Better inverses

doc/attr.xml

@@ -1205,6 +1205,17 @@ gap> GeneratorsSmallest(T);
 </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="OneInverseOfSemigroupElement">
+<ManSection>
+  <Attr Name="OneInverseOfSemigroupElement" Arg="S, x"/>
+  <Returns> One inverse of an element of a semigroup.</Returns>
+  <Description>
+    <C>OneInverseOfSemigroupElement</C> returns one inverse of the element 
+    <C>x</C> in the semigroup <A>S</A>.
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="IndecomposableElements">
 <ManSection>
   <Attr Name="IndecomposableElements" Arg="S"/>

gap/attributes/attr.gd

@@ -67,6 +67,10 @@ DeclareAttribute("UnderlyingSemigroupOfSemigroupWithAdjoinedZero",
 
 DeclareOperation("InversesOfSemigroupElementNC",
                  [IsSemigroup, IsMultiplicativeElement]);
+DeclareOperation("OneInverseOfSemigroupElementNC",
+                 [IsSemigroup, IsMultiplicativeElement]);
+DeclareOperation("OneInverseOfSemigroupElement",
+                 [IsSemigroup, IsMultiplicativeElement]);
 
 DeclareAttribute("IndecomposableElements", IsSemigroup);
 DeclareAttribute("NambooripadLeqRegularSemigroup", IsSemigroup);

gap/attributes/attr.gi

@@ -777,10 +777,29 @@ InstallMethod(InversesOfSemigroupElementNC,
 "for a semigroup and a multiplicative element",
 [IsSemigroup, IsMultiplicativeElement],
 function(S, x)
-  if not IsFinite(S) then
-    TryNextMethod();
-  fi;
-  return Filtered(EnumeratorSorted(S), y -> x * y * x = x and y * x * y = y);
+    if not IsFinite(S) then
+        TryNextMethod();
+    fi;
+    return Filtered(EnumeratorSorted(S), 
+    y -> x * y * x = x and y * x * y = y);
+end);
+
+InstallMethod(InversesOfSemigroupElementNC,
+"for CanUseFroidurePin and a multiplicative element",
+[CanUseFroidurePin, IsMultiplicativeElement],
+function(S, a)
+  local R, L, inverses, e, f, s;
+  R := RClass(S, a);
+  L := LClass(S, a);
+  inverses := EmptyPlist(NrIdempotents(R) * NrIdempotents(L));
+
+  for e in Idempotents(R) do
+    s := RightGreensMultiplierNC(S, a, e) * e;
+    for f in Idempotents(L) do
+      Add(inverses, f * s);
+    od;
+  od;
+  return inverses;
 end);
 
 InstallMethod(InversesOfSemigroupElement,
@@ -796,6 +815,48 @@ function(S, x)
   return InversesOfSemigroupElementNC(S, x);
 end);
 
+InstallMethod(OneInverseOfSemigroupElementNC,
+"for a semigroup and a multiplicative element",
+[IsSemigroup, IsMultiplicativeElement],
+function(S, x)
+    if not IsFinite(S) then
+        TryNextMethod();
+    fi;
+    return First(EnumeratorSorted(S), 
+    y -> x * y * x = x and y * x * y = y);
+end);
+
+InstallMethod(OneInverseOfSemigroupElementNC,
+"for CanUseFroidurePin and a multiplicative element",
+[CanUseFroidurePin, IsMultiplicativeElement],
+function(S, a)
+  local R, L, e, f, s;
+  R := RClass(S, a);
+  L := LClass(S, a);
+  e := Idempotents(R);
+  if IsEmpty(e) then
+    return fail;
+   fi;
+   e := e[1];
+  s := RightGreensMultiplierNC(S, a, e);
+  f := Idempotents(L)[1];
+  return  f * s * e;
+end);
+
+InstallMethod(OneInverseOfSemigroupElement,
+"for CanUseFroidurePin and a multiplicative element",
+[IsSemigroup, IsMultiplicativeElement],
+function(S, x)
+  if not IsFinite(S) then
+    TryNextMethod();
+  elif not x in S then
+    ErrorNoReturn("the 2nd argument (a mult. element) must belong to the 1st ",
+                  "argument (a semigroup)");
+  fi;
+  return OneInverseOfSemigroupElementNC(S, x);
+end);
+
+
 InstallMethod(UnderlyingSemigroupOfSemigroupWithAdjoinedZero,
 "for a semigroup",
 [IsSemigroup],

gap/greens/froidure-pin.gi

@@ -606,39 +606,47 @@ function(S)
 end);
 
 InstallMethod(LeftGreensMultiplierNC,
-"for a semigroup that can use Froidure-Pin and L-related elements with one",
+"for a semigroup that can use Froidure-Pin and L-related elements",
 [IsSemigroup and CanUseFroidurePin,
- IsMultiplicativeElementWithOne,
- IsMultiplicativeElementWithOne],
+ IsMultiplicativeElement,
+ IsMultiplicativeElement],
 function(S, a, b)
-  local gens, D, path;
+  local gens, D, path, a1, b1;
   gens := GeneratorsOfSemigroup(S);
   D := LeftCayleyDigraph(S);
-  a := PositionCanonical(S, a);
-  b := PositionCanonical(S, b);
-  path := NextIterator(IteratorOfPaths(D, a, b));
+  a1 := PositionCanonical(S, a);
+  b1 := PositionCanonical(S, b);
+  path := NextIterator(IteratorOfPaths(D, a1, b1));
   if path = fail then
     # This can occur when, for example, a = b and S is not a monoid.
-    return One(gens);
+    if IsMultiplicativeElementWithOne(a) and IsMultiplicativeElementWithOne(b) then
+      return One(gens);
+    else
+      return SEMIGROUPS.UniversalFakeOne;
+    fi;
   fi;
   return EvaluateWord(gens, Reversed(path[2]));
 end);
 
 InstallMethod(RightGreensMultiplierNC,
-"for a semigroup that can use Froidure-Pin and R-related elements with one",
+"for a semigroup that can use Froidure-Pin and R-related elements",
 [IsSemigroup and CanUseFroidurePin,
- IsMultiplicativeElementWithOne,
- IsMultiplicativeElementWithOne],
+ IsMultiplicativeElement,
+ IsMultiplicativeElement],
 function(S, a, b)
-  local gens, D, path;
+  local gens, D, path, a1, b1;
   gens := GeneratorsOfSemigroup(S);
   D := RightCayleyDigraph(S);
-  a := PositionCanonical(S, a);
-  b := PositionCanonical(S, b);
-  path := NextIterator(IteratorOfPaths(D, a, b));
+  a1 := PositionCanonical(S, a);
+  b1 := PositionCanonical(S, b);
+  path := NextIterator(IteratorOfPaths(D, a1, b1));
   if path = fail then
     # This can occur when, for example, a = b and S is not a monoid.
-    return One(gens);
+    if IsMultiplicativeElementWithOne(a) and IsMultiplicativeElementWithOne(b) then
+      return One(gens);
+    else
+      return SEMIGROUPS.UniversalFakeOne;
+    fi;
   fi;
   return EvaluateWord(gens, path[2]);
 end);

tst/standard/ideals/ideals.tst

@@ -230,7 +230,7 @@ gap> I := SemigroupIdeal(S,
 >  Matrix(IsBooleanMat, [[1, 0, 1], [0, 0, 0], [0, 1, 0]]),
 >  Matrix(IsBooleanMat, [[0, 1, 1], [0, 1, 1], [1, 0, 1]]));;
 gap> x := Matrix(IsBooleanMat, [[1, 0, 1], [0, 1, 0], [1, 0, 1]]);;
-gap> InversesOfSemigroupElement(I, x);
+gap> AsSet(InversesOfSemigroupElement(I, x));
 [ Matrix(IsBooleanMat, [[0, 0, 0], [0, 1, 0], [0, 0, 1]]), 
   Matrix(IsBooleanMat, [[0, 0, 0], [0, 1, 0], [1, 0, 0]]), 
   Matrix(IsBooleanMat, [[0, 0, 0], [0, 1, 0], [1, 0, 1]]),