Better inverses (#1021)

* Stub

* Add possibly better method for InversesOfSemigroupElementNC

* delete Nambooripad Method for Transformation Semigroup

* change inverses to a set

* change filter into IsMultiplicativeElementWithOne as it is required for the argument of RightGreensMultiplierNC.

* add the AsSet to the test rather than the method

* fix consecutive empty lines

* fix RightGreensMultiplierNC does not work on fp semigroups where there is no one but the method should work

* fix comment

* add OneInverseOfSemigroupElement and fix small issues

* fix declaration and filter problem

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* fix formatting for gaplint

* fix issue

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* fix issue

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* addnon-no-check version of OneInverse, and a NC version for cannot use froidure pin.

* fix small issues in the previous commit

* Update gap/attributes/attr.gi

Co-authored-by: James Mitchell <[email protected]>

* add declaration of OneInverseOfSemigroupElement

* add documentation  of OneInverseOfSemigroupElement

* fix formatting for gaplint

* fix syntax error

* add test for OneInverseOfSemigroupElement

* addMultiplicativeNeutralElement as a return

* addMultiplicativeNeutralElement as a return for the other one

* update documentation for LeftGreensMultiplier

* fix filter problem

* fix filter problem and also add test

* fix test problems

* fix gaplint

* fix test

* add example in documentation

* Update doc/attr.xml

Co-authored-by: James Mitchell <[email protected]>

* if there is no inverse

* delete merge remnants

* add to chapter

* fix small issue

* fix example format

---------

Co-authored-by: James D. Mitchell <[email protected]>
Co-authored-by: James Mitchell <[email protected]>

doc/attr.xml

@@ -1205,6 +1205,34 @@ 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> and returns fail if this
+    element has no inverse in <A>S</A>.
+    <A>x</A> in the semigroup <A>S</A>.
+  <Example><![CDATA[
+gap> S := FullTransformationMonoid(4);
+<full transformation monoid of degree 4>
+gap> s := Transformation([2, 3, 1, 1]);
+Transformation( [ 2, 3, 1, 1 ] )
+gap> OneInverseOfSemigroupElement(S, s);
+Transformation( [ 3, 1, 2, 2 ] )
+gap> e := IdentityTransformation;
+IdentityTransformation
+gap> OneInverseOfSemigroupElement(S, e);
+IdentityTransformation
+gap> F := FreeSemigroup(1);
+<free semigroup on the generators [ s1 ]>
+gap> OneInverseOfSemigroupElement(F, F.1);
+Error, the semigroup is not finite]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="IndecomposableElements">
 <ManSection>
   <Attr Name="IndecomposableElements" Arg="S"/>

doc/greens-generic.xml

@@ -1066,7 +1066,7 @@ gap> List(DClasses(S), SchutzenbergerGroup);
   element <C>s</C> is of the same type as the elements of <A>S</A> but may
   or may not be an element of <A>S</A>. In particular, if <A>S</A> is not a
   monoid and <C><A>a</A> = <A>b</A></C>, then
-  <C>One(GeneratorsOfSemigroup(S))</C> may be returned. Even if <C><A>a</A> &lt;>
+  <C>One(GeneratorsOfSemigroup(S))</C> or an adjoined identity may be returned. Even if <C><A>a</A> &lt;>
   <A>b</A></C>, then it is not guaranteed that the returned element <C>s</C>
   will belong to <A>S</A>. It is guaranteed that the left action of <C>s</C> on
   the elements of the &L;-class of <A>a</A> is the same as the left action of

doc/z-chap11.xml

@@ -58,6 +58,17 @@
   <!--**********************************************************************-->
   <!--**********************************************************************-->
 
+  <Section>
+
+    <Heading>
+       Operations for elements in a semigroup
+    </Heading>
+    <#Include Label = "OneInverseOfSemigroupElement">
+  </Section>
+
+  <!--**********************************************************************-->
+  <!--**********************************************************************-->
+
   <Section>
 
     <Heading>

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

@@ -759,7 +759,7 @@ end);
 
 InstallMethod(InversesOfSemigroupElementNC,
 "for a group as semigroup and a multiplicative element",
-[IsGroupAsSemigroup, IsMultiplicativeElement],
+[IsGroupAsSemigroup and CanUseFroidurePin, IsMultiplicativeElement],
 function(G, x)
   local i, iso, inv;
   if IsMultiplicativeElementWithInverse(x) then
@@ -774,26 +774,79 @@ function(G, x)
 end);
 
 InstallMethod(InversesOfSemigroupElementNC,
-"for a semigroup and a multiplicative element",
-[IsSemigroup, IsMultiplicativeElement],
+"for a semigroup that can use froidure-pin 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,
+"for a semigroup that can use froidure-pin and a multiplicative element",
+[CanUseFroidurePin, IsMultiplicativeElement],  # to beat the library method
 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 Filtered(EnumeratorSorted(S), y -> x * y * x = x and y * x * y = y);
+  return InversesOfSemigroupElementNC(S, x);
 end);
 
-InstallMethod(InversesOfSemigroupElement,
+InstallMethod(OneInverseOfSemigroupElementNC,
 "for a semigroup and a multiplicative element",
-[IsSemigroup, IsMultiplicativeElement], 1,  # to beat the library method
+[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];
+  f := Idempotents(L);
+  if IsEmpty(f) then
+    return fail;
+  fi;
+  f := f[1];
+  s := RightGreensMultiplierNC(S, a, e);
+  return  f * s * e;
+end);
+
+InstallMethod(OneInverseOfSemigroupElement,
+"for a semigroup and a multiplicative element",
+[IsSemigroup, IsMultiplicativeElement],
 function(S, x)
   if not IsFinite(S) then
-    TryNextMethod();
+    ErrorNoReturn("the semigroup is not finite");
   elif not x in S then
     ErrorNoReturn("the 2nd argument (a mult. element) must belong to the 1st ",
                   "argument (a semigroup)");
   fi;
-  return InversesOfSemigroupElementNC(S, x);
+  return OneInverseOfSemigroupElementNC(S, x);
 end);
 
 InstallMethod(UnderlyingSemigroupOfSemigroupWithAdjoinedZero,

gap/greens/froidure-pin.gi

@@ -606,39 +606,53 @@ 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);
+    elif MultiplicativeNeutralElement(S) <> fail then
+        return MultiplicativeNeutralElement(S);
+    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);
+    elif MultiplicativeNeutralElement(S) <> fail then
+        return MultiplicativeNeutralElement(S);
+    else
+        return SEMIGROUPS.UniversalFakeOne;
+    fi;
   fi;
   return EvaluateWord(gens, path[2]);
 end);

gap/semigroups/grpperm.gi

@@ -53,7 +53,7 @@ end);
 
 # fall back method, same method for ideals
 
-InstallMethod(IsomorphismPermGroup, "for a semigroup", [IsSemigroup],
+InstallMethod(IsomorphismPermGroup, "for a semigroup", [CanUseFroidurePin],
 function(S)
   local cay, deg, G, gen1, gen2, next, pos, iso, inv, i;
   if not IsFinite(S) then

tst/standard/attributes/attr.tst

@@ -1474,9 +1474,39 @@ gap> InversesOfSemigroupElement(S, IdentityTransformation);
 Error, usage: the 2nd argument must be an element of the 1st,
 gap> InversesOfSemigroupElementNC(S, S.1);
 Error, no method found! For debugging hints type ?Recovery from NoMethodFound
-Error, no 2nd choice method found for `InversesOfSemigroupElementNC' on 2 argu\
+Error, no 1st choice method found for `InversesOfSemigroupElementNC' on 2 argu\
 ments
 
+# attr: OneInverseOfSemigroupElement, for a semigroup
+gap> S := Semigroup([
+>  Matrix(IsMaxPlusMatrix, [[-2, 2, 0], [-1, 0, 0], [1, -3, 1]]),
+>  Matrix(IsMaxPlusMatrix, [[- infinity, 0, 0], [0, 1, 0], [1, -1, 0]])]);;
+gap> OneInverseOfSemigroupElement(S, S.1);
+Error, the semigroup is not finite
+gap> S := Semigroup(Transformation([2, 3, 1, 3, 3]));;
+gap> OneInverseOfSemigroupElement(S, Transformation([1, 3, 2]));
+Error, the 2nd argument (a mult. element) must belong to the 1st argument (a s\
+emigroup)
+gap> S := Semigroup([Matrix(IsBooleanMat, [[0, 0, 1], [0, 1, 1], [1, 0, 0]]),
+>  Matrix(IsBooleanMat, [[1, 0, 0], [1, 0, 1], [1, 1, 1]])]);;
+gap> OneInverseOfSemigroupElement(S, S.1);
+fail
+gap> OneInverseOfSemigroupElement(S, S.1 * S.2 * S.1);
+Matrix(IsBooleanMat, [[1, 1, 1], [1, 1, 1], [0, 0, 1]])
+
+# OneInverseOfSemigroupElement, for a semigroup that cannot use Froidure-Pin
+gap> S := Semigroup(SEMIGROUPS.UniversalFakeOne);;
+gap> OneInverseOfSemigroupElement(S, S.1);
+<universal fake one>
+
+# OneInverseOfSemigroupElement, for an infinite semigroup, 1
+gap> S := FreeSemigroup(1);
+<free semigroup on the generators [ s1 ]>
+gap> OneInverseOfSemigroupElementNC(S, S.1);
+Error, no method found! For debugging hints type ?Recovery from NoMethodFound
+Error, no 2nd choice method found for `OneInverseOfSemigroupElementNC' on 2 ar\
+guments
+
 # attr: IdempotentGeneratedSubsemigroup, 2
 gap> S := FullTransformationMonoid(3);;
 gap> I := IdempotentGeneratedSubsemigroup(S);;

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]]), 

tst/standard/semigroups/grpperm.tst

@@ -397,7 +397,7 @@ Group([ (1,3)(2,5)(4,6), (1,4,5)(2,6,3) ])
 # IsomorphismPermGroup, infinite 1 / 1
 gap> IsomorphismPermGroup(FreeMonoid(3));
 Error, no method found! For debugging hints type ?Recovery from NoMethodFound
-Error, no 3rd choice method found for `IsomorphismPermGroup' on 1 arguments
+Error, no 2nd choice method found for `IsomorphismPermGroup' on 1 arguments
 
 # IsomorphismPermGroup, for a block bijection semigroup
 gap> S := Semigroup(Bipartition([[1, 2, -3, -4], [3, 4, -1, -2]]));;

tst/standard/semigroups/semirms.tst

@@ -2629,8 +2629,7 @@ gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4),
 >  [(), (2, 4, 3), (1, 2)]]);
 <Rees 0-matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )>
 gap> IsomorphismPermGroup(R);
-Error, the underlying semigroup of the argument (a  subsemigroup of a Rees 0-m\
-atrix semigroup) does not satisfy IsGroupAsSemigroup
+Error, the argument (a semigroup) does not satisfy IsGroupAsSemigroup
 gap> S := Semigroup(MultiplicativeZero(R));;
 gap> IsomorphismPermGroup(S);
 <subsemigroup of 3x4 Rees 0-matrix semigroup with 1 generator> -> Group(())