, 7, 1, 4, 3, 2, 7 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageSetOfTransformation( f );</span>
[ 1, 2, 3, 4, 5, 6, 7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageSetOfTransformation( f, 10 );</span>
[ 1, 2, 3, 4, 5, 6, 7, 9, 10 ]
</pre></div>
<p><a id="X818EBB167C7EA37B" name="X818EBB167C7EA37B"></a></p>
<h5>53.5-4 RankOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">list</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The rank of a transformation.</p>
<p>When the arguments are a transformation <var class="Arg">f</var> and a positive integer <var class="Arg">n</var>, <code class="code">RankOfTransformation</code> returns the size of the set of images of the transformation <var class="Arg">f</var> in the range <code class="code">[ 1 .. <var class="Arg">n</var> ]</code>. If the optional second argument <var class="Arg">n</var> is not specified, then the degree of <var class="Arg">f</var> is used.</p>
<p>When the arguments are a transformation <var class="Arg">f</var> and a list <var class="Arg">list</var> of positive integers, this function returns the size of the set of images of the transformation <var class="Arg">f</var> on <var class="Arg">list</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 8, 5, 8, 2, 2, 8, 4, 7, 3, 1 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageSetOfTransformation( f );</span>
[ 1, 2, 3, 4, 5, 7, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RankOfTransformation( f );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">RankOfTransformation( f, 100 ); </span>
97
<span class="GAPprompt">gap></span> <span class="GAPinput">RankOfTransformation( f, [ 2, 5, 8 ] );</span>
3
</pre></div>
<p><a id="X844F00F982D5BD3C" name="X844F00F982D5BD3C"></a></p>
<h5>53.5-5 MovedPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MovedPoints</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MovedPoints</code>( <var class="Arg">coll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A set of positive integers.</p>
<p>When the argument is a transformation, <code class="code">MovedPoints</code> returns the set of positive integers <code class="code">i</code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code>.</p>
<p><code class="code">MovedPoints</code> returns the set of points moved by some element of the collection of transformations <var class="Arg">coll</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 10, 1, 4, 6, 5, 1, 2, 3, 3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MovedPoints( f ); </span>
[ 1, 2, 3, 5, 6, 7, 8, 9, 10 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f := IdentityTransformation; </span>
IdentityTransformation
<span class="GAPprompt">gap></span> <span class="GAPinput">MovedPoints( f );</span>
[ ]
</pre></div>
<p><a id="X7FA6A4B57FDA003D" name="X7FA6A4B57FDA003D"></a></p>
<h5>53.5-6 NrMovedPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrMovedPoints</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrMovedPoints</code>( <var class="Arg">coll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A positive integer.</p>
<p>When the argument is a transformation,<code class="code">NrMovedPoints</code> returns the number of positive integers <code class="code">i</code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code>.</p>
<p><code class="code">MovedPoints</code> returns the number of points which are moved by at least one element of the collection of transformations <var class="Arg">coll</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 7, 1, 4, 3, 2, 7, 7, 6, 6, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints( f );</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints( IdentityTransformation );</span>
0
</pre></div>
<p><a id="X86C0DDDC7881273A" name="X86C0DDDC7881273A"></a></p>
<h5>53.5-7 SmallestMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestMovedPoint</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestMovedPoint</code>( <var class="Arg">coll</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: A positive integer or <code class="keyw">infinity</code>.</p>
<p><code class="code">SmallestMovedPoint</code> returns the smallest positive integer <code class="code">i</code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code> if such an <code class="code">i</code> exists. If <var class="Arg">f</var> is the identity transformation, then <code class="keyw">infinity</code> is returned.</p>
<p>If the argument is a collection of transformations <var class="Arg">coll</var>, then the smallest point which is moved by at least one element of <var class="Arg">coll</var> is returned, if such a point exists. If <var class="Arg">coll</var> only contains identity transformations, then <code class="code">SmallestMovedPoint</code> returns <code class="keyw">infinity</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullTransformationSemigroup( 5 ); </span>
<full transformation monoid of degree 5>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestMovedPoint( S ); </span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( IdentityTransformation );</span>
<trivial transformation group of degree 0 with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestMovedPoint( S );</span>
infinity
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1, 2, 3, 6, 6, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestMovedPoint( f );</span>
4
</pre></div>
<p><a id="X8383A7727AC97724" name="X8383A7727AC97724"></a></p>
<h5>53.5-8 LargestMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestMovedPoint</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestMovedPoint</code>( <var class="Arg">coll</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: A positive integer.</p>
<p><code class="code">LargestMovedPoint</code> returns the largest positive integers <code class="code">i</code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code> if such an <code class="code">i</code> exists. If <var class="Arg">f</var> is the identity transformation, then <code class="code">0</code> is returned.</p>
<p>If the argument is a collection of transformations <var class="Arg">coll</var>, then the largest point which is moved by at least one element of <var class="Arg">coll</var> is returned, if such a point exists. If <var class="Arg">coll</var> only contains identity transformations, then <code class="code">LargestMovedPoint</code> returns <code class="code">0</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullTransformationSemigroup( 5 ); </span>
<full transformation monoid of degree 5>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( S );</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( IdentityTransformation );</span>
<trivial transformation group of degree 0 with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( S );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1, 2, 3, 6, 6, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( f ); </span>
5
</pre></div>
<p><a id="X7CCFE27E83676572" name="X7CCFE27E83676572"></a></p>
<h5>53.5-9 SmallestImageOfMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestImageOfMovedPoint</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestImageOfMovedPoint</code>( <var class="Arg">coll</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: A positive integer or <code class="keyw">infinity</code>.</p>
<p><code class="code">SmallestImageOfMovedPoint</code> returns the smallest positive integer <code class="code">i ^ <var class="Arg">f</var></code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code> if such an <code class="code">i</code> exists. If <var class="Arg">f</var> is the identity transformation, then <code class="keyw">infinity</code> is returned.</p>
<p>If the argument is a collection of transformations <var class="Arg">coll</var>, then the smallest integer which is the image a point moved by at least one element of <var class="Arg">coll</var> is returned, if such a point exists. If <var class="Arg">coll</var> only contains identity transformations, then <code class="code">SmallestImageOfMovedPoint</code> returns <code class="keyw">infinity</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullTransformationSemigroup( 5 ); </span>
<full transformation monoid of degree 5>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestImageOfMovedPoint( S ); </span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( IdentityTransformation );</span>
<trivial transformation group of degree 0 with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestImageOfMovedPoint( S );</span>
infinity
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1, 2, 3, 6, 6, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestImageOfMovedPoint( f );</span>
6
</pre></div>
<p><a id="X7E7172567C3A3E63" name="X7E7172567C3A3E63"></a></p>
<h5>53.5-10 LargestImageOfMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestImageOfMovedPoint</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestImageOfMovedPoint</code>( <var class="Arg">coll</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: A positive integer.</p>
<p><code class="code">LargestImageOfMovedPoint</code> returns the largest positive integer <code class="code">i ^ <var class="Arg">f</var></code> such that <code class="code">i ^ <var class="Arg">f</var> <> i</code> if such an <code class="code">i</code> exists. If <var class="Arg">f</var> is the identity transformation, then <code class="code">0</code> is returned.</p>
<p>If the argument is a collection of transformations <var class="Arg">coll</var>, then the largest integer which is the image a point moved by at least one element of <var class="Arg">coll</var> is returned, if such a point exists. If <var class="Arg">coll</var> only contains identity transformations, then <code class="code">LargestImageOfMovedPoint</code> returns <code class="code">0</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullTransformationSemigroup( 5 ); </span>
<full transformation monoid of degree 5>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestImageOfMovedPoint( S );</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( IdentityTransformation );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestImageOfMovedPoint( S );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1, 2, 3, 6, 6, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestImageOfMovedPoint( f ); </span>
6
</pre></div>
<p><a id="X8083794579274E87" name="X8083794579274E87"></a></p>
<h5>53.5-11 FlatKernelOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FlatKernelOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The flat kernel of a transformation.</p>
<p>If the kernel classes of the transformation <var class="Arg">f</var> on <code class="code">[ 1 .. <var class="Arg">n</var> ]</code> are <span class="SimpleMath">K_1, dots, K_r</span>, then <code class="code">FlatKernelOfTransformation</code> returns a list <code class="code">L</code> such that <code class="code">L[i] = j</code> for all <code class="code">i</code> in <span class="SimpleMath">K_j</span>. For a given transformation and positive integer <var class="Arg">n</var>, there is a unique such list.</p>
<p>If the optional second argument <var class="Arg">n</var> is not present, then the degree of <var class="Arg">f</var> is used by defualt.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 10, 3, 7, 10, 1, 5, 9, 2, 6, 10 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FlatKernelOfTransformation( f );</span>
[ 1, 2, 3, 1, 4, 5, 6, 7, 8, 1 ]
</pre></div>
<p><a id="X80FCB5048789CF75" name="X80FCB5048789CF75"></a></p>
<h5>53.5-12 KernelOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>, <var class="Arg">bool</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The kernel of a transformation.</p>
<p>When the arguments are a transformation <var class="Arg">f</var>, a positive integer <var class="Arg">n</var>, and <code class="keyw">true</code>, <code class="code">KernelOfTransformation</code> returns the kernel of the transformation <var class="Arg">f</var> on <code class="code">[ 1 .. <var class="Arg">n</var> ]</code> as a set of sets of positive integers. If the argument <var class="Arg">bool</var> is <code class="keyw">false</code>, then only the non-singleton classes are returned.</p>
<p>The second and third arguments are optional, the default values are the degree of <var class="Arg">f</var> and <code class="keyw">true</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 11, 1, 12, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation( f );</span>
[ [ 1, 4 ], [ 2, 5 ], [ 3 ], [ 6, 7 ], [ 8, 10 ], [ 9 ], [ 11 ],
[ 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation( f, 5 );</span>
[ [ 1, 4 ], [ 2, 5 ], [ 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation( f, 5, false );</span>
[ [ 1, 4 ], [ 2, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation( f, 15 );</span>
[ [ 1, 4 ], [ 2, 5 ], [ 3 ], [ 6, 7 ], [ 8, 10 ], [ 9 ], [ 11 ],
[ 12 ], [ 13 ], [ 14 ], [ 15 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfTransformation( f, false ); </span>
[ [ 1, 4 ], [ 2, 5 ], [ 6, 7 ], [ 8, 10 ] ]
</pre></div>
<p><a id="X860306EB7FAAD2D4" name="X860306EB7FAAD2D4"></a></p>
<h5>53.5-13 InverseOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InverseOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p><code class="code">InverseOfTransformation</code> returns a semigroup inverse of the transformation <var class="Arg">f</var> in the full transformation semigroup. An <em>inverse</em> of <var class="Arg">f</var> is any transformation <code class="code">g</code> such that <code class="code"><var class="Arg">f</var> * g * <var class="Arg">f</var> = <var class="Arg">f</var></code> and <code class="code">g * <var class="Arg">f</var> * g = g</code>. Every transformation has at least one inverse.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := InverseOfTransformation( f );</span>
Transformation( [ 8, 1, 1, 1, 10, 2, 3, 1, 6, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">f * g * f;</span>
Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 1, 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">g * f * g;</span>
Transformation( [ 8, 1, 1, 1, 10, 2, 3, 1, 6, 1 ] )
</pre></div>
<p><a id="X7BB9DB6E8558356D" name="X7BB9DB6E8558356D"></a></p>
<h5>53.5-14 Inverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Inverse</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p>If the transformation <var class="Arg">f</var> is a bijection, then <code class="code">Inverse</code> or <code class="code"><var class="Arg">f</var> ^ -1</code> returns the inverse of <var class="Arg">f</var>. If <var class="Arg">f</var> is not a bijection, then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Transformation( [ 3, 8, 12, 1, 11, 9, 9, 4, 10, 5, 10, 6 ] ) ^ -1;</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">Transformation( [ 2, 3, 1 ] ) ^ -1;</span>
Transformation( [ 3, 1, 2 ] )
</pre></div>
<p><a id="X863216CB7AF88BED" name="X863216CB7AF88BED"></a></p>
<h5>53.5-15 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">( function )</td></tr></table></div>
<p>Returns: A pair of positive integers.</p>
<p>Returns the least positive integers <code class="code">m</code> and <code class="code">r</code> such that <code class="code"><var class="Arg">f</var> ^ (m + r) = <var class="Arg">f</var> ^ m</code>, which are known as the <em>index</em> and <em>period</em> of the transformation <var class="Arg">f</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 3, 4, 4, 6, 1, 3, 3, 7, 1 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriodOfTransformation( f ); </span>
[ 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f ^ 2 = f ^ 5; </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriodOfTransformation( IdentityTransformation );</span>
[ 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriodOfTransformation( Transformation( [ 1, 2, 1 ] ) );</span>
[ 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriodOfTransformation( Transformation( [ 1, 2, 3 ] ) );</span>
[ 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriodOfTransformation( Transformation( [ 1, 3, 2 ] ) );</span>
[ 1, 2 ]
</pre></div>
<p><a id="X85FE9F20810BCC70" name="X85FE9F20810BCC70"></a></p>
<h5>53.5-16 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: A positive integer.</p>
<p>This function returns the least positive integer <code class="code">n</code> such that the transformation <code class="code"><var class="Arg">f</var> ^ n</code> is an idempotent. The smallest idempotent power of <var class="Arg">f</var> is the least multiple of the period of <var class="Arg">f</var> that is greater than or equal to the index of <var class="Arg">f</var>; see <code class="func">IndexPeriodOfTransformation</code> (<a href="chap53.html#X863216CB7AF88BED"><span class="RefLink">53.5-15</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 7, 4, 1, 7, 4, 6, 1, 3, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestIdempotentPower( f );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 6, 6, 2, 7, 1, 5, 3, 10, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestIdempotentPower( f );</span>
2
</pre></div>
<p><a id="X858E944481F6B591" name="X858E944481F6B591"></a></p>
<h5>53.5-17 ComponentsOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComponentsOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of lists of positive integers.</p>
<p><code class="code">ComponentsOfTransformation</code> returns a list of the components of the transformation <var class="Arg">f</var>. Each component is a subset of <code class="code">[ 1 .. DegreeOfTransformation( f ) ]</code>, and the union of the components is <code class="code">[ 1 .. DegreeOfTransformation( f ) ]</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] );</span>
Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentsOfTransformation( f ); </span>
[ [ 1, 6, 4, 9 ], [ 2, 12, 3, 11, 5, 7, 10 ], [ 8 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f := AsTransformation( (1,8,2,4,11,5,10)(3,7)(9,12) );</span>
Transformation( [ 8, 4, 7, 11, 10, 6, 3, 2, 12, 1, 5, 9 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentsOfTransformation( f ); </span>
[ [ 1, 8, 2, 4, 11, 5, 10 ], [ 3, 7 ], [ 6 ], [ 9, 12 ] ]</pre></div>
<p><a id="X8640AE1C79201470" name="X8640AE1C79201470"></a></p>
<h5>53.5-18 NrComponentsOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrComponentsOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A positive integer.</p>
<p><code class="code">NrComponentsOfTransformation</code> returns the number of components of the transformation <var class="Arg">f</var> on the range <code class="code">[ 1 .. DegreeOfTransformation( <var class="Arg">f</var> ) ]</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] );</span>
Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">NrComponentsOfTransformation( f );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">f := AsTransformation( (1,8,2,4,11,5,10)(3,7)(9,12) );</span>
Transformation( [ 8, 4, 7, 11, 10, 6, 3, 2, 12, 1, 5, 9 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">NrComponentsOfTransformation( f );</span>
4
</pre></div>
<p><a id="X784650B583CEAF7D" name="X784650B583CEAF7D"></a></p>
<h5>53.5-19 ComponentRepsOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComponentRepsOfTransformation</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of lists of positive integers.</p>
<p><code class="code">ComponentRepsOfTransformation</code> returns the representatives, in the following sense, of the components of the transformation <var class="Arg">f</var>. For every <code class="code">i</code> in <code class="code">[ 1 .. DegreeOfTransformation( f ) ]</code> there exists a representative <code class="code">j</code> and a positive integer <code class="code">k</code> such that <code class="code">i ^ (<var class="Arg">f</var> ^ k) = j</code>. The representatives returned by <code class="code">ComponentRepsOfTransformation</code> are partitioned according to the component they belong to. <code class="code">ComponentRepsOfTransformation</code> returns the least number of representatives.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] );</span>
Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentRepsOfTransformation( f );</span>
[ [ 3, 10 ], [ 9 ], [ 8 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f := AsTransformation( (1,8,2,4,11,5,10)(3,7)(9,12) );</span>
Transformation( [ 8, 4, 7, 11, 10, 6, 3, 2, 12, 1, 5, 9 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentRepsOfTransformation( f );</span>
[ [ 1 ], [ 3 ], [ 6 ], [ 9 ] ]
</pre></div>
<p><a id="X7EAA15557D55D93B" name="X7EAA15557D55D93B"></a></p>
<h5>53.5-20 CyclesOfTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclesOfTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">list</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of lists of positive integers.</p>
<p>When the arguments of this function are a transformation <var class="Arg">f</var> and a list <var class="Arg">list</var>, it returns a list of the cycles of the components of <var class="Arg">f</var> containing any element of <var class="Arg">list</var>.</p>
<p>If the optional second argument is not present, then the range <code class="code">[ 1 .. DegreeOfTransformation( <var class="Arg">f</var> ) ]</code> is used as the default value for <var class="Arg">list</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] );</span>
Transformation( [ 6, 12, 11, 1, 7, 6, 2, 8, 4, 7, 5, 12 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclesOfTransformation( f ); </span>
[ [ 6 ], [ 12 ], [ 8 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclesOfTransformation( f, [ 1, 2, 4 ] ); </span>
[ [ 6 ], [ 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclesOfTransformation( f, [ 1 .. 17 ] );</span>
[ [ 6 ], [ 12 ], [ 8 ], [ 13 ], [ 14 ], [ 15 ], [ 16 ], [ 17 ] ]
</pre></div>
<p><a id="X786EB02A829260DB" name="X786EB02A829260DB"></a></p>
<h5>53.5-21 CycleTransformationInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleTransformationInt</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of positive integers.</p>
<p>If <var class="Arg">f</var> is a transformation and <var class="Arg">n</var> is a positive integer, then <code class="code">CycleTransformationInt</code> returns the cycle of the component of <var class="Arg">f</var> containing <var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 6, 2, 8, 4, 7, 5, 8, 3, 5, 8 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleTransformationInt( f, 1 );</span>
[ 8, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleTransformationInt( f, 12 );</span>
[ 12 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleTransformationInt( f, 5 ); </span>
[ 8, 3 ]
</pre></div>
<p><a id="X845869E0815A6AA6" name="X845869E0815A6AA6"></a></p>
<h5>53.5-22 LeftOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftOne</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightOne</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A transformation.</p>
<p><code class="code">LeftOne</code> returns an idempotent transformation <code class="code">e</code> such that the kernel (with respect to the degree of <var class="Arg">f</var>) of <code class="code">e</code> equals the kernel of the transformation <var class="Arg">f</var> and <code class="code">e * <var class="Arg">f</var> = f</code>.</p>
<p><code class="code">RightOne</code> returns an idempotent transformation <code class="code">e</code> such that the image set (with respect to the degree of <var class="Arg">f</var>) of <code class="code">e</code> equals the image set of <var class="Arg">f</var> and <code class="code"><var class="Arg">f</var> * e = f</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 11, 10, 2, 11, 4, 4, 7, 6, 9, 10, 1, 11 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := RightOne( f );</span>
Transformation( [ 1, 2, 2, 4, 4, 6, 7, 7, 9, 10, 11, 11 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdempotent( e );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">f * e = f;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">e := LeftOne( f );</span>
Transformation( [ 1, 2, 3, 1, 5, 5, 7, 8, 9, 2, 11, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">e * f = f; </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdempotent( e );</span>
true
</pre></div>
<p><a id="X7F19C9C77F9F8981" name="X7F19C9C77F9F8981"></a></p>
<h5>53.5-23 TrimTransformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrimTransformation</code>( <var class="Arg">f</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: Nothing.</p>
<p>It can happen that the internal representation of a transformation uses more memory than necessary. For example, this can happen when composing transformations where it is possible that the resulting transformation <var class="Arg">f</var> belongs to <code class="code">IsTrans4Rep</code> and stores its images as 32-bit integers, while none of its moved points exceeds 65536. The purpose of <code class="code">TrimTransformation</code> is to change the internal representation of such an <var class="Arg">f</var> to remove the trailing fixed points in the internal representation of <var class="Arg">f</var>.</p>
<p>If the optional second argument <var class="Arg">n</var> is provided, then the internal representation of <var class="Arg">f</var> is reduced to the images of the first <var class="Arg">n</var> positive integers. Please note that it must be the case that <code class="code">i ^ <var class="Arg">f</var> <= n</code> for all <code class="code">i</code> in the range <code class="code">[ 1 .. <var class="Arg">n</var> ]</code> otherwise the resulting object will not define a transformation.</p>
<p>If the optional second argument is not included, then the degree of <var class="Arg">f</var> is used by default.</p>
<p>The transformation <var class="Arg">f</var> is changed in-place, and nothing is returned by this function.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 1 .. 2 ^ 16 ], x -> x + 1 );</span>
<transformation on 65537 pts with rank 65536>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Transformation( [ 1 .. 2 ^ 16 + 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">function( x )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if x = 1 or x = 65537 then </span>
<span class="GAPprompt">></span> <span class="GAPinput"> return x; </span>
<span class="GAPprompt">></span> <span class="GAPinput"> else </span>
<span class="GAPprompt">></span> <span class="GAPinput"> return x - 1; </span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi; </span>
<span class="GAPprompt">></span> <span class="GAPinput">end );</span>
<transformation on 65536 pts with rank 65535>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := g * f;</span>
Transformation( [ 2, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformation( h ); IsTrans4Rep( h ); MemoryUsage( h );</span>
65537
true
262188
<span class="GAPprompt">gap></span> <span class="GAPinput">TrimTransformation( h ); h;</span>
Transformation( [ 2, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformation( h ); IsTrans4Rep( h ); MemoryUsage( h );</span>
2
false
44
</pre></div>
<p><a id="X810D23017A5527B7" name="X810D23017A5527B7"></a></p>
<h4>53.6 <span class="Heading">Displaying transformations</span></h4>
<p>It is possible to change the way that <strong class="pkg">GAP</strong> displays transformations using the user preferences <code class="code">TransformationDisplayLimit</code> and <code class="code">NotationForTransformations</code>; see Section <code class="func">UserPreference</code> (<a href="chap3.html#X7B0AD104839B6C3C"><span class="RefLink">3.2-3</span></a>) for more information about user preferences.</p>
<p>If <code class="code">f</code> is a transformation where the degree <code class="code">n</code> of <code class="code">f</code> exceeds the value of the user preference <code class="code">TransformationDisplayLimit</code>, then <code class="code">f</code> is displayed as:</p>
<div class="example"><pre><transformation on n pts with rank r></pre></div>
<p>where <code class="code">r</code> is the rank of <code class="code">f</code> relative to <code class="code">n</code>. The idea is to abbreviate the display of transformations defined on many points. The default value for the <code class="code">TransformationDisplayLimit</code> is <code class="code">100</code>.</p>
<p>If the degree of <code class="code">f</code> does not exceed the value of <code class="code">TransformationDisplayLimit</code>, then how <code class="code">f</code> is displayed depends on the value of the user preference <code class="code">NotationForTransformations</code>.</p>
<p>There are two possible values for <code class="code">NotationForTransformations</code>:</p>
<dl>
<dt><strong class="Mark">input</strong></dt>
<dd><p>With this option a transformation <var class="Arg">f</var> is displayed in as: <code class="code">Transformation( ImageListOfTransformation( <var class="Arg">f</var>, n ) )</code> where <code class="code">n</code> is the degree of <var class="Arg">f</var>. The only exception is the identity transformation, which is displayed as: <code class="code">IdentityTransformation</code>.</p>
</dd>
<dt><strong class="Mark">fr</strong></dt>
<dd><p>With this option a transformation <var class="Arg">f</var> is displayed in as: <code class="code"><transformation: ImageListOfTransformation( <var class="Arg">f</var>, n )></code> where <code class="code">n</code> is the largest moved point of <var class="Arg">f</var>. The only exception is the identity transformation, which is displayed as: <code class="code"><identity transformation></code>.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "TransformationDisplayLimit", 12 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 3, 8, 12, 1, 11, 9, 9, 4, 10, 5, 10, 6 ] );</span>
<transformation on 12 pts with rank 10>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "TransformationDisplayLimit", 100 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f;</span>
Transformation( [ 3, 8, 12, 1, 11, 9, 9, 4, 10, 5, 10, 6 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">SetUserPreference( "NotationForTransformations", "fr" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f;</span>
<transformation: 3,8,12,1,11,9,9,4,10,5,10,6>
</pre></div>
<p><a id="X7B51CE257B814B09" name="X7B51CE257B814B09"></a></p>
<h4>53.7 <span class="Heading">Semigroups of transformations</span></h4>
<p>As mentioned at the start of the chapter, every semigroup is isomorphic to a semigroup of transformations, and in this section we describe the functions in <strong class="pkg">GAP</strong> specific to transformation semigroups. For more information about semigroups in general see Chapter <a href="chap51.html#X8665D8737FDD5B10"><span class="RefLink">51</span></a>.</p>
<p>The <strong class="pkg">Semigroups</strong> package contains many additional functions and methods for computing with semigroups of transformations. In particular, <strong class="pkg">Semigroups</strong> contains more efficient methods than those available in the <strong class="pkg">GAP</strong> library (and in many cases more efficient than any other software) for creating semigroups of transformations, calculating their Green's classes, size, elements, group of units, minimal ideal, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and more. Since a transformation semigroup is also a transformation collection, there are special methods for <code class="func">MovedPoints</code> (<a href="chap53.html#X844F00F982D5BD3C"><span class="RefLink">53.5-5</span></a>), <code class="func">NrMovedPoints</code> (<a href="chap53.html#X7FA6A4B57FDA003D"><span class="RefLink">53.5-6</span></a>), <code class="func">LargestMovedPoint</code> (<a href="chap53.html#X8383A7727AC97724"><span class="RefLink">53.5-8</span></a>), <code class="func">SmallestMovedPoint</code> (<a href="chap53.html#X86C0DDDC7881273A"><span class="RefLink">53.5-7</span></a>), <code class="func">LargestImageOfMovedPoint</code> (<a href="chap53.html#X7E7172567C3A3E63"><span class="RefLink">53.5-10</span></a>), and <code class="func">SmallestImageOfMovedPoint</code> (<a href="chap53.html#X7CCFE27E83676572"><span class="RefLink">53.5-9</span></a>), when applied to a transformation semigroup.</p>
<p><a id="X7EAF835D7FE4026F" name="X7EAF835D7FE4026F"></a></p>
<h5>53.7-1 IsTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformationSemigroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformationMonoid</code>( <var class="Arg">obj</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>A <em>transformation semigroup</em> is simply a semigroup consisting of transformations. An object <var class="Arg">obj</var> is a transformation semigroup in <strong class="pkg">GAP</strong> if it satisfies <code class="func">IsSemigroup</code> (<a href="chap51.html#X7B412E5B8543E9B7"><span class="RefLink">51.1-1</span></a>) and <code class="func">IsTransformationCollection</code> (<a href="chap53.html#X7A6747CE85F2E6EA"><span class="RefLink">53.1-2</span></a>).</p>
<p>A <em>transformation monoid</em> is a monoid consisting of transformations. An object <var class="Arg">obj</var> is a transformation monoid in <strong class="pkg">GAP</strong> if it satisfies <code class="func">IsMonoid</code> (<a href="chap51.html#X861C523483C6248C"><span class="RefLink">51.2-1</span></a>) and <code class="func">IsTransformationCollection</code> (<a href="chap53.html#X7A6747CE85F2E6EA"><span class="RefLink">53.1-2</span></a>).</p>
<p>Note that it is possible for a transformation semigroup to have a multiplicative neutral element (i.e. an identity element) but not to satisfy <code class="code">IsTransformationMonoid</code>. For example,</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation( [ 2, 6, 7, 2, 6, 9, 9, 1, 1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( f, One( f ) );</span>
<commutative transformation monoid of degree 10 with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonoid( S );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransformationMonoid( S );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( </span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 3, 8, 1, 4, 5, 6, 7, 1, 10, 10 ] ), </span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 10 ] ) );</span>
<transformation semigroup of degree 10 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">One( S );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">MultiplicativeNeutralElement( S );</span>
Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 10 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonoid( S );</span>
false
</pre></div>
<p>In this example <code class="code">S</code> cannot be converted into a monoid using <code class="func">AsMonoid</code> (<a href="chap51.html#X7B22038F832B9C0F"><span class="RefLink">51.2-5</span></a>) since the <code class="func">One</code> (<a href="chap31.html#X8046262384895B2A"><span class="RefLink">31.10-2</span></a>) of any element in <code class="code">S</code> differs from the multiplicative neutral element.</p>
<p>For more details see <code class="func">IsMagmaWithOne</code> (<a href="chap35.html#X86071DE7835F1C7C"><span class="RefLink">35.1-2</span></a>).</p>
<p><a id="X7EA699C687952544" name="X7EA699C687952544"></a></p>
<h5>53.7-2 DegreeOfTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfTransformationSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A non-negative integer.</p>
<p>The <em>degree</em> of a transformation semigroup <var class="Arg">S</var> is just the maximum of the degrees of the elements of <var class="Arg">S</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 3, 8, 1, 4, 5, 6, 7, 1, 10, 10, 11 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 11 ] ) );</span>
<transformation semigroup of degree 10 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTransformationSemigroup( S );</span>
10
</pre></div>
<p><a id="X7D2B0685815B4053" name="X7D2B0685815B4053"></a></p>
<h5>53.7-3 FullTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullTransformationSemigroup</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">‣ FullTransformationMonoid</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The full transformation semigroup of degree <var class="Arg">n</var>.</p>
<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">FullTransformationSemigroup</code> returns the monoid consisting of all transformations with degree at most <var class="Arg">n</var>, called the <em>full transformation semigroup</em>.</p>
<p>The full transformation semigroup is regular, has <code class="code"><var class="Arg">n</var> ^ <var class="Arg">n</var></code> elements, and is generated by any set containing transformations that generate the symmetric group on <var class="Arg">n</var> points and any transformation of rank <code class="code"><var class="Arg">n</var> - 1</code>.</p>
<p><code class="code">FulTransformationMonoid</code> is a synonym for <code class="code">FullTransformationSemigroup</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FullTransformationSemigroup( 1234 ); </span>
<full transformation monoid of degree 1234>
</pre></div>
<p><a id="X85C58E1E818C838C" name="X85C58E1E818C838C"></a></p>
<h5>53.7-4 IsFullTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullTransformationSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullTransformationMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>If the transformation semigroup <var class="Arg">S</var> of degree <code class="code">n</code> contains every transformation of degree at most <code class="code">n</code>, then <code class="code">IsFullTransformationSemigroup</code> returns <code class="keyw">true</code> and otherwise it returns <code class="keyw">false</code>.</p>
<p><code class="code">IsFullTransformationMonoid</code> is a synonym of <code class="code">IsFullTransformationSemigroup</code>. It is common in the literature for the full transformation monoid to be referred to as the full transformation semigroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( AsTransformation( (1,3,4,2), 5 ), </span>
<span class="GAPprompt">></span> <span class="GAPinput"> AsTransformation( (1,3,5), 5 ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation( [ 1, 1, 2, 3, 4 ] ) );</span>
<transformation semigroup of degree 5 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullTransformationSemigroup( S );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S;</span>
<full transformation monoid of degree 5>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullTransformationMonoid( S );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullTransformationSemigroup( 5 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullTransformationSemigroup( S );</span>
true
</pre></div>
<p><a id="X78F29C817CF6827F" name="X78F29C817CF6827F"></a></p>
<h5>53.7-5 IsomorphismTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismTransformationSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismTransformationMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: An isomorphism to a transformation semigroup or monoid.</p>
<p>Returns an isomorphism from the finite semigroup <var class="Arg">S</var> to a transformation semigroup. For most types of objects in <strong class="pkg">GAP</strong> the degree of this transformation semigroup will be equal to the size of <var class="Arg">S</var> plus <code class="code">1</code>.</p>
<p>Let <code class="code"><var class="Arg">S</var> ^ 1</code> denote the monoid obtained from <var class="Arg">S</var> by adjoining an identity element. Then <var class="Arg">S</var> acts faithfully on <code class="code"><var class="Arg">S</var> ^ 1</code> by right multiplication, i.e. every element of <var class="Arg">S</var> describes a transformation on <code class="code">1, .. , |S| + 1</code>. The isomorphism from <var class="Arg">S</var> to the transformation semigroup described in this way is called the <em>right regular representation</em> of <var class="Arg">S</var>. In most cases, <code class="code">IsomorphismTransformationSemigroup</code> will return the right regular representation of <var class="Arg">S</var>.</p>
<p>As exceptions, if <var class="Arg">S</var> is a permutation group or a partial perm semigroup, then the elements of <var class="Arg">S</var> act naturally and faithfully by transformations on the values from <code class="code">1</code> to the largest moved point of <var class="Arg">S</var>.</p>
<p>If <var class="Arg">S</var> is a finitely presented semigroup, then the Todd-Coxeter approach will be attempted.</p>
<p><code class="code">IsomorphismTransformationMonoid</code> differs from <code class="code">IsomorphismTransformationSemigroup</code> only in that its range is a transformation monoid, and not only a semigroup, when the semigroup <var class="Arg">S</var> is a monoid.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( [ [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ^ 0 ] ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ [ Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3) ] ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( S );</span>
81
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTransformationSemigroup( S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := SymmetricInverseSemigroup( 4 );</span>
<symmetric inverse monoid of degree 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTransformationMonoid( S );</span>
MappingByFunction( <symmetric inverse monoid of degree 4>,
<transformation monoid of degree 5 with 4 generators>
, function( x ) ... end, <Operation "AsPartialPerm"> )
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( (1,2,3) );</span>
Group([ (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTransformationMonoid( G );</span>
MappingByFunction( Group([ (1,2,3) ]), <commutative transformation
monoid of degree 3 with 1 generator>
, function( x ) ... end, function( x ) ... end )</pre></div>
<p><a id="X820ECE00846E480F" name="X820ECE00846E480F"></a></p>
<h5>53.7-6 AntiIsomorphismTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AntiIsomorphismTransformationSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: An anti-isomorphism.</p>
<p>If <var class="Arg">S</var> is a semigroup, then <code class="code">AntiIsomorphismTransformationSemigroup</code> returns an anti-isomorphism from <var class="Arg">S</var> to a transformation semigroup. At present, the degree of the resulting transformation semigroup equals the size of <var class="Arg">S</var> plus <span class="SimpleMath">1</span>, and, consequently, this function is of limited use.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup( Transformation( [ 5, 5, 1, 1, 3 ] ), </span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation( [ 2, 4, 1, 5, 5 ] ) );</span>
<transformation semigroup of degree 5 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( S );</span>
172
<span class="GAPprompt">gap></span> <span class="GAPinput">AntiIsomorphismTransformationSemigroup( S );</span>
MappingByFunction( <transformation semigroup of size 172, degree 5
with 2 generators>, <transformation semigroup of degree 173 with 2
generators>, function( x ) ... end, function( x ) ... end )
</pre></div>
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