Method selection loop for `OneInverseOfSemigroupElementNC` when applied to groups

[reiniscirpons]

Did some root-causing of an issue raised by @JosephDayngerRuiz in his Cartan Matrix PR, minimal reproducible example below:

gap> t := Transformation([3, 3, 4, 2]);;
gap> S := Semigroup(t);;
gap> OneInverseOfSemigroupElementNC(S, t); # No error
Transformation( [ 4, 4, 2, 3 ] )
gap> IsGroupAsSemigroup(S);
gap> OneInverseOfSemigroupElementNC(S, t); # Loop!
Error, recursion depth trap (5000) in
  HasLibsemigroupsFroidurePin( S ) at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/main/froidure-pin.gi:26 called from 
HasFroidurePin( S ) at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/main/acting.gi:76 called from
x in S at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/greens/acting.gi:865 called from
RClass( S, a ) at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/attributes/attr.gi:823 called from
OneInverseOfSemigroupElementNC( S, x ) at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/attributes/attr.gi:849 called from
OneInverseOfSemigroupElement( S, a ) at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/greens/generic.gi:484 called from
...  at *stdin*:7
you may 'return;'
brk>

Tracing the applicable method stack of the second call we get

gap> ApplicableMethod(OneInverseOfSemigroupElementNC, [S, t], 6);
#I  Searching Method for OneInverseOfSemigroupElementNC with 2 arguments:
#I  Total: 2 entries
#I  Method 1: ``OneInverseOfSemigroupElementNC: for CanUseFroidurePin and a multiplicative element'' at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/attributes/attr.gi:
818, value: 22
#I  Function Body:
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;
endfunction( S, a ) ... end

Which resolves to a call to RightGreensMultiplierNC with e := Transformation( [ 2, 2 ] );, tracing this call reveals

gap> ApplicableMethod(RightGreensMultiplierNC, [S, t, e], 6);                                                
#I  Searching Method for RightGreensMultiplierNC with 3 arguments:
#I  Total: 4 entries
#I  Method 1: ``RightGreensMultiplierNC: for a semigroup and R-related elements'' at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/greens/generic.gi:486, value: 53
#I   - 1st argument needs [ "IsMagmaWithInverses" ]
#I  Method 2: ``RightGreensMultiplierNC: for a semigroup and R-related elements'' at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/greens/generic.gi:481, value: 45
#I  Function Body:
function ( S, a, b )
    return OneInverseOfSemigroupElement( S, a ) * b;
endfunction( S, a, b ) ... end

Finally, OneInverseOfSemigroupElement then calls OneInverseOfSemigroupElementNC:

gap> ApplicableMethod(OneInverseOfSemigroupElement, [S, t], 6);  
#I  Searching Method for OneInverseOfSemigroupElement with 2 arguments:
#I  Total: 1 entry
#I  Method 1: ``OneInverseOfSemigroupElement: for a semigroup and a multiplicative element'' at /home/rc234/Desktop/Source/gap/pkg/Semigroups/gap/attributes/attr.gi:839, value: 18
#I  Function Body:
function ( S, x )
    if not IsFinite( S ) then
        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 OneInverseOfSemigroupElementNC( S, x );
endfunction( S, x ) ... end

This completes the loop.

One quick fix might be just to add a higher priority method for OneInverseOfSemigroupElement for groups which just calls the groups Inverse function, but the underlying loop is still a bit of an issue.

[james-d-mitchell]

Thanks for the detailed report @reiniscirpons, I'll endeavour to fix this at some point.