NC versions of PreImages, PreImagesSet, PreImagesElm and PreImagesRepresentative

doc/ref/grpfp.xml

@@ -466,7 +466,7 @@ gap> hom:=IsomorphismFpGroup(u);
 [ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ] -> [ F1 ]
 gap> new:=Range(hom);
 <fp group on the generators [ F1 ]>
-gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentative(hom,i));
+gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentativeNC(hom,i));
 [ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ]
 ]]></Example>
 <P/>
@@ -484,7 +484,7 @@ behave like ordinary words and no extra treatment should be necessary.
 <Example><![CDATA[
 gap> r:=Range(hom).1^10;
 F1^10
-gap> p:=PreImagesRepresentative(hom,r);
+gap> p:=PreImagesRepresentativeNC(hom,r);
 <[ [ 1, 10 ] ]|(f2^-1*f1^-1)^10>
 ]]></Example>
 

doc/ref/mapping.xml

@@ -1,6 +1,6 @@
 <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
 <!-- %% -->
-<!-- %A  mapping.xml                  GAP documentation              Thomas Breuer -->
+<!-- %A  mapping.xml               GAP documentation        Thomas Breuer -->
 <!-- %% -->
 <!-- %% -->
 <!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->

doc/ref/rws.xml

@@ -174,7 +174,7 @@ gap> red:=ReducedForm(k,UnderlyingElement(melm));
 f1^-1^3*f2^-1*f1^2
 gap> melm:=ElementOfFpMonoid(FamilyObj(One(mon)),red);
 f1^-1^3*f2^-1*f1^2
-gap> gpelm:=PreImagesRepresentative(mhom,melm);
+gap> gpelm:=PreImagesRepresentativeNC(mhom,melm);
 f1^-3*f2^-1*f1^2
 gap> r=gpelm;
 true

doc/tut/group.xml

@@ -1,12 +1,12 @@
-<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
+<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
 <!-- %% -->
-<!-- %W  group.tex                 GAP documentation             Thomas Breuer -->
-<!-- %W                                                         & Frank Celler -->
-<!-- %W                                                     & Martin Schönert -->
-<!-- %W                                                       & Heiko Theißen -->
+<!-- %W  group.tex           GAP documentation               Thomas Breuer -->
+<!-- %W                                                     & Frank Celler -->
+<!-- %W                                                  & Martin Schönert -->
+<!-- %W                                                    & Heiko Theißen -->
 <!-- %% -->
 <!-- %% -->
-<!-- %Y  Copyright 1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,   Germany -->
+<!-- %Y  Copyright 1997,  Lehrstuhl D für Mathematik, RWTH Aachen, Germany -->
 <!-- %% -->
 <!-- %%  This file contains a tutorial introduction to groups. -->
 <!-- %% -->
@@ -168,10 +168,9 @@ action domain of <C>norm</C>. (It only happens that both are subsets of the
 natural numbers.) We can now form  images and preimages under the natural
 homomorphism. The set of  preimages of an element under  <C>hom</C> is a coset
 modulo <C>elab</C>.
-We use the function <Ref Func="PreImages" BookName="ref"/> here because
-<C>hom</C> is not
-a bijection, so an  element of the range can   have several preimages  or
-none at all.
+We use the function <Ref Func="PreImages" BookName="ref"/> here 
+because <C>hom</C> is not a bijection, 
+so an element of the range can have several preimages or none at all.
 <P/>
 <Example><![CDATA[
 gap> ker:= Kernel( hom );
@@ -942,18 +941,20 @@ gap> PreImage( hom, TrivialSubgroup(s3) );  # the kernel
 Group([ (1,4)(2,3), (1,3)(2,4) ])
 ]]></Example>
 <P/>
-This homomorphism  from <M>S_4</M> onto  <M>S_3</M>  is well known  from elementary
-group theory.  Images   of elements and  subgroups  under   <C>hom</C> can  be
+This homomorphism  from <M>S_4</M> onto <M>S_3</M> is well known from 
+elementary group theory.  
+Images of elements and subgroups under <C>hom</C> can be
 calculated with the function <Ref Func="Image" BookName="ref"/>.
-But since the mapping <C>hom</C> is  not bijective, we  cannot use the
-function <Ref Func="PreImage" BookName="ref"/> for  preimages   of
-elements  (they can have   several preimages). Instead,   we have  to use
-<Ref Func="PreImagesRepresentative" BookName="ref"/>,
-which  returns  one  preimage if at  least one
-exists (and would  return <K>fail</K> if none  exists, which  cannot occur for
+But since the mapping <C>hom</C> is not bijective, we cannot use the
+function <Ref Func="PreImage" BookName="ref"/> for preimages of elements 
+(they can have several preimages).  
+Instead, we have to use 
+<Ref Func="PreImagesRepresentative" BookName="ref"/> 
+which returns one preimage if at least one exists 
+(and would return <K>fail</K> if none exists, which cannot occur for
 our surjective <C>hom</C>).
-On the other hand, we  can use <Ref Func="PreImage" BookName="ref"/> for the
-preimage of a set (which always exists, even if it  is empty).
+On the other hand, we can use <Ref Func="PreImage" BookName="ref"/> for the
+preimage of a set (which always exists, even if it is empty).
 <P/>
 Suppose we mistype the input when trying to construct a homomorphism as below.
 <P/>

lib/algebra.gi

@@ -3672,7 +3672,7 @@ InstallMethod( CentralIdempotentsOfAlgebra,
 
       until k>Length(ideals);
 
-      id:= List( ids, e -> PreImagesRepresentative( hom, e ) );
+      id:= List( ids, e -> PreImagesRepresentativeNC( hom, e ) );
 
       # Now we lift the idempotents to the big algebra `A'. The
       # first idempotent is lifted as follows:

lib/algfp.gi

@@ -705,7 +705,7 @@
       if hom = fail then
         TryNextMethod();
       fi;
-      return PreImagesRepresentative( hom, r );
+      return PreImagesRepresentativeNC( hom, r );
       end ) );
 
 

lib/alghom.gi

@@ -577,26 +577,26 @@
 
 #############################################################################
 ##
-#M  PreImagesRepresentative( <map>, <elm> ) . . . . . .  for algebra g.m.b.i.
+#M  PreImagesRepresentativeNC( <map>, <elm> ) . . . . .  for algebra g.m.b.i.
 ##
-InstallMethod( PreImagesRepresentative,
+InstallMethod( PreImagesRepresentativeNC,
     "for algebra g.m.b.i., and element",
     FamRangeEqFamElm,
     [ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep,
       IsObject ],
     function( map, elm )
-    return PreImagesRepresentative( AsLeftModuleGeneralMappingByImages(map),
-                                    elm );
+        return PreImagesRepresentativeNC(
+                   AsLeftModuleGeneralMappingByImages(map), elm );
     end );
 
-InstallMethod( PreImagesRepresentative,
+InstallMethod( PreImagesRepresentativeNC,
     "for algebra g.m.b.i. knowing inverse, and element",
     FamRangeEqFamElm,
     [ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep
       and HasInverseGeneralMapping,
       IsObject ],
     function( map, elm )
-    return ImagesRepresentative( InverseGeneralMapping(map), elm );
+        return ImagesRepresentative( InverseGeneralMapping(map), elm );
     end );
 
 
@@ -915,7 +915,7 @@
 
 #############################################################################
 ##
-#M  PreImagesRepresentative( <ophom>, <mat> )
+#M  PreImagesRepresentativeOperationAlgebraHomomorphism( <ophom>, <mat> )
 ##
 BindGlobal( "PreImagesRepresentativeOperationAlgebraHomomorphism", function( ophom, mat )
     if not IsBound( ophom!.basisImage ) then
@@ -928,7 +928,7 @@
     return mat;
 end );
 
-InstallMethod( PreImagesRepresentative,
+InstallMethod( PreImagesRepresentativeNC,
     "for an operation algebra homomorphism, and an element",
     FamRangeEqFamElm,
     [ IsOperationAlgebraHomomorphismDefaultRep, IsMatrix ],
@@ -1090,9 +1090,9 @@
 
 #############################################################################
 ##
-#M  PreImagesRepresentative( <ophom>, <mat> )
+#M  PreImagesRepresentativeNC( <ophom>, <mat> )
 ##
-InstallMethod( PreImagesRepresentative,
+InstallMethod( PreImagesRepresentativeNC,
     "for an alg. hom. from f. p. algebra, and an element",
     FamRangeEqFamElm,
     [ IsAlgebraHomomorphismFromFpRep, IsMatrix ],

lib/alglie.gi

@@ -34,7 +34,7 @@
       # under the natural homomorphism.
       Add( S, C );
       hom:= NaturalHomomorphismByIdeal( L, C );
-      C:= PreImages( hom, LieCentre( Range( hom ) ) );
+      C:= PreImagesNC( hom, LieCentre( Range( hom ) ) );
 #T we would like to get ideals!
 #T is it possible to teach the hom. that the preimage of an ideal is an ideal?
 
@@ -1707,7 +1707,7 @@
       quo:= ImagesSource( hom );
       r1:= LieSolvableRadical( quo );
       B:= BasisVectors( Basis( r1 ) );
-      B:= List( B, x -> PreImagesRepresentative( hom, x ) );
+      B:= List( B, x -> PreImagesRepresentativeNC( hom, x ) );
       Append( B, BasisVectors( Basis( n ) ) );
 
     fi;
@@ -2100,7 +2100,7 @@
       SetRadicalOfAlgebra( Q, Subalgebra( Q, [ Zero( Q ) ] ) );
 
       id:= List( CentralIdempotentsOfAlgebra( Q ),
-                                x->PreImagesRepresentative(hom,x));
+                                x->PreImagesRepresentativeNC(hom,x));
 
       # Now we lift the idempotents to the big algebra `A'. The
       # first idempotent is lifted as follows:
@@ -4029,9 +4029,9 @@
 
 ###########################################################################
 ##
-#M   PreImagesRepresentative( f, x )
+#M   PreImagesRepresentativeNC( f, x )
 ##
-InstallMethod( PreImagesRepresentative,
+InstallMethod( PreImagesRepresentativeNC,
     "for Fp to SCA mapping, and element",
     FamRangeEqFamElm,
     [ IsFptoSCAMorphism, IsSCAlgebraObj ], 0,
@@ -5643,8 +5643,8 @@
     T:= EmptySCTable( dim , Zero(F) , "antisymmetric" );
     pimgs := [];
     for i in [1..dim] do
-        a:= PreImagesRepresentative( Homs[pos[i]] ,
-                    PreImagesRepresentative( hom_pcg[pos[i]], gens[i] ) );
+        a:= PreImagesRepresentativeNC( Homs[pos[i]] ,
+                    PreImagesRepresentativeNC( hom_pcg[pos[i]], gens[i] ) );
 
         # calculate the p-th power image of `a':
 
@@ -5661,8 +5661,8 @@
                # Calculate the commutator [a,b], and map the result into
                # the correct homogeneous component.
 
-                b:= PreImagesRepresentative( Homs[pos[j]],
-                         PreImagesRepresentative( hom_pcg[pos[j]], gens[j] ));
+                b:= PreImagesRepresentativeNC( Homs[pos[j]],
+                      PreImagesRepresentativeNC( hom_pcg[pos[j]], gens[j] ));
                 c:= Image( hom_pcg[pos[i] + pos[j]],
                            Image(Homs[pos[i] + pos[j]], a^-1*b^-1*a*b) );
                 e:= ExtRepOfObj(c);
@@ -5839,8 +5839,8 @@
     T:= EmptySCTable( dim , Zero(F) , "antisymmetric" );
     pimgs := [];
     for i in [1..dim] do
-        a:= PreImagesRepresentative( Homs[pos[i]] ,
-                    PreImagesRepresentative( hom_pcg[pos[i]], gens[i] ) );
+        a:= PreImagesRepresentativeNC( Homs[pos[i]] ,
+                    PreImagesRepresentativeNC( hom_pcg[pos[i]], gens[i] ) );
 
 
         # calculate the p-th power image of `a':
@@ -5858,8 +5858,8 @@
                # Calculate the commutator [a,b], and map the result into
                # the correct homogeneous component.
 
-                b:= PreImagesRepresentative( Homs[pos[j]],
-                         PreImagesRepresentative( hom_pcg[pos[j]], gens[j] ));
+                b:= PreImagesRepresentativeNC( Homs[pos[j]],
+                      PreImagesRepresentativeNC( hom_pcg[pos[j]], gens[j] ));
                 c:= Image( hom_pcg[pos[i] + pos[j]],
                            Image(Homs[pos[i] + pos[j]], a^-1*b^-1*a*b) );
                 e:= ExtRepOfObj(c);

lib/algrep.gd

@@ -697,10 +697,12 @@ DeclareOperation( "ModuleByRestriction", [ IsAlgebraModule, IsAlgebra ] );
 ##  <Oper Name="NaturalHomomorphismBySubAlgebraModule" Arg='V, W'/>
 ##
 ##  <Description>
-##  Here <A>V</A> must be a sub-algebra module of <A>V</A>. This function returns
-##  the projection from <A>V</A> onto <C><A>V</A>/<A>W</A></C>. It is a linear map, that is
-##  also a module homomorphism. As usual images can be formed with
-##  <C>Image( f, v )</C> and pre-images with <C>PreImagesRepresentative( f, u )</C>.
+##  Here <A>V</A> must be a sub-algebra module of <A>V</A>.
+##  This function returns
+##  the projection from <A>V</A> onto <C><A>V</A>/<A>W</A></C>.
+##  It is a linear map, that is also a module homomorphism.
+##  As usual images can be formed with <C>Image( f, v )</C>
+##  and pre-images with <C>PreImagesRepresentativeNC( f, u )</C>.
 ##  <P/>
 ##  The quotient module can also be formed
 ##  by entering <C><A>V</A>/<A>W</A></C>.
@@ -726,7 +728,7 @@ DeclareOperation( "ModuleByRestriction", [ IsAlgebraModule, IsAlgebra ] );
 ##  18 over Rationals>>
 ##  gap> v:= Basis( quo )[1];
 ##  [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
-##  gap> PreImagesRepresentative( f, v );
+##  gap> PreImagesRepresentativeNC( f, v );
 ##  v.4
 ##  gap> Basis( C )[4]^v;
 ##  [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]

lib/algrep.gi

@@ -1212,24 +1212,24 @@
     if IsLeftAlgebraModuleElementCollection( V ) then
         if IsRightAlgebraModuleElementCollection( V ) then
             left_op:= function( x, v )
-                 return ImagesRepresentative( f, x^PreImagesRepresentative( f, v ) );
+                 return ImagesRepresentative( f, x^PreImagesRepresentativeNC( f, v ) );
             end;
             right_op:= function( v, x )
-                 return ImagesRepresentative( f, PreImagesRepresentative( f, v )^x );
+                 return ImagesRepresentative( f, PreImagesRepresentativeNC( f, v )^x );
             end;
             qmod:= BiAlgebraModule( LeftActingAlgebra( V ),
                            RightActingAlgebra( V ),
                            left_op, right_op, quot );
         else
             left_op:= function( x, v )
-                 return ImagesRepresentative( f, x^PreImagesRepresentative( f, v ) );
+                 return ImagesRepresentative( f, x^PreImagesRepresentativeNC( f, v ) );
             end;
             qmod:= LeftAlgebraModule( LeftActingAlgebra( V ),
                            left_op, quot);
         fi;
     else
         right_op:= function( v, x )
-             return ImagesRepresentative( f, PreImagesRepresentative( f, v )^x );
+             return ImagesRepresentative( f, PreImagesRepresentativeNC( f, v )^x );
         end;
         qmod:= RightAlgebraModule( RightActingAlgebra( V ),
                        right_op, quot );