|
|
|
|
Quelle chap4_mj.html
Sprache: HTML
|
|
| products/Sources/formale Sprachen/GAP/pkg/homalg/doc/chap4_mj.html |
 |
<?xml version= "1.0" encoding= "UTF-8"?>
<! DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
< html xmlns= "http://www.w3.org/1999/xhtml" xml:lang= "en">
< head>
< script type= "text/javascript"
src= "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</ script>
< title>GAP (homalg) - Chapter 4: Morphisms</ title>
< meta http-equiv= "content-type" content= "text/html; charset=UTF-8" />
< meta name= "generator" content= "GAPDoc2HTML" />
< link rel= "stylesheet" type= "text/css" href= "manual.css" />
< script src= "manual.js" type= "text/javascript"></ script>
< script type= "text/javascript">overwriteStyle();</ script>
</ head>
< body class= "chap4" onload= "jscontent()">
< div class= "chlinktop">< span class= "chlink1">Goto Chapter: </ span>< a href= "chap0_mj.html">Top< /a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chap12_mj.html">12</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapC_mj.html">C</a> <a href="chapD_mj.html">D</a> <a href="chapE_mj.html">E</a> <a href="chapF_mj.html">F</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap3_mj.html">[Previous Chapter]</a> <a href="chap5_mj.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p>
<p><a id="X7BEB6C617FED52DA" name="X7BEB6C617FED52DA"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X7BEB6C617FED52DA">4 <span class="Heading">Morphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7DE206257C909BDE">4.1 <span class="Heading">Morphisms: Categories and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D0F89828196DFF0">4.1-1 IsHomalgMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81458CA5836D582F">4.1-2 IsHomalgStaticMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7933C51A842ABA32">4.1-3 IsHomalgEndomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X823580787F23EB10">4.1-4 IsMorphismOfFinitelyGeneratedObjectsRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84A97E897C74B492">4.1-5 IsStaticMorphismOfFinitelyGeneratedObjectsRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X86A95A9B85D8B58B">4.2 <span class="Heading">Morphisms: Constructors</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7B0B60BD79756A00">4.3 <span class="Heading">Morphisms: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F66120A814DC16B">4.3-1 IsMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B7206EC7F584F25">4.3-2 IsGeneralizedMorphismWithFullDomain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AD32A427B247366">4.3-3 IsGeneralizedEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83C68AEA7FE4AA29">4.3-4 IsGeneralizedMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83F05F467DA5EA4D">4.3-5 IsGeneralizedIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X814D78347858EC13">4.3-6 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CB5896082D29173">4.3-7 IsIdempotent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78AD1FDD7F53932C">4.3-8 IsMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8724CEF182DC4064">4.3-9 IsEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7DFACF1F7D7F7EE9">4.3-10 IsSplitMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80A66EFA862E56BC">4.3-11 IsSplitEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E07BBF57B92BA56">4.3-12 IsIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F30E3D37E9D7F37">4.3-13 IsAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X806EEA4685A4A3F3">4.4 <span class="Heading">Morphisms: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7DE8173F80E07AB1">4.4-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X829F76BB80BD55DB">4.4-2 Range</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F3927E287087B64">4.4-3 CokernelEpi</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D71AE8E838712D7">4.4-4 CokernelNaturalGeneralizedIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87C00FFB79FA93A8">4.4-5 KernelSubobject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X82672DB279FAEFCC">4.4-6 KernelEmb</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X82FB6A4687E778D5">4.4-7 ImageSubobject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X85FA7C19800F72B2">4.4-8 ImageObjectEmb</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86E3E1BA7BCE4D66">4.4-9 ImageObjectEpi</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X823682157C6B4D63">4.4-10 MorphismAid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X85F22F4084EA7D31">4.4-11 InverseOfGeneralizedMorphismWithFullDomain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8500C49A784C8EDC">4.4-12 DegreeOfMorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X789623548056F7B7">4.5 <span class="Heading">Morphisms: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B4F9EF27A241520">4.5-1 ByASmallerPresentation</a></span>
</div></div>
</div>
<h3>4 <span class="Heading">Morphisms</span></h3>
<p><a id="X7DE206257C909BDE" name="X7DE206257C909BDE"></a></p>
<h4>4.1 <span class="Heading">Morphisms: Categories and Representations</span></h4>
<p><a id="X7D0F89828196DFF0" name="X7D0F89828196DFF0"></a></p>
<h5>4.1-1 IsHomalgMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgMorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>This is the super <strong class="pkg">GAP</strong>-category which will include the <strong class="pkg">GAP</strong>-categories <code class="func">IsHomalgStaticMorphism</code> (<a href="chap4_mj.html#X81458CA5836D582F"><span class="RefLink">4.1-2</span></a>) and <code class="func">IsHomalgChainMorphism</code> (<a href="chap7_mj.html#X7CB62E188027B7C5"><span class="RefLink">7.1-1</span></a>). We need this <strong class="pkg">GAP</strong>-category to be able to build complexes with *objects* being objects of <strong class="pkg">homalg</strong> categories or again complexes. We need this GAP-category to be able to build chain morphisms with *morphisms* being morphisms of <strong class="pkg">homalg</strong> categories or again chain morphisms. <br /> CAUTION: Never let <strong class="pkg">homalg</strong> morphisms (which are not endomorphisms) be multiplicative elements!!</p>
<div class="example"><pre>
DeclareCategory( "IsHomalgMorphism",
IsHomalgStaticObjectOrMorphism and
IsAdditiveElementWithInverse );
</pre></div>
<p><a id="X81458CA5836D582F" name="X81458CA5836D582F"></a></p>
<h5>4.1-2 IsHomalgStaticMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgStaticMorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>This is the super <strong class="pkg">GAP</strong>-category which will include the <strong class="pkg">GAP</strong>-categories <code class="code">IsHomalgMap</code>, etc. <br /> CAUTION: Never let homalg morphisms (which are not endomorphisms) be multiplicative elements!!</p>
<div class="example"><pre>
DeclareCategory( "IsHomalgStaticMorphism",
IsHomalgMorphism );
</pre></div>
<p><a id="X7933C51A842ABA32" name="X7933C51A842ABA32"></a></p>
<h5>4.1-3 IsHomalgEndomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgEndomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>This is the super <strong class="pkg">GAP</strong>-category which will include the <strong class="pkg">GAP</strong>-categories <code class="code">IsHomalgSelfMap</code>, <code class="func">IsHomalgChainEndomorphism</code> (<a href="chap7_mj.html#X853BD37084BFC602"><span class="RefLink">7.1-2</span></a>), etc. be multiplicative elements!!</p>
<div class="example"><pre>
DeclareCategory( "IsHomalgEndomorphism",
IsHomalgMorphism and
IsMultiplicativeElementWithInverse );
</pre></div>
<p><a id="X823580787F23EB10" name="X823580787F23EB10"></a></p>
<h5>4.1-4 IsMorphismOfFinitelyGeneratedObjectsRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMorphismOfFinitelyGeneratedObjectsRep</code>( <var class="Arg">phi</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <strong class="pkg">GAP</strong> representation of morphisms of finitley generated <strong class="pkg">homalg</strong> objects.</p>
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgMorphism</code> (<a href="chap4_mj.html#X7D0F89828196DFF0"><span class="RefLink">4.1-1</span></a>).)</p>
<div class="example"><pre>
DeclareRepresentation( "IsMorphismOfFinitelyGeneratedObjectsRep",
IsHomalgMorphism,
[ ] );
</pre></div>
<p><a id="X84A97E897C74B492" name="X84A97E897C74B492"></a></p>
<h5>4.1-5 IsStaticMorphismOfFinitelyGeneratedObjectsRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsStaticMorphismOfFinitelyGeneratedObjectsRep</code>( <var class="Arg">phi</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <strong class="pkg">GAP</strong> representation of static morphisms of finitley generated <strong class="pkg">homalg</strong> static objects.</p>
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticMorphism</code> (<a href="chap4_mj.html#X81458CA5836D582F"><span class="RefLink">4.1-2</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="func">IsMorphismOfFinitelyGeneratedObjectsRep</code> (<a href="chap4_mj.html#X823580787F23EB10"><span class="RefLink">4.1-4</span></a>).)</p>
<div class="example"><pre>
DeclareRepresentation( "IsStaticMorphismOfFinitelyGeneratedObjectsRep",
IsHomalgStaticMorphism and
IsMorphismOfFinitelyGeneratedObjectsRep,
[ ] );
</pre></div>
<p><a id="X86A95A9B85D8B58B" name="X86A95A9B85D8B58B"></a></p>
<h4>4.2 <span class="Heading">Morphisms: Constructors</span></h4>
<p><a id="X7B0B60BD79756A00" name="X7B0B60BD79756A00"></a></p>
<h4>4.3 <span class="Heading">Morphisms: Properties</span></h4>
<p><a id="X7F66120A814DC16B" name="X7F66120A814DC16B"></a></p>
<h5>4.3-1 IsMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p><code class="code">IsMorphism</code>=<code class="code">true</code> means one of the following:</p>
<ul>
<li><p>The property method <code class="code">IsMorphism</code>(<var class="Arg">phi</var>) was explicitly invoked by the user and it returned <code class="code">true</code>, where prior to the invocation <code class="code">HasIsMorphism</code>(<var class="Arg">phi</var>) was <code class="code">false</code>. The method is meant to check the integrity of the data structure at the time of it invocation. What this precisely means depends on the specific <strong class="pkg">homalg</strong>-based package.</p>
</li>
<li><p>The user has explicitly <code class="code">SetIsMorphism</code>(<var class="Arg">phi</var>, <code class="code">true</code>).</p>
</li>
<li><p>The morphism <var class="Arg">phi</var> is output of a categorical procedure where <code class="code">IsMorphism</code> has become <code class="code">true</code> for all morphisms in the input.</p>
</li>
<li><p>The morphism <var class="Arg">phi</var> is output of a categorical procedure which gurantees the integrity of the data structure of its output independent of its input.</p>
</li>
</ul>
<p><a id="X7B7206EC7F584F25" name="X7B7206EC7F584F25"></a></p>
<h5>4.3-2 IsGeneralizedMorphismWithFullDomain</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneralizedMorphismWithFullDomain</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if <var class="Arg">phi</var> is a generalized morphism.</p>
<p><a id="X7AD32A427B247366" name="X7AD32A427B247366"></a></p>
<h5>4.3-3 IsGeneralizedEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneralizedEpimorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if <var class="Arg">phi</var> is a generalized epimorphism.</p>
<p><a id="X83C68AEA7FE4AA29" name="X83C68AEA7FE4AA29"></a></p>
<h5>4.3-4 IsGeneralizedMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneralizedMonomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if <var class="Arg">phi</var> is a generalized monomorphism.</p>
<p><a id="X83F05F467DA5EA4D" name="X83F05F467DA5EA4D"></a></p>
<h5>4.3-5 IsGeneralizedIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneralizedIsomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if <var class="Arg">phi</var> is a generalized isomorphism.</p>
<p><a id="X814D78347858EC13" name="X814D78347858EC13"></a></p>
<h5>4.3-6 IsOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsOne</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is the identity morphism.</p>
<p><a id="X7CB5896082D29173" name="X7CB5896082D29173"></a></p>
<h5>4.3-7 IsIdempotent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIdempotent</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is an automorphism.</p>
<p><a id="X78AD1FDD7F53932C" name="X78AD1FDD7F53932C"></a></p>
<h5>4.3-8 IsMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMonomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is a monomorphism.</p>
<p><a id="X8724CEF182DC4064" name="X8724CEF182DC4064"></a></p>
<h5>4.3-9 IsEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsEpimorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is an epimorphism.</p>
<p><a id="X7DFACF1F7D7F7EE9" name="X7DFACF1F7D7F7EE9"></a></p>
<h5>4.3-10 IsSplitMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSplitMonomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is a split monomorphism. <br /></p>
<p><a id="X80A66EFA862E56BC" name="X80A66EFA862E56BC"></a></p>
<h5>4.3-11 IsSplitEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSplitEpimorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is a split epimorphism. <br /></p>
<p><a id="X7E07BBF57B92BA56" name="X7E07BBF57B92BA56"></a></p>
<h5>4.3-12 IsIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIsomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is an isomorphism.</p>
<p><a id="X7F30E3D37E9D7F37" name="X7F30E3D37E9D7F37"></a></p>
<h5>4.3-13 IsAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsAutomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> is an automorphism.</p>
<p><a id="X806EEA4685A4A3F3" name="X806EEA4685A4A3F3"></a></p>
<h4>4.4 <span class="Heading">Morphisms: Attributes</span></h4>
<p><a id="X7DE8173F80E07AB1" name="X7DE8173F80E07AB1"></a></p>
<h5>4.4-1 Source</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Source</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> object</p>
<p>The source of the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var>.</p>
<p><a id="X829F76BB80BD55DB" name="X829F76BB80BD55DB"></a></p>
<h5>4.4-2 Range</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Range</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> object</p>
<p>The target (range) of the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var>.</p>
<p><a id="X7F3927E287087B64" name="X7F3927E287087B64"></a></p>
<h5>4.4-3 CokernelEpi</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CokernelEpi</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The natural epimorphism from the <code class="code">Range</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span> onto the <code class="code">Cokernel</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span>.</p>
<p><a id="X7D71AE8E838712D7" name="X7D71AE8E838712D7"></a></p>
<h5>4.4-4 CokernelNaturalGeneralizedIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CokernelNaturalGeneralizedIsomorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The natural generalized isomorphism from the <code class="code">Cokernel</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span> onto the <code class="code">Range</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span>.</p>
<p><a id="X87C00FFB79FA93A8" name="X87C00FFB79FA93A8"></a></p>
<h5>4.4-5 KernelSubobject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; KernelSubobject</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>This constructor returns the finitely generated kernel of the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> as a subobject of the <strong class="pkg">homalg</strong> object <code class="code">Source</code>(<var class="Arg">phi</var>) with generators given by the syzygies of <var class="Arg">phi</var>.</p>
<p><a id="X82672DB279FAEFCC" name="X82672DB279FAEFCC"></a></p>
<h5>4.4-6 KernelEmb</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; KernelEmb</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The natural embedding of the <code class="code">Kernel</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span> into the <code class="code">Source</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span>.</p>
<p><a id="X82FB6A4687E778D5" name="X82FB6A4687E778D5"></a></p>
<h5>4.4-7 ImageSubobject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageSubobject</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>This constructor returns the finitely generated image of the <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> as a subobject of the <strong class="pkg">homalg</strong> object <code class="code">Range</code>(<var class="Arg">phi</var>) with generators given by <var class="Arg">phi</var> applied to the generators of its source object.</p>
<p><a id="X85FA7C19800F72B2" name="X85FA7C19800F72B2"></a></p>
<h5>4.4-8 ImageObjectEmb</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageObjectEmb</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The natural embedding of the <code class="code">ImageObject</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span> into the <code class="code">Range</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span>.</p>
<p><a id="X86E3E1BA7BCE4D66" name="X86E3E1BA7BCE4D66"></a></p>
<h5>4.4-9 ImageObjectEpi</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageObjectEpi</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The natural epimorphism from the <code class="code">Source</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span> onto the <code class="code">ImageObject</code><span class="SimpleMath">\((\)</span><var class="Arg">phi</var><span class="SimpleMath">\()\)</span>.</p>
<p><a id="X823682157C6B4D63" name="X823682157C6B4D63"></a></p>
<h5>4.4-10 MorphismAid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MorphismAid</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The morphism aid map of a true generalized map. <br /> (no method installed)</p>
<p><a id="X85F22F4084EA7D31" name="X85F22F4084EA7D31"></a></p>
<h5>4.4-11 InverseOfGeneralizedMorphismWithFullDomain</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; InverseOfGeneralizedMorphismWithFullDomain</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>
<p>The generalized inverse of the epimorphism <var class="Arg">phi</var> (cf. <a href="chapBib_mj.html#biBBaSF">[Bar09, Cor. 4.8]</a>)).</p>
<p><a id="X8500C49A784C8EDC" name="X8500C49A784C8EDC"></a></p>
<h5>4.4-12 DegreeOfMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DegreeOfMorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an integer</p>
<p>The degree of the morphism <var class="Arg">phi</var> between graded objects. <br /> (no method installed)</p>
<p><a id="X789623548056F7B7" name="X789623548056F7B7"></a></p>
<h4>4.5 <span class="Heading">Morphisms: Operations and Functions</span></h4>
<p><a id="X7B4F9EF27A241520" name="X7B4F9EF27A241520"></a></p>
<h5>4.5-1 ByASmallerPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ByASmallerPresentation</code>( <var class="Arg">phi</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkg">homalg</strong> (static) objects.</p>
<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
"for homalg morphisms",
[ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],
function( phi )
ByASmallerPresentation( Source( phi ) );
ByASmallerPresentation( Range( phi ) );
return DecideZero( phi );
end );
</pre></div>
<p>This method performs side effects on its argument <var class="Arg">phi</var> and returns it.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );</span>
<A 2 x 3 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( M );</span>
<A non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );</span>
<A 2 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span>
<A non-torsion left module presented by 2 relations for 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">1, 0, -2, -4, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 1, 4, 7, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">1, 0, -2, -4 \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, zz );</span>
<A 3 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( mat, M, N );</span>
<A "homomorphism" of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span>
<A homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( phi );</span>
[ [ 1, 0, -2, -4 ],
[ 0, 1, 4, 7 ],
[ 1, 0, -2, -4 ] ]
the map is currently represented by the above 3 x 4 matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( phi );</span>
<A non-zero homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( phi );</span>
[ [ 0, 0, 0 ],
[ 1, -1, -2 ] ]
the map is currently represented by the above 2 x 3 matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">M;</span>
<A rank 1 left module presented by 1 relation for 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M );</span>
Z/< 3 > + Z^(1 x 1)
<span class="GAPprompt">gap></span> <span class="GAPinput">N;</span>
<A rank 2 left module presented by 1 relation for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( N );</span>
Z/< 4 > + Z^(1 x 2)
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap3_mj.html">[Previous Chapter]</a> <a href="chap5_mj.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chap12_mj.html">12</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapC_mj.html">C</a> <a href="chapD_mj.html">D</a> <a href="chapE_mj.html">E</a> <a href="chapF_mj.html">F</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
¤ Dauer der Verarbeitung: 0.33 Sekunden
¤
*© Formatika GbR, Deutschland
|
|
2026-11-04
|
|
|
|
|
Neuigkeiten |
| Aktuelles |
| Motto des Tages |
|
Open Source Software |
|
|
|
Jenseits des Üblichen ....
|
|
Besucherstatistik |
|
|
| Statistik |
| #Sources=1104857 |
| #Domains=438801 |
|
|
