Fix InjectionZeroMagma claiming to be bijective

doc/ref/mgmadj.xml

@@ -49,7 +49,7 @@ gap> x:=Transformation( [ 7, 7, 7, 3, 7, 3, 7 ] );;
 gap> x^map;
 <monoid with 0 adjoined elt: Transformation( [ 7, 7, 7, 3, 7, 3, 7 ]
  )>
-gap> PreImage(map, x^map)=x;
+gap> PreImagesRepresentative(map, x^map)=x;
 true
 ]]></Example>
   </Description>
@@ -72,7 +72,7 @@ gap> T:=MagmaWithZeroAdjoined(S);
 gap> UnderlyingInjectionZeroMagma(T);
 MappingByFunction( <transformation monoid of degree 8 with 2
  generators>, <<transformation monoid of degree 8 with 2 generators>
-  with 0 adjoined>, function( elt ) ... end, function( x ) ... end )
+  with 0 adjoined>, function( elt ) ... end )
 ]]></Example>
   </Description>
 </ManSection>

lib/mgmadj.gi

@@ -176,7 +176,10 @@ function(m)
   fi;
   SetGeneratorsOfMagma(out, gens);
 
-  inj:=MappingByFunction(m, out, inj, x-> x!.elt);
+  # the zero is not in the image, so <inj> is not bijective
+  inj:=MappingByFunction(m, out, inj, false, x-> x!.elt);
+  SetIsInjective(inj, true);
+  SetIsSurjective(inj, false);
   SetUnderlyingInjectionZeroMagma(out, inj);
 
   return inj;

lib/reesmat.gi

@@ -1306,7 +1306,7 @@ end);
 
 BindGlobal("_InjectionPrincipalFactor",
 function(D, constructor)
-  local map, inv, G, mat, rep, R, L, x, RR, LL, rms, iso, hom, i, j;
+  local map, inv, G, mat, rep, R, L, x, RR, LL, rms, iso, prefun, hom, i, j;
 
   map := IsomorphismPermGroup(GroupHClassOfGreensDClass(D));
   inv := InverseGeneralMapping(map);
@@ -1366,15 +1366,21 @@ function(D, constructor)
                      [i, (rep * RR[i] * x * LL[j]) ^ map, j, mat]);
   end;
 
-  hom := MappingByFunction(D, rms, iso,
-                           function(x)
-                             if x![1] = 0 then
-                               return fail;
-                             fi;
-                             return R[x![1]] * (x![2] ^ inv) * L[x![3]];
-                           end);
-  SetIsInjective(hom, true);
-  SetIsTotal(hom, true);
+  prefun := function(x)
+    if x![1] = 0 then
+      return fail;
+    fi;
+    return R[x![1]] * (x![2] ^ inv) * L[x![3]];
+  end;
+
+  if IsReesZeroMatrixSemigroup(rms) then
+    # the zero of <rms> is not in the image
+    hom := MappingByFunction(D, rms, iso, false, prefun);
+    SetIsInjective(hom, true);
+    SetIsSurjective(hom, false);
+  else
+    hom := MappingByFunction(D, rms, iso, prefun);
+  fi;
   return hom;
 end);
 
@@ -1431,7 +1437,7 @@ function(S)
     if x = MultiplicativeZero(Range(map)) then
       return MultiplicativeZero(S);
     fi;
-    return x ^ InverseGeneralMapping(map);
+    return PreImagesRepresentativeNC(map, x);
   end;
 
   return MagmaIsomorphismByFunctionsNC(S,

tst/testbugfix/2026-10-18-InjectionZeroMagma.tst

@@ -0,0 +1,16 @@
+# InjectionZeroMagma is injective but not surjective
+gap> map := InjectionZeroMagma( SymmetricGroup( 2 ) );;
+gap> [ IsInjective( map ), IsSurjective( map ), IsBijective( map ) ];
+[ true, false, false ]
+gap> x := (1,2) ^ map;;
+gap> PreImagesRepresentative( map, x );
+(1,2)
+gap> PreImagesElm( map, x );
+[ (1,2) ]
+gap> z := MultiplicativeZero( Range( map ) );;
+gap> PreImagesRepresentative( map, z );
+fail
+gap> PreImagesElm( map, z );
+[  ]
+gap> PreImage( map, z );
+Error, <map> must be an injective and surjective mapping

tst/testinstall/reesmat.tst

@@ -831,8 +831,9 @@ gap> ForAll(S, s -> (s ^ iso) ^ inv = s);
 true
 gap> D := GreensDClassOfElement(S, S.1);;
 gap> iso := _InjectionPrincipalFactor(D, ReesZeroMatrixSemigroup);;
-gap> inv := InverseGeneralMapping(iso);;
-gap> MultiplicativeZero(Range(iso)) ^ inv;
+gap> IsSurjective(iso);
+false
+gap> PreImagesRepresentative(iso, MultiplicativeZero(Range(iso)));
 fail
 
 # AssociatedReesMatrixSemigroupOfDClass: for a D-class of a finite semigroup

tst/testinstall/semigrp.tst

@@ -125,7 +125,7 @@ gap> g := f/[f.1^4];;
 gap> phi := InjectionZeroMagma(g);
 MappingByFunction( <fp group on the generators 
 [ a ]>, <<fp group on the generators 
-[ a ]> with 0 adjoined>, function( elt ) ... end, function( x ) ... end )
+[ a ]> with 0 adjoined>, function( elt ) ... end )
 gap> m := Range(phi);
 <<fp group on the generators [ a ]> with 0 adjoined>
 gap> el := Elements(m);;