attrinv: fix NaturalPartialOrder method

Previously we pre-sorted the elements of the inverse semigroup
by a Greens D > function, but we should use a Greens D < function.

Resolves semigroups#389.

gap/attributes/attrinv.gi

@@ -36,25 +36,29 @@ function(S)
   if not IsFinite(S) then
     ErrorNoReturn("Semigroups: NaturalPartialOrder: usage,\n",
                   "the argument is not a finite semigroup,");
-  fi;
-
-  if not IsInverseSemigroup(S) then
+  elif not IsInverseSemigroup(S) then
     ErrorNoReturn("Semigroups: NaturalPartialOrder: usage,\n",
                   "the argument is not an inverse semigroup,");
+  elif IsPartialPermSemigroup(S) then
+    # Use the library method for partial perm inverse semigroups.
+    return NaturalPartialOrder(S);
   fi;
 
   Info(InfoWarning, 2, "NaturalPartialOrder: this method ",
                        "fully enumerates its argument!");
 
   elts := ShallowCopy(Elements(S));
-  p    := Sortex(elts, IsGreensDGreaterThanFunc(S)) ^ -1;
+  p    := Sortex(elts, {x, y} -> IsGreensDGreaterThanFunc(S)(y, x)) ^ -1;
   func := NaturalLeqInverseSemigroup(S);
   out  := List([1 .. Size(S)], x -> []);
 
+  # <elts> is sorted so that D_elts[i] < D_elts[j] => i < j.
+  # Thus NaturalLeqInverseSemigroup(S)(i, j) => i <= j.
+
   for i in [1 .. Size(S)] do
     for j in [i + 1 .. Size(S)] do
-      if func(elts[j], elts[i]) then
-        AddSet(out[i ^ p], j ^ p);
+      if func(elts[i], elts[j]) then
+        AddSet(out[j ^ p], i ^ p);
       fi;
     od;
   od;

tst/standard/attrinv.tst

@@ -582,11 +582,61 @@ gap> CharacterTableOfInverseSemigroup(S[10]);
 gap> S := InverseSemigroup([Bipartition([[1, -3], [2, -1], [3, 4, -2, -4]]),
 > Bipartition([[1, -1], [2, -3], [3, -2], [4, -4]])]);
 <inverse block bijection semigroup of degree 4 with 2 generators>
-gap> NaturalPartialOrder(AsSemigroup(IsTransformationSemigroup, S));
+gap> S := AsSemigroup(IsTransformationSemigroup, S);
+<transformation semigroup of size 20, degree 20 with 3 generators>
+gap> n := Size(S);;
+gap> elts := Elements(S);;
+gap> NaturalPartialOrder(S);
 [ [ 2, 8, 9, 15, 16, 19 ], [ 9, 16, 19 ], [ 4, 9, 11 ], [ 9 ], [ 9, 16, 18 ], 
   [ 5, 9, 10, 14, 16, 18 ], [ 9, 13, 20 ], [ 9 ], [  ], [ 9 ], [ 9 ], 
   [ 9, 11, 13 ], [ 9 ], [ 9, 10, 16 ], [ 8, 9, 16 ], [ 9 ], [ 4, 9, 20 ], 
   [ 9 ], [ 9 ], [ 9 ] ]
+gap> List([1 .. n],
+>         i -> Filtered([1 .. n],
+>                       j -> i <> j and ForAny(Idempotents(S),
+>                                              e -> e * elts[i] = elts[j])));
+[ [ 2, 8, 9, 15, 16, 19 ], [ 9, 16, 19 ], [ 4, 9, 11 ], [ 9 ], [ 9, 16, 18 ], 
+  [ 5, 9, 10, 14, 16, 18 ], [ 9, 13, 20 ], [ 9 ], [  ], [ 9 ], [ 9 ], 
+  [ 9, 11, 13 ], [ 9 ], [ 9, 10, 16 ], [ 8, 9, 16 ], [ 9 ], [ 4, 9, 20 ], 
+  [ 9 ], [ 9 ], [ 9 ] ]
+gap> last = last2;
+true
+
+#T# attrinv: NaturalPartialOrder (for a semigroup), works, 2
+gap> S := Semigroup(SymmetricInverseMonoid(3), rec(acting := true));;
+gap> es := IdempotentGeneratedSubsemigroup(S);;
+gap> n := Size(es);;
+gap> elts := Elements(es);
+[ <empty partial perm>, <identity partial perm on [ 1 ]>, 
+  <identity partial perm on [ 2 ]>, <identity partial perm on [ 1, 2 ]>, 
+  <identity partial perm on [ 3 ]>, <identity partial perm on [ 2, 3 ]>, 
+  <identity partial perm on [ 1, 3 ]>, <identity partial perm on [ 1, 2, 3 ]> 
+ ]
+gap> NaturalPartialOrder(es);
+[ [  ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1 ], [ 1, 3, 5 ], [ 1, 2, 5 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+gap> List([1 .. n],
+>         i -> Filtered([1 .. n], j -> elts[j] = elts[j] * elts[i] and i <> j));
+[ [  ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1 ], [ 1, 3, 5 ], [ 1, 2, 5 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+gap> last = last2;
+true
+
+#T# attrinv: NaturalPartialOrder (for a semigroup), works, 3
+gap> S := Semigroup(SymmetricInverseMonoid(3), rec(acting := true));;
+gap> es := IdempotentGeneratedSubsemigroup(S);;
+gap> es := AsSemigroup(IsBlockBijectionSemigroup, es);;
+gap> n := Size(es);;
+gap> elts := Elements(es);;
+gap> NaturalPartialOrder(es);
+[ [  ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+gap> List([1 .. n],
+>         i -> Filtered([1 .. n], j -> elts[j] = elts[j] * elts[i] and i <> j));
+[ [  ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+gap> last = last2;
+true
 
 #T# attrinv: NaturalPartialOrder (for a semigroup), error, 1/2
 gap> S := Semigroup(

tst/standard/display.tst

@@ -693,7 +693,7 @@ TABLE BORDER=\"0\" CELLBORDER=\"1\" CELLPADDING=\"10\" CELLSPACING=\"0\" PORT=\
  BGCOLOR=\"gray\" PORT=\"e1\">*</TD></TR>\n</TABLE>>];\n1 -> 2\n2 -> 3\n3 -> 4\
 \nedge [color=blue,arrowhead=none,style=dashed]\n3:e2 -> 4:e1\n3:e3 -> 4:e1\n2\
 :e4 -> 3:e2\n2:e4 -> 3:e3\n3:e5 -> 4:e1\n2:e6 -> 3:e3\n2:e6 -> 3:e5\n2:e7 -> 3\
-:e5\n1:e8 -> 2:e4\n1:e8 -> 2:e6\n1:e8 -> 2:e7\n }"
+:e2\n2:e7 -> 3:e5\n1:e8 -> 2:e4\n1:e8 -> 2:e6\n1:e8 -> 2:e7\n }"
 
 # DotSemilatticeOfIdempotents
 gap> S := Semigroup(SymmetricInverseMonoid(3), rec(acting := true));;

tst/testinstall.tst

@@ -1684,6 +1684,17 @@ gap> Size(S);
 gap> Elements(S);
 [ ONE, (1,(),1) ]
 
+#T# Issue 389: NaturalPartialOrder
+gap> S := Semigroup(SymmetricInverseMonoid(3), rec(acting := true));;
+gap> es := IdempotentGeneratedSubsemigroup(S);; 
+gap> NaturalPartialOrder(es);
+[ [  ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1 ], [ 1, 3, 5 ], [ 1, 2, 5 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+gap> es := AsSemigroup(IsBlockBijectionSemigroup, es);;
+gap> NaturalPartialOrder(es);
+[ [  ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], 
+  [ 1, 2, 3, 4, 5, 6, 7 ] ]
+
 #T# SEMIGROUPS_UnbindVariables
 gap> Unbind(B);
 gap> Unbind(D);