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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap31.html">[Previous Chapter]</a> <a href="chap33.html">[Next Chapter]</a> </div>
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<p><a id="X7C9734B880042C73" name="X7C9734B880042C73"></a></p>
<div class="ChapSects"><a href="chap32.html#X7C9734B880042C73">32 <span class="Heading">Mappings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X859A13548515A5D7">32.1 <span class="Heading">Direct Products and their Elements</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87FD9FE787023FF0">32.1-1 IsDirectProductElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X78F8A1168280E06D">32.1-2 DirectProductFamily</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7CF6FEFB8290D5CB">32.2 <span class="Heading">Creating Mappings</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X79D0D2F07A14D039">32.2-1 GeneralMappingByElements</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D55E1977ED70E01">32.2-2 <span class="Heading">MappingByFunction</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X865FC25A87D36F3D">32.2-3 InverseGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7BD2D5A87CD6B213">32.2-4 RestrictedInverseGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7ED1E4E27CCE2DCA">32.2-5 CompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86486B687B7077AC">32.2-6 CompositionMapping2</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7A926D167C3155F6">32.2-7 IsCompositionMappingRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87775B438008DCA5">32.2-8 ConstituentsCompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X795FF8DC785F110A">32.2-9 ZeroMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7EBAE0368470A603">32.2-10 IdentityMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86452F8587CBAEA0">32.2-11 <span class="Heading">Embedding</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8769E8DA80BC96C1">32.2-12 <span class="Heading">Projection</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X800014D683A81009">32.2-13 RestrictedMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E5A430D7F838F1C">32.3 <span class="Heading">Properties and Attributes of (General) Mappings</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X83C7494E828CC9C8">32.3-1 IsTotal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86D44C8A78BF1981">32.3-2 IsSingleValued</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7CC95EB282854385">32.3-3 IsMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7F065FD7822C0A12">32.3-4 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X784ECE847E005B8F">32.3-5 IsSurjective</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X878F56AB7B342767">32.3-6 IsBijective</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7B6FD7277CDE9FCB">32.3-7 Range</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7DE8173F80E07AB1">32.3-8 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X784F871383FB599B">32.3-9 UnderlyingRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X786581DE871A47D0">32.3-10 UnderlyingGeneralMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X83B4FF15847F06FC">32.4 <span class="Heading">Images under Mappings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D23C1CE863DACD8">32.4-1 ImagesSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X85ADB89B7C8DD7D0">32.4-2 ImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D51184B7EE5B2CF">32.4-3 ImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8781348F7F5796A0">32.4-4 ImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7CFAB0157BFB1806">32.4-5 ImageElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87F4D35A826599C6">32.4-6 <span class="Heading">Image</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86114B2E7E77488C">32.4-7 <span class="Heading">Images</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X79BB1EC07C828667">32.5 <span class="Heading">Preimages under Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X78EF1FE77B0973C0">32.5-1 PreImagesRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7FBB830C8729E995">32.5-2 PreImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D212F727CAE971A">32.5-3 PreImageElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7AE24A1586B7DE79">32.5-4 PreImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X856BAFC87B2D2811">32.5-5 PreImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X836FAEAC78B55BF4">32.5-6 <span class="Heading">PreImage</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X85C8590E832002EF">32.5-7 <span class="Heading">PreImages</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E2E16277940FA0B">32.6 <span class="Heading">Arithmetic Operations for General Mappings</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X834E02BB7D4B4AE5">32.7 <span class="Heading">Mappings which are Compatible with Algebraic Structures</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X8008FCCC7F4C731F">32.8 <span class="Heading">Magma Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7DC72CF28539A251">32.8-1 IsMagmaHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8181676787E760A2">32.8-2 MagmaHomomorphismByFunctionNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X79D0216E871B7051">32.8-3 NaturalHomomorphismByGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X806F892C862F29F9">32.9 <span class="Heading">Mappings that Respect Multiplication</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7BEFF95883EAEC78">32.9-1 RespectsMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7EE4DA097AE9CBC1">32.9-2 RespectsOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7F27AE9C84A4DF90">32.9-3 RespectsInverses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X819DD174829BF3AE">32.9-4 IsGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X81A5A5CF846E5FBF">32.9-5 KernelOfMultiplicativeGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7F09B6E28080DCB4">32.9-6 CoKernelOfMultiplicativeGeneralMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X8455A5A67C35178B">32.10 <span class="Heading">Mappings that Respect Addition</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7A3321E878925C3A">32.10-1 RespectsAddition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8130D8907B92F746">32.10-2 RespectsAdditiveInverses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D342736781EB280">32.10-3 RespectsZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7B99EF287A8A0BD9">32.10-4 IsAdditiveGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7EC0E9907D6631D6">32.10-5 KernelOfAdditiveGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X813C6D7980213F41">32.10-6 CoKernelOfAdditiveGeneralMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7C24431C81532575">32.11 <span class="Heading">Linear Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87842ED97FA19973">32.11-1 RespectsScalarMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X780BE6307A3271A9">32.11-2 IsLeftModuleGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7F6841107E59107F">32.11-3 IsLinearMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E88C32A82E942DA">32.12 <span class="Heading">Ring Homomorphisms</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7C8DA031799B79D5">32.12-1 IsRingGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7988102883675606">32.12-2 IsRingWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86B14F908601DEA9">32.12-3 IsAlgebraGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X842AD44679C5BDC2">32.12-4 IsAlgebraWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8324DA78879DF4D7">32.12-5 IsFieldHomomorphism</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E4A55567BED0F88">32.13 <span class="Heading">General Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8656AB8A7D672CAE">32.13-1 IsGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X791690817E23D90C">32.13-2 IsConstantTimeAccessGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X81CFF5F87BBEA8AD">32.13-3 IsEndoGeneralMapping</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7D6F78587C00CDD0">32.14 <span class="Heading">Technical Matters Concerning General Mappings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7D28581F82481163">32.14-1 IsSPGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X80D02AD183E01F16">32.14-2 IsGeneralMappingFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X86CFADBA7F2FE446">32.14-3 FamilyRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7C3736E281A9E505">32.14-4 FamilySource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7AE54FB67E2E6374">32.14-5 FamiliesOfGeneralMappingsAndRanges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7E1E26E37C413F6F">32.14-6 GeneralMappingsFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7CF92CC37A6BBDA5">32.14-7 TypeOfDefaultGeneralMapping</a></span>
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<h3>32 <span class="Heading">Mappings</span></h3>
<p>A <em>mapping</em> in <strong class="pkg">GAP</strong> is what is called a <q>function</q> in mathematics. <strong class="pkg">GAP</strong> also implements <em>generalized mappings</em> in which one element might have several images, these can be imagined as subsets of the cartesian product and are often called <q>relations</q>.</p>
<p>Most operations are declared for general mappings and therefore this manual often refers to <q>(general) mappings</q>, unless you deliberately need the generalization you can ignore the <q>general</q> bit and just read it as <q>mappings</q>.</p>
<p>A <em>general mapping</em> <span class="SimpleMath">F</span> in <strong class="pkg">GAP</strong> is described by its source <span class="SimpleMath">S</span>, its range <span class="SimpleMath">R</span>, and a subset <span class="SimpleMath">Rel</span> of the direct product <span class="SimpleMath">S × R</span>, which is called the underlying relation of <span class="SimpleMath">F</span>. <span class="SimpleMath">S</span>, <span class="SimpleMath">R</span>, and <span class="SimpleMath">Rel</span> are generalized domains (see <a href="chap12.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a>). The corresponding attributes for general mappings are <code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>), <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>), and <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>).</p>
<p>Note that general mappings themselves are <em>not</em> domains. One reason for this is that two general mappings with same underlying relation are regarded as equal only if also the sources are equal and the ranges are equal. Other, more technical, reasons are that general mappings and domains have different basic operations, and that general mappings are arithmetic objects (see <a href="chap32.html#X7E2E16277940FA0B"><span class="RefLink">32.6</span></a>); both should not apply to domains.</p>
<p>Each element of an underlying relation of a general mapping lies in the category of direct product elements (see <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>)).</p>
<p>For each <span class="SimpleMath">s ∈ S</span>, the set <span class="SimpleMath">{ r ∈ R | (s,r) ∈ Rel }</span> is called the set of <em>images</em> of <span class="SimpleMath">s</span>. Analogously, for <span class="SimpleMath">r ∈ R</span>, the set <span class="SimpleMath">{ s ∈ S | (s,r) ∈ Rel }</span> is called the set of <em>preimages</em> of <span class="SimpleMath">r</span>.</p>
<p>The <em>ordering</em> of general mappings via <code class="code"><</code> is defined by the ordering of source, range, and underlying relation. Specifically, if the source and range domains of <var class="Arg">map1</var> and <var class="Arg">map2</var> are the same, then one considers the union of the preimages of <var class="Arg">map1</var> and <var class="Arg">map2</var> as a strictly ordered set. The smaller of <var class="Arg">map1</var> and <var class="Arg">map2</var> is the one whose image is smaller on the first point of this sequence where they differ.</p>
<p>For mappings which preserve an algebraic structure a <em>kernel</em> is defined. Depending on the structure preserved the operation to compute this kernel is called differently, see Section <a href="chap32.html#X834E02BB7D4B4AE5"><span class="RefLink">32.7</span></a>.</p>
<p>Some technical details of general mappings are described in section <a href="chap32.html#X7E4A55567BED0F88"><span class="RefLink">32.13</span></a>.</p>
<p><a id="X859A13548515A5D7" name="X859A13548515A5D7"></a></p>
<h4>32.1 <span class="Heading">Direct Products and their Elements</span></h4>
<p><a id="X87FD9FE787023FF0" name="X87FD9FE787023FF0"></a></p>
<h5>32.1-1 IsDirectProductElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDirectProductElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p><code class="func">IsDirectProductElement</code> is a subcategory of the meet of <code class="func">IsDenseList</code> (<a href="chap21.html#X870AA9D8798C93DD"><span class="RefLink">21.1-2</span></a>), <code class="func">IsMultiplicativeElementWithInverse</code> (<a href="chap31.html#X7FDB14E57814FA3B"><span class="RefLink">31.14-13</span></a>), <code class="func">IsAdditiveElementWithInverse</code> (<a href="chap31.html#X7C0E4AE883947778"><span class="RefLink">31.14-7</span></a>), and <code class="func">IsCopyable</code> (<a href="chap12.html#X811EFD727EBD1ADC"><span class="RefLink">12.6-1</span></a>), where the arithmetic operations (addition, zero, additive inverse, multiplication, powering, one, inverse) are defined componentwise.</p>
<p>Note that each of these operations will cause an error message if its result for at least one component cannot be formed.</p>
<p>For an object in the filter <code class="func">IsDirectProductElement</code>, <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>) returns a mutable plain list with the same entries. The sum and the product of a direct product element and a list in <code class="func">IsListDefault</code> (<a href="chap21.html#X7BAD12E67BFC90DE"><span class="RefLink">21.12-3</span></a>) is the list of sums and products, respectively. The sum and the product of a direct product element and an object that is neither a list nor a collection is the direct product element of componentwise sums and products, respectively.</p>
<p><a id="X78F8A1168280E06D" name="X78F8A1168280E06D"></a></p>
<h5>32.1-2 DirectProductFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProductFamily</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">args</var> must be a dense list of <code class="func">CollectionsFamily</code> (<a href="chap30.html#X84E5A67E87D8DD66"><span class="RefLink">30.2-1</span></a>) families, otherwise the function raises an error.</p>
<p><code class="func">DirectProductFamily</code> returns <code class="code">fam</code>, a collections family of <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>) objects.</p>
<p><code class="code">fam</code> is the <code class="func">CollectionsFamily</code> (<a href="chap30.html#X84E5A67E87D8DD66"><span class="RefLink">30.2-1</span></a>) of <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>) objects whose <code class="code">i</code>-th component is in <code class="code">ElementsFamily(args[i])</code>.</p>
<p>Note that a collection in <code class="code">fam</code> may not itself be a direct product; it just is a subcollection of a direct product.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D8 := DihedralGroup(IsPermGroup, 8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FamilyObj(D8) = CollectionsFamily(PermutationsFamily);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">fam := DirectProductFamily([FamilyObj(D8), FamilyObj(D8)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComponentsOfDirectProductElementsFamily(ElementsFamily(fam));</span>
[ <Family: "PermutationsFamily">, <Family: "PermutationsFamily"> ]
</pre></div>
<p>Also note that not all direct products in <strong class="pkg">GAP</strong> are created via these families. For example if the arguments to <code class="func">DirectProduct</code> (<a href="chap49.html#X861BA02C7902A4F4"><span class="RefLink">49.1-1</span></a>) are permutation groups, then it returns a permutation group as well, whose elements are not <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>) objects.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam = FamilyObj(DirectProduct(D8, D8));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">D4 := DihedralGroup(IsPcGroup, 4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam2 := DirectProductFamily([FamilyObj(D8), FamilyObj(D4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam2 = FamilyObj(DirectProduct(D8, D4));</span>
true
</pre></div>
<p><a id="X7CF6FEFB8290D5CB" name="X7CF6FEFB8290D5CB"></a></p>
<h4>32.2 <span class="Heading">Creating Mappings</span></h4>
<p><a id="X79D0D2F07A14D039" name="X79D0D2F07A14D039"></a></p>
<h5>32.2-1 GeneralMappingByElements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralMappingByElements</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">elms</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the general mapping with source <var class="Arg">S</var> and range <var class="Arg">R</var>, and with underlying relation consisting of the collection <var class="Arg">elms</var> of direct product elements.</p>
<p><a id="X7D55E1977ED70E01" name="X7D55E1977ED70E01"></a></p>
<h5>32.2-2 <span class="Heading">MappingByFunction</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappingByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>[, <var class="Arg">invfun</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">‣ MappingByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>, <var class="Arg">false</var>, <var class="Arg">prefun</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">MappingByFunction</code> returns a mapping <code class="code">map</code> with source <var class="Arg">S</var> and range <var class="Arg">R</var>, such that each element <span class="SimpleMath">s</span> of <var class="Arg">S</var> is mapped to the element <var class="Arg">fun</var><span class="SimpleMath">( s )</span>, where <var class="Arg">fun</var> is a <strong class="pkg">GAP</strong> function.</p>
<p>If the argument <var class="Arg">invfun</var> is bound then <code class="code">map</code> is a bijection between <var class="Arg">S</var> and <var class="Arg">R</var>, and the preimage of each element <span class="SimpleMath">r</span> of <var class="Arg">R</var> is given by <var class="Arg">invfun</var><span class="SimpleMath">( r )</span>, where <var class="Arg">invfun</var> is a <strong class="pkg">GAP</strong> function.</p>
<p>If five arguments are given and the fourth argument is <code class="keyw">false</code> then the <strong class="pkg">GAP</strong> function <var class="Arg">prefun</var> can be used to compute a single preimage also if <code class="code">map</code> is not bijective.</p>
<p>The mapping returned by <code class="func">MappingByFunction</code> lies in the filter <code class="func">IsNonSPGeneralMapping</code> (<a href="chap32.html#X7D28581F82481163"><span class="RefLink">32.14-1</span></a>), see <a href="chap32.html#X7D6F78587C00CDD0"><span class="RefLink">32.14</span></a>.</p>
<p><a id="X865FC25A87D36F3D" name="X865FC25A87D36F3D"></a></p>
<h5>32.2-3 InverseGeneralMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InverseGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>inverse general mapping</em> of a general mapping <var class="Arg">map</var> is the general mapping whose underlying relation (see <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>)) contains a pair <span class="SimpleMath">(r,s)</span> if and only if the underlying relation of <var class="Arg">map</var> contains the pair <span class="SimpleMath">(s,r)</span>.</p>
<p>See the introduction to Chapter <a href="chap32.html#X7C9734B880042C73"><span class="RefLink">32</span></a> for the subtleties concerning the difference between <code class="func">InverseGeneralMapping</code> and <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>).</p>
<p>Note that the inverse general mapping of a mapping <var class="Arg">map</var> is in general only a general mapping. If <var class="Arg">map</var> knows to be bijective its inverse general mapping will know to be a mapping. In this case also <code class="code">Inverse( <var class="Arg">map</var> )</code> works.</p>
<p><a id="X7BD2D5A87CD6B213" name="X7BD2D5A87CD6B213"></a></p>
<h5>32.2-4 RestrictedInverseGeneralMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedInverseGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>restricted inverse general mapping</em> of a general mapping <var class="Arg">map</var> is the general mapping whose underlying relation (see <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>)) contains a pair <span class="SimpleMath">(r,s)</span> if and only if the underlying relation of <var class="Arg">map</var> contains the pair <span class="SimpleMath">(s,r)</span>, and whose domain is restricted to the image of <var class="Arg">map</var> and whose range is the domain of <var class="Arg">map</var>.</p>
<p><a id="X7ED1E4E27CCE2DCA" name="X7ED1E4E27CCE2DCA"></a></p>
<h5>32.2-5 CompositionMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompositionMapping</code>( <var class="Arg">map1</var>, <var class="Arg">map2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">CompositionMapping</code> allows one to compose arbitrarily many general mappings, and delegates each step to <code class="func">CompositionMapping2</code> (<a href="chap32.html#X86486B687B7077AC"><span class="RefLink">32.2-6</span></a>). The result is a map that maps an element first under the last argument, then under the penultimate argument and so forth.</p>
<p>Additionally, the properties <code class="func">IsInjective</code> (<a href="chap32.html#X7F065FD7822C0A12"><span class="RefLink">32.3-4</span></a>) and <code class="func">IsSingleValued</code> (<a href="chap32.html#X86D44C8A78BF1981"><span class="RefLink">32.3-2</span></a>) are maintained. If the range of the <span class="SimpleMath">i+1</span>-th argument is identical to the range of the <span class="SimpleMath">i</span>-th argument, also <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>) and <code class="func">IsSurjective</code> (<a href="chap32.html#X784ECE847E005B8F"><span class="RefLink">32.3-5</span></a>) are maintained. (So one should not call <code class="func">CompositionMapping2</code> (<a href="chap32.html#X86486B687B7077AC"><span class="RefLink">32.2-6</span></a>) directly if one wants to maintain these properties.)</p>
<p>Depending on the types of <var class="Arg">map1</var> and <var class="Arg">map2</var>, the returned mapping might be constructed completely new (for example by giving domain generators and their images, this is for example the case if both mappings preserve the same algebraic structures and <strong class="pkg">GAP</strong> can decompose elements of the source of <var class="Arg">map2</var> into generators) or as an (iterated) composition (see <code class="func">IsCompositionMappingRep</code> (<a href="chap32.html#X7A926D167C3155F6"><span class="RefLink">32.2-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := GroupHomomorphismByImages(CyclicGroup(IsPermGroup, 2),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CyclicGroup(IsPermGroup, 1));</span>
[ (1,2) ] -> [ () ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g := GroupHomomorphismByImages(CyclicGroup(IsPermGroup, 6),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CyclicGroup(IsPermGroup, 2));</span>
[ (1,2,3,4,5,6) ] -> [ (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMapping(f, g);</span>
[ (1,2,3,4,5,6) ] -> [ () ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMapping(g, f);</span>
[ (1,2) ] -> [ () ]
</pre></div>
<p><a id="X86486B687B7077AC" name="X86486B687B7077AC"></a></p>
<h5>32.2-6 CompositionMapping2</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompositionMapping2</code>( <var class="Arg">map2</var>, <var class="Arg">map1</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">‣ CompositionMapping2General</code>( <var class="Arg">map2</var>, <var class="Arg">map1</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">CompositionMapping2</code> returns the composition of <var class="Arg">map2</var> and <var class="Arg">map1</var>, this is the general mapping that maps an element first under <var class="Arg">map1</var>, and then maps the images under <var class="Arg">map2</var>.</p>
<p>(Note the reverse ordering of arguments in the composition via the multiplication <code class="func">\*</code> (<a href="chap31.html#X8481C9B97B214C23"><span class="RefLink">31.12-1</span></a>).</p>
<p><code class="func">CompositionMapping2General</code> is the method that forms a composite mapping with two constituent mappings. (This is used in some algorithms.)</p>
<p><a id="X7A926D167C3155F6" name="X7A926D167C3155F6"></a></p>
<h5>32.2-7 IsCompositionMappingRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCompositionMappingRep</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Mappings in this representation are stored as composition of two mappings, (pre)images of elements are computed in a two-step process. The constituent mappings of the composition can be obtained via <code class="func">ConstituentsCompositionMapping</code> (<a href="chap32.html#X87775B438008DCA5"><span class="RefLink">32.2-8</span></a>).</p>
<p><a id="X87775B438008DCA5" name="X87775B438008DCA5"></a></p>
<h5>32.2-8 ConstituentsCompositionMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstituentsCompositionMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">map</var> is stored in the representation <code class="func">IsCompositionMappingRep</code> (<a href="chap32.html#X7A926D167C3155F6"><span class="RefLink">32.2-7</span></a>) as composition of two mappings <var class="Arg">map1</var> and <var class="Arg">map2</var>, this function returns the two constituent mappings in a list <code class="code">[ <var class="Arg">map1</var>, <var class="Arg">map2</var> ]</code>.</p>
<p><a id="X795FF8DC785F110A" name="X795FF8DC785F110A"></a></p>
<h5>32.2-9 ZeroMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZeroMapping</code>( <var class="Arg">S</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A zero mapping is a total general mapping that maps each element of its source to the zero element of its range.</p>
<p>(Each mapping with empty source is a zero mapping.)</p>
<p><a id="X7EBAE0368470A603" name="X7EBAE0368470A603"></a></p>
<h5>32.2-10 IdentityMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityMapping</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the bijective mapping with source and range equal to the collection <var class="Arg">D</var>, which maps each element of <var class="Arg">D</var> to itself.</p>
<p><a id="X86452F8587CBAEA0" name="X86452F8587CBAEA0"></a></p>
<h5>32.2-11 <span class="Heading">Embedding</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Embedding</code>( <var class="Arg">S</var>, <var class="Arg">T</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">‣ Embedding</code>( <var class="Arg">S</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the embedding of the domain <var class="Arg">S</var> in the domain <var class="Arg">T</var>, or in the second form, some domain indexed by the positive integer <var class="Arg">i</var>. The precise natures of the various methods are described elsewhere: for Lie algebras, see <code class="func">LieFamily</code> (<a href="chap64.html#X8725993C7BF386EE"><span class="RefLink">64.1-3</span></a>); for group products, see <a href="chap49.html#X798FDA1386A0EAC6"><span class="RefLink">49.6</span></a> for a general description, or for examples see <a href="chap49.html#X7D39232A84CD8DBD"><span class="RefLink">49.1</span></a> for direct products, <a href="chap49.html#X87FE512E7DB7346C"><span class="RefLink">49.2</span></a> for semidirect products, or <a href="chap49.html#X7DF2AEBC8518FFA4"><span class="RefLink">49.4</span></a> for wreath products; or for magma rings see <a href="chap65.html#X80366F1480ACD8DF"><span class="RefLink">65.3</span></a>.</p>
<p><a id="X8769E8DA80BC96C1" name="X8769E8DA80BC96C1"></a></p>
<h5>32.2-12 <span class="Heading">Projection</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Projection</code>( <var class="Arg">S</var>, <var class="Arg">T</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">‣ Projection</code>( <var class="Arg">S</var>, <var class="Arg">i</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">‣ Projection</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the projection of the domain <var class="Arg">S</var> onto the domain <var class="Arg">T</var>, or in the second form, some domain indexed by the positive integer <var class="Arg">i</var>, or in the third form some natural quotient domain of <var class="Arg">S</var>. Various methods are defined for group products; see <a href="chap49.html#X798FDA1386A0EAC6"><span class="RefLink">49.6</span></a> for a general description, or for examples see <a href="chap49.html#X7D39232A84CD8DBD"><span class="RefLink">49.1</span></a> for direct products, <a href="chap49.html#X87FE512E7DB7346C"><span class="RefLink">49.2</span></a> for semidirect products, <a href="chap49.html#X815AFC537B215D7B"><span class="RefLink">49.3</span></a> for subdirect products, or <a href="chap49.html#X7DF2AEBC8518FFA4"><span class="RefLink">49.4</span></a> for wreath products.</p>
<p><a id="X800014D683A81009" name="X800014D683A81009"></a></p>
<h5>32.2-13 RestrictedMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedMapping</code>( <var class="Arg">map</var>, <var class="Arg">subdom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">subdom</var> is a subdomain of the source of the general mapping <var class="Arg">map</var>, this operation returns the restriction of <var class="Arg">map</var> to <var class="Arg">subdom</var>.</p>
<p><a id="X7E5A430D7F838F1C" name="X7E5A430D7F838F1C"></a></p>
<h4>32.3 <span class="Heading">Properties and Attributes of (General) Mappings</span></h4>
<p><a id="X83C7494E828CC9C8" name="X83C7494E828CC9C8"></a></p>
<h5>32.3-1 IsTotal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTotal</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> has images, i.e., <span class="SimpleMath">s^<var class="Arg">map</var> ≠ ∅</span> for all <span class="SimpleMath">s ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X86D44C8A78BF1981" name="X86D44C8A78BF1981"></a></p>
<h5>32.3-2 IsSingleValued</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSingleValued</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> has at most one image, i.e., <span class="SimpleMath">|s^<var class="Arg">map</var>| ≤ 1</span> for all <span class="SimpleMath">s ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>
<p>Equivalently, <code class="code">IsSingleValued( <var class="Arg">map</var> )</code> is <code class="keyw">true</code> if and only if the preimages of different elements in <span class="SimpleMath">R</span> are disjoint.</p>
<p><a id="X7CC95EB282854385" name="X7CC95EB282854385"></a></p>
<h5>32.3-3 IsMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>A <em>mapping</em> <var class="Arg">map</var> is a general mapping that assigns to each element <code class="code">elm</code> of its source a unique element <code class="code">Image( <var class="Arg">map</var>, elm )</code> of its range.</p>
<p>Equivalently, the general mapping <var class="Arg">map</var> is a mapping if and only if it is total and single-valued (see <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>), <code class="func">IsSingleValued</code> (<a href="chap32.html#X86D44C8A78BF1981"><span class="RefLink">32.3-2</span></a>)).</p>
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<h5>32.3-4 IsInjective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInjective</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if the images of different elements in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> are disjoint, i.e., <span class="SimpleMath">x^<var class="Arg">map</var> ∩ y^<var class="Arg">map</var> = ∅</span> for <span class="SimpleMath">x ≠ y ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>
<p>Equivalently, <code class="code">IsInjective( <var class="Arg">map</var> )</code> is <code class="keyw">true</code> if and only if each element in the range of <var class="Arg">map</var> has at most one preimage in <span class="SimpleMath">S</span>.</p>
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<h5>32.3-5 IsSurjective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSurjective</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the range <span class="SimpleMath">R</span> of the general mapping <var class="Arg">map</var> has preimages in the source <span class="SimpleMath">S</span> of <var class="Arg">map</var>, i.e., <span class="SimpleMath">{ s ∈ S ∣ x ∈ s^<var class="Arg">map</var> } ≠ ∅</span> for all <span class="SimpleMath">x ∈ R</span>, and <code class="keyw">false</code> otherwise.</p>
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<h5>32.3-6 IsBijective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBijective</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></d | |
