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7 Technical Notes
 7.1 Many object structures
 7.2 Many object homomorphisms

7 Technical Notes

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

7.1 Many object structures

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

as well as various category collections. For the various structures we have:

Among the groupoids constructed earlier are the single piece Gd8 and the five component union U5:


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", "GeneratorsOfMagmaWithInverses", "ObjectList", "Pieces" ]
gap> KnownTruePropertiesOfObject( Gd8 ); 
[ "IsNonTrivial", "IsFinite", "CanEasilyCompareElements", 
  "CanEasilySortElements", "IsDuplicateFree", "IsAssociative", 
  "IsSinglePieceDomain", "IsDirectProductWithCompleteGraphDomain" ]
gap> RepresentationsOfObject( Gd8 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsMWOSinglePieceRep" ]
gap> RepresentationsOfObject( U5 ); 
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPiecesRep" ]

Similarly, for arrows, we have:


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 ] )

7.2 Many object homomorphisms

Homomorphisms of structures with many objects have a similar heirarchy. A few examples:

Two forms of representation are used: for mappings to a single piece; and for unions of such mappings:

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


gap> FamilyObj(hom1);
NewFamily( "GeneralMappingWithObjectsFamily", [ 2726 ], 
[ 79, 80, 81, 82, 116, 119, 122, 126, 130, 149, 412, 2726 ] )
gap> KnownAttributesOfObject( hom1 );
[ "Range", "Source", "PieceImages" ]
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", "PieceImages", "ImagesOfObjects", "ImageElementsOfRays", 
  "ObjectTransformationOfGroupoidHomomorphism", "RootGroupHomomorphism" ]
gap> KnownAttributesOfObject( aut3 );
[ "Order", "Range", "Source", "PieceImages", "ImagesOfObjects", 
  "ImageElementsOfRays", "ObjectTransformationOfGroupoidHomomorphism", 
  "RootGroupHomomorphism" ]

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