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<?xml version="1.0" encoding="UTF-8"?>

<Chapter Label="chap-tecnotes">
<Heading>Technical Notes</Heading>

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

<Section Label="sec-tec-mwo">
<Heading>Many object structures</Heading>

Structures with many objects, and their elements, are defined in a manner similar to the single object case. For elements we have:
<List>
<Item>
<C>DeclareCategory( "IsMultiplicativeElementWithObjects",
    IsMultiplicativeElement ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsMultiplicativeElementWithObjectsAndOnes",
    IsMultiplicativeElementWithObjects ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsMultiplicativeElementWithObjectsAndInverses",
    IsMultiplicativeElementWithObjectsAndOnes ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsGroupoidElement", </C>
<P/>
<C>IsMultiplicativeElementWithObjectsAndInverses ); </C>
</Item>
</List>
as well as various category collections.
For the various structures we have:
<List>
<Item>
<C>DeclareCategory( "IsDomainWithObjects", IsDomain ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsMagmaWithObjects", IsDomainWithObjects and
    IsMultiplicativeElementWithObjectsCollection );</C>
</Item>
<Item>
<C>DeclareCategory( "IsSemigroupWithObjects",
    IsMagmaWithObjects and IsAssociative ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsMonoidWithObjects",
    IsSemigroupWithObjects and
    IsMultiplicativeElementWithObjectsAndOnesCollection ); </C>
<P/>
<C>IsMultiplicativeElementWithObjectsAndInversesCollection ); </C>
</Item>
<Item>
<C>DeclareCategory( "IsGroupoid", IsMonoidWithObjects and
    IsGroupoidElementCollection ); </C>
</Item>
</List>

Among the groupoids constructed earlier are the single piece
<C>Gd8</C> and the five component union <C>U5</C>:
<P/>
<Example>
<![CDATA[
gap> CategoriesOfObject( Gd8 );
[ "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" ]
gap> FamilyObj( Gd8 );     ## these numbers vary from one run to another
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 ] )
gap> KnownAttributesOfObject( Gd8 );
[ "Name", "Size", "ObjectList", "GeneratorsOfMagmaWithObjects",
  "GeneratorsOfGroupoid" ]
gap> KnownTruePropertiesOfObject( Gd8 );
[ "IsNonTrivial", "IsFinite", "IsDuplicateFree", "IsAssociative",
  "IsSinglePieceDomain", "IsDirectProductWithCompleteDigraphDomain" ]
gap> RepresentationsOfObject( Gd8 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsMWOSinglePieceRep" ]
gap> RepresentationsOfObject( U5 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPiecesRep" ]
]]>
</Example>

Similarly, for arrows, we have:
<P/>
<Example>
<![CDATA[
gap> [ a78, e2 ];
[ [m2 : -7 -> -8], [(1,3) : -8 -> -7] ]
gap> CategoriesOfObject(a78);
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithObjects" ]
gap> FamilyObj( a78 );     ## again these numbers vary
NewFamily( "MultiplicativeElementWithObjectsFamily", [ 2702 ],
[ 79, 80, 81, 82, 116, 119, 122, 2702 ] )
gap> CategoriesOfObject(e2);
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithObjects",
  "IsMultiplicativeElementWithObjectsAndOnes",
  "IsMultiplicativeElementWithObjectsAndInverses",
  "IsGroupoidElement" ]
gap> FamilyObj( e2 );
NewFamily( "GroupoidElementFamily", [ 2714 ],
[ 79, 80, 81, 82, 116, 119, 122, 2702, 2706, 2710, 2714 ] )
]]>
</Example>
</Section>

<Section Label="sec-tec-mwohom">
<Heading>Many object homomorphisms</Heading>

Homomorphisms of structures with many objects have a similar heirarchy.
A few examples:
<List>
<Item>
<C>DeclareCategory( "IsGeneralMappingWithObjects", IsGeneralMapping ); </C>
</Item>
<Item>
<C>DeclareSynonymAttr( "IsMagmaWithObjectsGeneralMapping",
    IsGeneralMappingWithObjects and RespectsMultiplication ); </C>
</Item>
<Item>
<C>DeclareSynonymAttr( "IsMagmaWithObjectsHomomorphism",
    IsMagmaWithObjectsGeneralMapping and IsMapping ); </C>
</Item>
<Item>
<C>DeclareCategory("IsGroupoidHomomorphism",IsMagmaWithObjectsHomomorphism);
</C>
</Item>
</List>
<P/>
Two forms of representation are used: for mappings to a single piece;
and for unions of such mappings:
<List>
<Item>
<C>DeclareRepresentation( "IsMappingToSinglePieceRep",
    IsMagmaWithObjectsHomomorphism and IsAttributeStoringRep
        and IsGeneralMapping, [ "Source", "Range", "SinglePieceMappingData" ] ); </C>
</Item>
<Item>
<C>DeclareRepresentation( "IsMappingWithObjectsRep",
    IsMagmaWithObjectsHomomorphism and IsAttributeStoringRep
        and IsGeneralMapping, [ "Source", "Range", "PiecesOfMapping" ] ); </C>
</Item>
</List>
<P/>
In previous chapters, <C>hom1</C> was an endofunction on <C>M78</C>;
<C>homd8</C> was a homomorphism from <C>Gd8</C> to <C>Gs3</C>;
and  <C>aut3</C> was an automorphism of <C>Ga4</C>.
All homomorphisms have family <C>GeneralMappingWithObjectsFamily</C>.
Perhaps it would be better to have separate families for each structure?
<Example>
<![CDATA[
gap> FamilyObj(hom1);
NewFamily( "GeneralMappingWithObjectsFamily", [ 2726 ],
[ 79, 80, 81, 82, 116, 119, 122, 126, 130, 149, 412, 2726 ] )
gap> KnownAttributesOfObject( hom1 );
[ "Range", "Source", "SinglePieceMappingData" ]
gap> KnownTruePropertiesOfObject( hom1 );
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal",
  "IsSingleValued", "RespectsMultiplication", "IsGeneralMappingToSinglePiece",
  "IsGeneralMappingFromSinglePiece", "IsInjectiveOnObjects",
  "IsSurjectiveOnObjects" ]
gap> CategoriesOfObject( homd8 );
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithOne", "IsMultiplicativeElementWithInverse",
  "IsAssociativeElement", "IsGeneralMapping", "IsGeneralMappingWithObjects",
  "IsGroupoidHomomorphism" ]
gap> KnownAttributesOfObject( homd8 );
[ "Range", "Source", "SinglePieceMappingData", "ImagesOfObjects", "ImageElementsOfRays",
  "ObjectTransformationOfGroupoidHomomorphism", "RootGroupHomomorphism" ]
gap> KnownAttributesOfObject( aut3 );
[ "Order", "Range", "Source", "SinglePieceMappingData", "ImagesOfObjects",
  "ImageElementsOfRays", "ObjectTransformationOfGroupoidHomomorphism",
  "RootGroupHomomorphism" ]
]]>
</Example>

</Section>

</Chapter>

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