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### 3 Mappings 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 $$N$$,

• a map $$f_A$$ from the arrows of $$M$$ to those of $$N$$.

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 the underlying magma of $$M$$ is a monoid or group, the map $$f_A$$ is required to preserve 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 )
 ‣ MappingToSinglePieceData( mwohom ) ( attribute )
 ‣ PiecesOfMapping( mwohom ) ( attribute )
 ‣ IsomorphismNewObjects( src, objlist ) ( operation )

There are a variety of homomorphism constructors.

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

• a magma homomorphism hom from the underlying magma of $$M$$ to the underlying magma of $$N$$,

• a list imobs of the images of the objects of $$M$$.

In the first example we construct endomappings of m and M78.


gap> tup1 := [ DirectProductElement([m1,m2]), DirectProductElement([m2,m1]),
>              DirectProductElement([m3,m4]), DirectProductElement([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> ib87 := ImageElm( i56, b87 );
[m4 : -5 -> -6]
gap> M65 := Range( i56);;
gap> SetName( M65, "M65" );
gap> j56 := InverseGeneralMapping( i56 );;
gap> ImagesOfObjects( j56 );
[ -7, -8 ]
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 := [ MappingToSinglePieceData( hom1 ),
> MappingToSinglePieceData( j56 ) ];
[ [ <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 )

When $$f : M \to N$$ 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> ImageElm( h14, a78 );
[m1 : -7 -> -8]
gap> ImageElm( h14, r31 );
[Transformation( [ 4, 1, 3, 3 ] ) : -11 -> -13]



##### 3.3-2 IsInjectiveOnObjects
 ‣ IsInjectiveOnObjects( mwohom ) ( property )
 ‣ IsSurjectiveOnObjects( mwohom ) ( property )
 ‣ IsBijectiveOnObjects( mwohom ) ( property )
 ‣ IsEndomorphismWithObjects( mwohom ) ( property )
 ‣ IsAutomorphismWithObjects( mwohom ) ( property )

The meaning of these five properties is obvious.


gap> IsInjectiveOnObjects( h14 );
true
gap> IsSurjectiveOnObjects( h14 );
true
gap> IsBijectiveOnObjects( h14 );
true
gap> IsEndomorphismWithObjects( h14 );
false
gap> IsAutomorphismWithObjects( h14 );
false



#### 3.4 Mappings defined by a function

##### 3.4-1 MappingWithObjectsByFunction
 ‣ MappingWithObjectsByFunction( src, rng, fun, imobs ) ( operation )
 ‣ IsMappingWithObjectsByFunction( map ) ( property )
 ‣ UnderlyingFunction( map ) ( attribute )

More general mappings, which need not preserve multiplication, are available using this operation. See section 5.6 for an application.


gap> flip := function(a) return Arrow(M78,a!,a!,a!); end;
function( a ) ... end
gap> flipmap := MappingWithObjectsByFunction( M78, M78, flip, [-8,-7] );
magma with objects mapping by function : M78 -> M78
function: function ( a )
return Arrow( M78, a!, a!, a! );
end
gap> a78; ImageElm( flipmap, a78 );
[m2 : -7 -> -8]
[m2 : -8 -> -7]


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