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<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="chap4_mj.html">[Previous Chapter]</a>    <a href="chap6_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap5.html">[MathJax off]</a></p>
<p><a id="X79B130FC7906FB4C" name="X79B130FC7906FB4C"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X79B130FC7906FB4C">5 <span class="Heading">Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7FBC2FC77E93856C">5.1 <span class="Heading">Elements: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X784BBB2A782DB774">5.1-1 IsHomalgElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87F3740E85E9AA51">5.1-2 IsElementOfAnObjectGivenByAMorphismRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8159567F8721ADCA">5.2 <span class="Heading">Elements: Constructors</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7C36DB5C81520E55">5.3 <span class="Heading">Elements: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D16A0BB80BA69DC">5.3-1 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7886344B7A8B9304">5.3-2 IsCyclicGenerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X80C6B26284721409">5.3-3 IsTorsion</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X875351A77DEB949A">5.4 <span class="Heading">Elements: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8557F75878DEEA58">5.4-1 Annihilator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X865C489C7FE750A9">5.5 <span class="Heading">Elements: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A4ED0528640EEFE">5.5-1 in</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Elements</span></h3>

<p>An element of an object <span class="SimpleMath">\(M\)</span> is internally represented by a morphism from the "structure object" to the object <span class="SimpleMath">\(M\)</span>. In particular, the data structure for object elements automatically profits from the intrinsic realization of morphisms in the <strong class="pkg">homalg</strong> project.</p>

<p><a id="X7FBC2FC77E93856C" name="X7FBC2FC77E93856C"></a></p>

<h4>5.1 <span class="Heading">Elements: Category and Representations</span></h4>

<p><a id="X784BBB2A782DB774" name="X784BBB2A782DB774"></a></p>

<h5>5.1-1 IsHomalgElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgElement</code>( <var class="Arg">M</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>The <strong class="pkg">GAP</strong> category of object elements.</p>

<p><a id="X87F3740E85E9AA51" name="X87F3740E85E9AA51"></a></p>

<h5>5.1-2 IsElementOfAnObjectGivenByAMorphismRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsElementOfAnObjectGivenByAMorphismRep</code>( <var class="Arg">M</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 elements of finitley presented objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgElement</code> (<a href="chap5_mj.html#X784BBB2A782DB774"><span class="RefLink">5.1-1</span></a>).)</p>

<p><a id="X8159567F8721ADCA" name="X8159567F8721ADCA"></a></p>

<h4>5.2 <span class="Heading">Elements: Constructors</span></h4>

<p><a id="X7C36DB5C81520E55" name="X7C36DB5C81520E55"></a></p>

<h4>5.3 <span class="Heading">Elements: Properties</span></h4>

<p><a id="X7D16A0BB80BA69DC" name="X7D16A0BB80BA69DC"></a></p>

<h5>5.3-1 IsZero</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZero</code>( <var class="Arg">m</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 object element <var class="Arg">m</var> is zero.</p>

<p><a id="X7886344B7A8B9304" name="X7886344B7A8B9304"></a></p>

<h5>5.3-2 IsCyclicGenerator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclicGenerator</code>( <var class="Arg">m</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 object element <var class="Arg">m</var> is a cyclic generator.</p>

<p><a id="X80C6B26284721409" name="X80C6B26284721409"></a></p>

<h5>5.3-3 IsTorsion</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTorsion</code>( <var class="Arg">m</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 object element <var class="Arg">m</var> is a torsion element.</p>

<p><a id="X875351A77DEB949A" name="X875351A77DEB949A"></a></p>

<h4>5.4 <span class="Heading">Elements: Attributes</span></h4>

<p><a id="X8557F75878DEEA58" name="X8557F75878DEEA58"></a></p>

<h5>5.4-1 Annihilator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Annihilator</code>( <var class="Arg">e</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>

<p>The annihilator of the object element <var class="Arg">e</var> as a subobject of the structure object.</p>

<p><a id="X865C489C7FE750A9" name="X865C489C7FE750A9"></a></p>

<h4>5.5 <span class="Heading">Elements: Operations and Functions</span></h4>

<p><a id="X7A4ED0528640EEFE" name="X7A4ED0528640EEFE"></a></p>

<h5>5.5-1 in</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ in</code>( <var class="Arg">m</var>, <var class="Arg">N</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Is the element <var class="Arg">m</var> of the object <span class="SimpleMath">\(M\)</span> included in the subobject <var class="Arg">N</var><span class="SimpleMath">\(\leq M\)</span>, i.e., does the morphism (with the unit object as source and <span class="SimpleMath">\(M\)</span> as target) underling the element <var class="Arg">m</var> of <span class="SimpleMath">\(M\)</span> factor over the subobject morphism <var class="Arg">N</var><span class="SimpleMath">\(\to M\)</span>?</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 := 2 * zz;</span>
<A free left module of rank 2 on free generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := HomalgModuleElement( "[ 6, 0 ]", M );</span>
( 6, 0 )
<span class="GAPprompt">gap></span> <span class="GAPinput">N := Subobject( HomalgMap( "[ 2, 0 ]", 1 * zz, M ) );</span>
<A free left submodule given by a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">K := Subobject( HomalgMap( "[ 4, 0 ]", 1 * zz, M ) );</span>
<A free left submodule given by a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">a in M;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a in N;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a in UnderlyingObject( N );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a in K;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">a in UnderlyingObject( K );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">a in 3 * zz;</span>
false 
</pre></div>


<div class="example"><pre>
InstallMethod( \in,
        "for homalg elements",
        [ IsHomalgElement, IsStaticFinitelyPresentedSubobjectRep ],
        
  function( m, N )
    local phi, psi;
    
    phi := UnderlyingMorphism( m );
    
    psi := MorphismHavingSubobjectAsItsImage( N );
    
    if not IsIdenticalObj( Range( phi ), Range( psi ) ) then
        Error( "the super object of the subobject and the range ",
               "of the morphism underlying the element do not coincide\n" );
    fi;
    
    return IsZero( PreCompose( phi, CokernelEpi( psi ) ) );
    
end );
</pre></div>


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