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.
‣ 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 -> 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 )[1], > MappingToSinglePieceData( 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]
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] ]
‣ HomomorphismByUnion( src, rng, homs ) | ( operation ) |
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> ImageElm( h14, a78 ); [m1 : -7 -> -8] gap> ImageElm( h14, r31 ); [Transformation( [ 4, 1, 3, 3 ] ) : -11 -> -13]
‣ 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
‣ 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![1],a![3],a![2]); 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![1], a![3], a![2] ); end gap> a78; ImageElm( flipmap, a78 ); [m2 : -7 -> -8] [m2 : -8 -> -7]
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