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<div class="ChapSects"><a href="chap9_mj.html#X7F2BA6DC7AD84FBC">9 <span class="Heading">Technical Notes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7C5B725A7AED3C8D">9.1 <span class="Heading">Many object structures</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7A4760E887C9C968">9.2 <span class="Heading">Many object homomorphisms</span></a>
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<h3>9 <span class="Heading">Technical Notes</span></h3>

<p>This short chapter is included for the benefit of anyone wishing to implement some other variety of many-object structures, for example <em>ringoids</em>, which are rings with many objects; <em>Lie groupoids</em>, which are Lie groups with many objects; and so on.</p>

<p><a id="X7C5B725A7AED3C8D" name="X7C5B725A7AED3C8D"></a></p>

<h4>9.1 <span class="Heading">Many object structures</span></h4>

<p>Structures with many objects, and their elements, are defined in a manner similar to the single object case. For elements we have:</p>


<ul>
<li><p><code class="code">DeclareCategory( "IsMultiplicativeElementWithObjects", IsMultiplicativeElement ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsMultiplicativeElementWithObjectsAndOnes", IsMultiplicativeElementWithObjects ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsMultiplicativeElementWithObjectsAndInverses", IsMultiplicativeElementWithObjectsAndOnes ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsGroupoidElement", </code></p>

<p><code class="code">IsMultiplicativeElementWithObjectsAndInverses ); </code></p>

</li>
</ul>
<p>as well as various category collections. For the various structures we have:</p>


<ul>
<li><p><code class="code">DeclareCategory( "IsDomainWithObjects", IsDomain ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsMagmaWithObjects", IsDomainWithObjects and IsMultiplicativeElementWithObjectsCollection );</code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsSemigroupWithObjects", IsMagmaWithObjects and IsAssociative ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsMonoidWithObjects", IsSemigroupWithObjects and IsMultiplicativeElementWithObjectsAndOnesCollection ); </code></p>

<p><code class="code">IsMultiplicativeElementWithObjectsAndInversesCollection ); </code></p>

</li>
<li><p><code class="code">DeclareCategory( "IsGroupoid", IsMonoidWithObjects and IsGroupoidElementCollection ); </code></p>

</li>
</ul>
<p>Among the groupoids constructed earlier are the single piece <code class="code">Gd8</code> and the five component union <code class="code">U5</code>:</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">CategoriesOfObject( Gd8 );</span>
[ "IsListOrCollection", "IsCollection", "IsExtLElement", 
  "CategoryCollections(IsExtLElement)", "IsExtRElement", 
  "CategoryCollections(IsExtRElement)", 
  "CategoryCollections(IsMultiplicativeElement)", "IsGeneralizedDomain", 
  "IsMagma", "IsDomainWithObjects", 
  "CategoryCollections(IsMultiplicativeElementWithObjects)", 
  "CategoryCollections(IsMultiplicativeElementWithObjectsAndOnes)", 
  "CategoryCollections(IsMultiplicativeElementWithObjectsAndInverses)\
", "CategoryCollections(IsGroupoidElement)", "IsMagmaWithObjects",
  "IsMagmaWithObjectsAndOnes", "IsMagmaWithObjectsAndInverses", 
  "IsGroupoid" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FamilyObj( Gd8 );     ## these numbers vary from one run to another</span>
NewFamily( "GroupoidFamily", [ 2722 ], [ 53, 54, 79, 80, 81, 82, 92, 93, 116, 
  117, 119, 120, 123, 205, 501, 2690, 2703, 2707, 2711, 2715, 2718, 2720, 
  2721, 2722 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject( Gd8 ); </span>
[ "Name", "Size", "ObjectList", "GeneratorsOfMagmaWithObjects", 
  "GeneratorsOfGroupoid" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownTruePropertiesOfObject( Gd8 ); </span>
[ "IsNonTrivial", "IsFinite", "IsDuplicateFree", "IsAssociative", 
  "IsSinglePieceDomain", "IsDirectProductWithCompleteDigraphDomain" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentationsOfObject( Gd8 );</span>
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsMWOSinglePieceRep" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentationsOfObject( U5 ); </span>
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPiecesRep" ]

</pre></div>

<p>Similarly, for arrows, we have:</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">[ a78, e2 ];</span>
[ [m2 : -7 -> -8], [(1,3) : -8 -> -7] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CategoriesOfObject(a78);</span>
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement", 
  "IsMultiplicativeElementWithObjects" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FamilyObj( a78 );     ## again these numbers vary</span>
NewFamily( "MultiplicativeElementWithObjectsFamily", [ 2702 ], 
[ 79, 80, 81, 82, 116, 119, 122, 2702 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">CategoriesOfObject(e2); </span>
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement", 
  "IsMultiplicativeElementWithObjects", 
  "IsMultiplicativeElementWithObjectsAndOnes", 
  "IsMultiplicativeElementWithObjectsAndInverses", 
  "IsGroupoidElement" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FamilyObj( e2 );        </span>
NewFamily( "GroupoidElementFamily", [ 2714 ], 
[ 79, 80, 81, 82, 116, 119, 122, 2702, 2706, 2710, 2714 ] )

</pre></div>

<p><a id="X7A4760E887C9C968" name="X7A4760E887C9C968"></a></p>

<h4>9.2 <span class="Heading">Many object homomorphisms</span></h4>

<p>Homomorphisms of structures with many objects have a similar heirarchy. A few examples:</p>


<ul>
<li><p><code class="code">DeclareCategory( "IsGeneralMappingWithObjects", IsGeneralMapping ); </code></p>

</li>
<li><p><code class="code">DeclareSynonymAttr( "IsMagmaWithObjectsGeneralMapping", IsGeneralMappingWithObjects and RespectsMultiplication ); </code></p>

</li>
<li><p><code class="code">DeclareSynonymAttr( "IsMagmaWithObjectsHomomorphism", IsMagmaWithObjectsGeneralMapping and IsMapping ); </code></p>

</li>
<li><p><code class="code">DeclareCategory("IsGroupoidHomomorphism",IsMagmaWithObjectsHomomorphism); </code></p>

</li>
</ul>
<p>Two forms of representation are used: for mappings to a single piece; and for unions of such mappings:</p>


<ul>
<li><p><code class="code">DeclareRepresentation( "IsMappingToSinglePieceRep", IsMagmaWithObjectsHomomorphism and IsAttributeStoringRep and IsGeneralMapping, [ "Source", "Range", "SinglePieceMappingData" ] ); </code></p>

</li>
<li><p><code class="code">DeclareRepresentation( "IsMappingWithObjectsRep", IsMagmaWithObjectsHomomorphism and IsAttributeStoringRep and IsGeneralMapping, [ "Source", "Range", "PiecesOfMapping" ] ); </code></p>

</li>
</ul>
<p>In previous chapters, <code class="code">hom1</code> was an endofunction on <code class="code">M78</code>; <code class="code">homd8</code> was a homomorphism from <code class="code">Gd8</code> to <code class="code">Gs3</code>; and <code class="code">aut3</code> was an automorphism of <code class="code">Ga4</code>. All homomorphisms have family <code class="code">GeneralMappingWithObjectsFamily</code>. Perhaps it would be better to have separate families for each structure?</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">FamilyObj(hom1);</span>
NewFamily( "GeneralMappingWithObjectsFamily", [ 2726 ], 
[ 79, 80, 81, 82, 116, 119, 122, 126, 130, 149, 412, 2726 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject( hom1 );</span>
[ "Range", "Source", "SinglePieceMappingData" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownTruePropertiesOfObject( hom1 );</span>
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", 
  "IsSingleValued", "RespectsMultiplication", "IsGeneralMappingToSinglePiece",
  "IsGeneralMappingFromSinglePiece", "IsInjectiveOnObjects", 
  "IsSurjectiveOnObjects" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CategoriesOfObject( homd8 );</span>
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement", 
  "IsMultiplicativeElementWithOne", "IsMultiplicativeElementWithInverse", 
  "IsAssociativeElement", "IsGeneralMapping", "IsGeneralMappingWithObjects", 
  "IsGroupoidHomomorphism" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject( homd8 );</span>
[ "Range", "Source", "SinglePieceMappingData", "ImagesOfObjects", "ImageElementsOfRays", 
  "ObjectTransformationOfGroupoidHomomorphism", "RootGroupHomomorphism" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject( aut3 );</span>
[ "Order", "Range", "Source", "SinglePieceMappingData", "ImagesOfObjects", 
  "ImageElementsOfRays", "ObjectTransformationOfGroupoidHomomorphism", 
  "RootGroupHomomorphism" ]

</pre></div>


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