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3 Homomorphisms of many-object structures
 3.1 Homomorphisms of magmas with objects
 3.2 Homomorphisms of semigroups and monoids with objects
 3.3 Homomorphisms to more than one piece

3 Homomorphisms of many-object structures

A homomorphism f from a magma with objects M to a magma with objects N consists of

The map f_A is required to be compatible with the tail and head maps and to preserve multiplication:

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

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

When M is a monoid or group, the map f_A is required to preserve object identities and inverses.

3.1 Homomorphisms of magmas with objects

3.1-1 MagmaWithObjectsHomomorphism
‣ MagmaWithObjectsHomomorphism( args )( function )
‣ HomomorphismFromSinglePiece( src, rng, hom, imobs )( operation )
‣ HomomorphismToSinglePiece( src, rng, images )( operation )
‣ PieceImages( mwohom )( attribute )
‣ HomsOfMapping( mwohom )( attribute )
‣ PiecesOfMapping( mwohom )( attribute )
‣ IsomorphismNewObjects( src, objlist )( operation )

There are a variety of homomorphism constructors.

The simplest construction gives a homomorphism M -> N with both M and N connected. It is implemented as IsMappingToSinglePieceRep with attributes Source, Range and PieceImages. The operation requires the following information:

In the first 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 ); 
gap> IsMagmaHomomorphism( f1 ); 
true
gap> hom1 := MagmaWithObjectsHomomorphism( M78, M78, f1, [-8,-7] );; 
gap> Display( hom1 );
homomorphism to single piece magma: M78 -> M78
magma hom: <mapping: m -> m >, object map: [ -8, -7 ] -> [ -8, -7 ]
gap> [ Source( hom1 ), Range( hom1 ) ]; 
[ M78, M78 ]
gap> b87;
[m4 : -8 -> -7]
gap> im1 := ImageElm( hom1, b87 );
[m3 : -8 -> -7]
gap> i56 := IsomorphismNewObjects( M78, [-5,-6] ); 
magma with objects homomorphism : 
[ [ IdentityMapping( m ), [ -5, -6 ] ] ]
gap> M65 := Range( i56);; 
gap> SetName( M65, "M65" ); 
gap> j56 := InverseGeneralMapping( i56 );; 
gap> ImagesOfObjects( j56 ); 
[ -7, -8 ]
gap> ib87 := ImageElm( i56, b87 );
[m4 : -5 -> -6]
gap> comp := j56 * hom1;
magma with objects homomorphism : M65 -> M78
[ [ <mapping: m -> m >, [ -7, -8 ] ] ]
gap> ImageElm( comp, ib87 );
[m3 : -8 -> -7]

A homomorphism to a connected magma with objects may have a source with several pieces, and so is a union of homomorphisms from single pieces.


gap> M4 := UnionOfPieces( [ M78, M65 ] );;
gap> images := [ PieceImages( hom1 )[1], PieceImages( j56 )[1] ]; 
[ [ <mapping: m -> m >, [ -8, -7 ] ], [ IdentityMapping( m ), [ -7, -8 ] ] ]
gap> map4 := HomomorphismToSinglePiece( M4, M78, images ); 
magma with objects homomorphism : 
[ [ <mapping: m -> m >, [ -8, -7 ] ], [ IdentityMapping( m ), [ -7, -8 ] ] ]
gap> ImageElm( map4, b87 );
[m3 : -8 -> -7]
gap> ImageElm( map4, ib87 );
[m4 : -8 -> -7]

3.2 Homomorphisms of semigroups and monoids with objects

The next example exhibits a homomorphism between transformation semigroups with objects.


gap> t2 := Transformation( [2,2,4,1] );; 
gap> s2 := Transformation( [1,1,4,4] );;
gap> r2 := Transformation( [4,1,3,3] );; 
gap> sgp2 := Semigroup( [ t2, s2, r2 ] );;
gap> SetName( sgp2, "sgp<t2,s2,r2>" );
gap> ##  apparently no method for transformation semigroups available for: 
gap> ##  nat := NaturalHomomorphismByGenerators( sgp, sgp2 );  so we use: 
gap> ##  in the function flip below t is a transformation on [1..n] 
gap> flip := function( t ) 
>     local i, j, k, L, L2, n; 
>     n := DegreeOfTransformation( t );  
>     L := ImageListOfTransformation( t ); 
>     if IsOddInt(n) then n:=n+1; L1:=Concatenation(L,[n]); 
>                    else L1:=L; fi; 
>     L2 := ShallowCopy( L1 );
>     for i in [1..n] do 
>         if IsOddInt(i) then j:=i+1; else j:=i-1; fi; 
>         k := L1[j]; 
>         if IsOddInt(k) then L2[i]:=k+1; else L2[i]:=k-1; fi; 
>     od; 
>     return( Transformation( L2 ) ); 
> end;; 
gap> smap := MappingByFunction( sgp, sgp2, flip );; 
gap> ok := RespectsMultiplication( smap ); 
true
gap> [ t, ImageElm( smap, t ) ]; 
[ Transformation( [ 1, 1, 2, 3 ] ), Transformation( [ 2, 2, 4, 1 ] ) ]
gap> [ s, ImageElm( smap, s ) ]; 
[ Transformation( [ 2, 2, 3, 3 ] ), Transformation( [ 1, 1, 4, 4 ] ) ]
gap> [ r, ImageElm( smap, r ) ]; 
[ Transformation( [ 2, 3, 4, 4 ] ), Transformation( [ 4, 1, 3, 3 ] ) ]
gap> SetName( smap, "smap" ); 
gap> T123 := SemigroupWithObjects( sgp2, [-13,-12,-11] );; 
gap> shom := MagmaWithObjectsHomomorphism( S123, T123, smap, [-11,-12,-13] );; 
gap> it12 := ImageElm( shom, t12 );;  [ t12, it12 ]; 
[ [Transformation( [ 1, 1, 2, 3 ] ) : -1 -> -2], 
  [Transformation( [ 2, 2, 4, 1 ] ) : -13 -> -12] ]
gap> is23 := ImageElm( shom, s23 );;  [ s23, is23 ]; 
[ [Transformation( [ 2, 2, 3, 3 ] ) : -2 -> -3], 
  [Transformation( [ 1, 1, 4, 4 ] ) : -12 -> -11] ]
gap> ir31 := ImageElm( shom, r31 );;  [ r31, ir31 ]; 
[ [Transformation( [ 2, 3, 4, 4 ] ) : -3 -> -1], 
  [Transformation( [ 4, 1, 3, 3 ] ) : -11 -> -13] ]

3.3 Homomorphisms to more than one piece

3.3-1 HomomorphismByUnion
‣ HomomorphismByUnion( src, rng, homs )( operation )
‣ IsInjectiveOnObjects( mwohom )( property )
‣ IsSurjectiveOnObjects( mwohom )( property )
‣ IsBijectiveOnObjects( mwohom )( property )
‣ IsEndomorphismWithObjects( mwohom )( property )
‣ IsAutomorphismWithObjects( mwohom )( property )

When f : M -> N and N has more than one connected component, then f is a union of homomorphisms, one for each piece in the range.


gap> N4 := UnionOfPieces( [ M78, T123 ] );
magma with objects having 2 pieces :-
1: semigroup with objects :-
    magma = sgp<t2,s2,r2>
  objects = [ -13, -12, -11 ]
2: M78
gap> h14 := HomomorphismByUnionNC( N1, N4, [ hom1, shom ] );
magma with objects homomorphism : 
[ magma with objects homomorphism : M78 -> M78
    [ [ <mapping: m -> m >, [ -8, -7 ] ] ], magma with objects homomorphism : 
    [ [ smap, [ -11, -12, -13 ] ] ] ]
gap> IsInjectiveOnObjects( h14 );
true
gap> IsSurjectiveOnObjects( h14 );
true
gap> IsBijectiveOnObjects( h14 ); 
true
gap> ImageElm( h14, t12 );
[Transformation( [ 2, 2, 4, 1 ] ) : -13 -> -12]
gap> h45 := IsomorphismNewObjects( N4, [ [-103,-102,-101], [-108,-107] ] );
magma with objects homomorphism : 
[ magma with objects homomorphism : 
    [ [ IdentityMapping( m ), [ -108, -107 ] ] ], 
  magma with objects homomorphism : 
    [ [ IdentityMapping( sgp<t2,s2,r2> ), [ -103, -102, -101 ] ] ] ]
gap> N5 := Range( h45 );;  SetName( N5, "N5" ); 
gap> h15 := h14 * h45;
magma with objects homomorphism : 
[ magma with objects homomorphism : [ [ <mapping: m -> m >, [ -108, -107 ] ] ]
    , magma with objects homomorphism : [ [ smap, [ -101, -102, -103 ] ] ] ]
gap> ImageElm( h15, t12 ); 
[Transformation( [ 2, 2, 4, 1 ] ) : -103 -> -102]

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