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3 Homomorphisms of many-object structures
 3.1 Homomorphisms of magmas with objects
  3.1-1 MagmaWithObjectsHomomorphism

3 Homomorphisms of many-object structures

A homomorphism f from a magma with objects M to a magma with objects N consists of a map f_O from the objects of M to those of N together with a map f_A from the arrows of M to those of N which is compatible with tail and head and which preserves multiplication:

f_A((a : u \to v)*f(b : v \to w)) ~=~ f_A(a*b : u \to w)

with tail f_O(u) and head f_O(v).

3.1 Homomorphisms of magmas with objects

3.1-1 MagmaWithObjectsHomomorphism
> MagmaWithObjectsHomomorphism( args )( function )
> MagmaHomomorphismFromSinglePiece( src, rng, hom, imobs )( operation )
> HomomorphismToSinglePiece( src, rng, images )( operation )
> HomomorphismByUnion( src, rng, homs )( operation )

As usual, there are a variety of homomorphism constructors. The basic construction is a homomorphism M -> N with both M and N connected, which is implemented as IsHomomorphismToSinglePieceRep with attributes Source, Range and PieceImages. We require the following information:

In the example we construct endomappings of m and M78.


gap> tup1 := [ Tuple([m1,m2]), Tuple([m2,m1]), Tuple([m3,m4]), Tuple([m4,m3]) ]; 
gap> f1 := GeneralMappingByElements( m, m, tup1 ); 
f1 = <general mapping: m -> m >
gap> IsMagmaHomomorphism( f1 ); 
true
gap> tup2 := [ Tuple([m1,m1]), Tuple([m2,m1]), Tuple([m3,m1]), Tuple([m4,m1]) ];; 
gap> f2 := GeneralMappingByElements( m, m, tup2 );; 
gap> IsMagmaHomomorphism( f2 );
true 
gap> map1 := HomomorphismFromSinglePiece( M78, M78, [-8,-7], f1 ); 
magma with objects homomorphism : M78 -> M78
gap> Display( map1 );
Mapping to single piece magma:
[ M78 ] -> [ M78 ]
magma mapping: <mapping: m -> m >
   object map: [ -8, -7 ] -> [ -8, -7 ]
Homomorphism to connected magma:
[ M78 ] -> [ M78 ]
  object map = [ [ -8, -7 ], [ -8, -7 ] ]
homomorphism = <homomorphism: m -> m >
gap> idm := f1*f1;; 
gap> idmap := HomomorphismFromSinglePiece( M78, M78, idm, [-7,-8] ); 
gap> map2 := HomomorphismFromSinglePiece( M78, M78, f2, [-7,-8] ); 

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