acting: add OnePseudoInverseOfSemigroupElement for more types

Author: james-d-mitchell

doc/attr.xml

@@ -1207,13 +1207,19 @@ gap> GeneratorsSmallest(T);
 
 <#GAPDoc Label="OneInverseOfSemigroupElement">
 <ManSection>
-  <Attr Name="OneInverseOfSemigroupElement" Arg="S, x"/>
-  <Returns> One inverse of an element of a semigroup.</Returns>
+  <Oper Name="OneInverseOfSemigroupElement" Arg="S, x"/>
+  <Returns> One inverse of an element of a semigroup or <K>fail</K>.</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>.
+    <C>x</C> in the semigroup <A>S</A> and returns <K>fail</K> if this
+    element has no inverse in <A>S</A>.<P/>
+
+    An <E>inverse</E> of an element <A>x</A> in a semigroup <A>S</A> is an
+    element <C>y</C> of <A>S</A> such that <C><A>x</A> * y * <A>x</A> =
+    <A>x</A></C> and <C>y * <A>x</A> * y = <A>y</A></C>.
+
+    See also <Ref Oper="OnePseudoInverseOfSemigroupElement"/>.
+
   <Example><![CDATA[
 gap> S := FullTransformationMonoid(4);
 <full transformation monoid of degree 4>
@@ -1233,6 +1239,70 @@ Error, the semigroup is not finite]]></Example>
 </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="OnePseudoInverseOfSemigroupElement">
+<ManSection>
+  <Oper Name="OnePseudoInverseOfSemigroupElement" Arg="x"/>
+  <Returns> One pseudo-inverse of an element of a semigroup.</Returns>
+  <Description>
+    <C>OnePseudoInverseOfSemigroupElement</C> returns one pseudo-inverse of the
+    element <A>x</A>. This is an element of the same type as <A>x</A> belonging
+    to some semigroup or monoid containing <A>x</A>.
+    <P/>
+
+    An <E>pseudo-inverse</E> of an element <A>x</A> is an element <C>y</C> such
+    that <C><A>x</A> * y * <A>x</A> = <A>x</A></C>.
+    <P/>
+
+    Methods are installed for this operation for the following types of elements:
+    <List>
+    <Mark>transformations</Mark>
+    <Item>
+    <Ref Prop="IsTransformation" BookName="ref"/>;
+    </Item>
+    <Mark>partial perms</Mark>
+    <Item>
+    <Ref Prop="IsPartialPerm" BookName="ref"/>;
+    </Item>
+    <Mark>bipartitions</Mark>
+    <Item>
+    <Ref Prop="IsBipartition"/>;
+    </Item>
+    <Mark>matrices over finite fields</Mark>
+    <Item>
+    &GAP; matrices with entries in a finite field;
+    </Item>
+    <Mark>elements of McAlister triple semigroups</Mark>
+    <Item>
+    <Ref Prop="IsMcAlisterTripleSemigroupElement"/>;
+    </Item>
+    <Mark>elements of regular Rees 0-matrix semigroups over groups</Mark>
+    <Item>
+    <Ref Prop="IsReesZeroMatrixSemigroupElement" BookName="ref"/>.
+    </Item>
+    </List>
+
+    See also <Ref Oper="OneInverseOfSemigroupElement"/>.
+
+  <Example><![CDATA[
+gap> s := Transformation([2, 3, 1, 1]);
+Transformation( [ 2, 3, 1, 1 ] )
+gap> OnePseudoInverseOfSemigroupElement(s);
+Transformation( [ 3, 1, 2, 1 ] )
+gap> e := IdentityTransformation;
+IdentityTransformation
+gap> OnePseudoInverseOfSemigroupElement(e);
+IdentityTransformation
+gap> F := FreeSemigroup(1);
+<free semigroup on the generators [ s1 ]>
+gap> OnePseudoInverseOfSemigroupElement(F, F.1);
+Error, no method found! For debugging hints type ?Recovery from NoMeth\
+odFound
+Error, no 1st choice method found for `OnePseudoInverseOfSemigroupElem\
+ent' on 2 arguments]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="IndecomposableElements">
 <ManSection>
   <Attr Name="IndecomposableElements" Arg="S"/>

doc/maxplusmat.xml

@@ -12,7 +12,6 @@
   <ManSection Label = "IsXMatrix">
   <Heading>Matrix filters</Heading>
     <Filt Name = "IsBooleanMat"              Arg = "obj" Type = "Category"/>
-    <Filt Name = "IsMatrixOverFiniteField"   Arg = "obj" Type = "Category"/>
     <Filt Name = "IsMaxPlusMatrix"           Arg = "obj" Type = "Category"/>
     <Filt Name = "IsMinPlusMatrix"           Arg = "obj" Type = "Category"/>
     <Filt Name = "IsTropicalMatrix"          Arg = "obj" Type = "Category"/>

doc/z-chap11.xml

@@ -64,6 +64,7 @@
        Operations for elements in a semigroup
     </Heading>
     <#Include Label = "OneInverseOfSemigroupElement">
+    <#Include Label = "OnePseudoInverseOfSemigroupElement">
   </Section>
 
   <!--**********************************************************************-->

gap/greens/acting.gi

@@ -1749,7 +1749,8 @@ function(S, a, b)
   p := LambdaPerm(S)(result * a, b);
   # Apply the inverse of the bijection \Psi in Proposition 3.11 of the Citrus
   # paper, to convert p from acting on the right to action on the left
-  return result * StabilizerAction(S)(a, p) * WeakInverse(a);
+  return result * StabilizerAction(S)(a, p)
+    * OnePseudoInverseOfSemigroupElement(a);
 end);
 
 InstallMethod(RightGreensMultiplierNC,
@@ -1771,7 +1772,8 @@ function(S, a, b)
   # At this point StabilizerAction(a * result, LambdaPerm(S)(a * result, b)) =
   # b.
   p := LambdaPerm(S)(a * result, b);
-  return WeakInverse(a) * StabilizerAction(S)(a * result, p);
+  return OnePseudoInverseOfSemigroupElement(a)
+    * StabilizerAction(S)(a * result, p);
 end);
 
 #############################################################################

gap/main/setup.gd

@@ -58,4 +58,8 @@ DeclareOperation("ConvertToInternalElement",
                  [IsSemigroup, IsMultiplicativeElement]);
 DeclareOperation("ConvertToExternalElement",
                  [IsSemigroup, IsMultiplicativeElement]);
-DeclareOperation("WeakInverse", [IsMultiplicativeElement]);
+
+DeclareOperation("OnePseudoInverseOfSemigroupElement",
+                 [IsMultiplicativeElement]);
+DeclareOperation("OnePseudoInverseOfSemigroupElementNC",
+                 [IsMultiplicativeElement]);

gap/main/setup.gi

@@ -1084,11 +1084,63 @@ function(S, mat)
   return Matrix(Unpack(mat){[1 .. n]}{[1 .. n]}, mat);
 end);
 
-InstallMethod(WeakInverse, "for a transformation",
+InstallMethod(OnePseudoInverseOfSemigroupElement,
+"for a multiplicative element",
+[IsMultiplicativeElement], OnePseudoInverseOfSemigroupElementNC);
+
+InstallMethod(OnePseudoInverseOfSemigroupElementNC, "for a transformation",
 [IsTransformation], InverseOfTransformation);
 
-InstallMethod(WeakInverse, "for a partial perm",
+InstallMethod(OnePseudoInverseOfSemigroupElementNC, "for a partial perm",
 [IsPartialPerm], InverseMutable);
 
-InstallMethod(WeakInverse, "for a bipartition",
+InstallMethod(OnePseudoInverseOfSemigroupElementNC, "for a bipartition",
 [IsBipartition], Star);
+
+InstallMethod(OnePseudoInverseOfSemigroupElementNC, "for a matrix over ff",
+[IsMatrixObjOverFiniteField], RightInverse);
+
+InstallMethod(OnePseudoInverseOfSemigroupElementNC,
+"for a McAlister triple semigroup element",
+[IsMcAlisterTripleSemigroupElement], INV);
+
+InstallMethod(OnePseudoInverseOfSemigroupElement,
+"for a Rees 0-matrix semigroup",
+[IsReesZeroMatrixSemigroupElement],
+function(x)
+  local R;
+
+  R := ReesMatrixSemigroupOfFamily(FamilyObj(x));
+  if not IsRegularSemigroup(R) then
+    ErrorNoReturn("the parent semigroup of the 1st argument ",
+                  "(an element of a Rees 0-matrix semigroup) ",
+                  "is not regular");
+  elif not IsGroup(UnderlyingSemigroup(R)) then
+    ErrorNoReturn("the underlying semigroup of the parent semigroup of ",
+                  "the 1st argument (an element of a Rees 0-matrix ",
+                  "semigroup) is not a group");
+  fi;
+  return OnePseudoInverseOfSemigroupElementNC(x);
+
+end);
+
+InstallMethod(OnePseudoInverseOfSemigroupElementNC,
+"for a Rees 0-matrix semigroup",
+[IsReesZeroMatrixSemigroupElement],
+function(x)
+  local R, mat, i, j, k, l;
+
+  R := ReesMatrixSemigroupOfFamily(FamilyObj(x));
+  if IsMultiplicativeZero(R, x) then
+    return x;
+  fi;
+
+  mat := Matrix(R);
+  i := x[1];
+  j := x[3];
+
+  k := First(Rows(R), k -> mat[j][k] <> 0);
+  l := First(Columns(R), l -> mat[l][i] <> 0);
+
+  return RMSElement(R, k, mat[j][k] ^ -1 * x[2] ^ -1 * mat[l][i] ^ -1, l);
+end);