- pairs, # list of 2-tuples with entries in [1..]
- pos, # position of in unmodified
- p, # a pair that corresponds to a non-zero product in
- rels, # relations of output semigroup isomorphic to
- q; # loop counter, element in modified
-
- erz := MinimalGeneratingSet(sgrp);
- d := Length(erz);
-
- f := FreeSemigroup(d);
- x := GeneratorsOfSemigroup( f );
-
- pairs := Reversed(Tuples([1..d], 2));
-
- z := MultiplicativeZero(sgrp);
- pos := PositionProperty(pairs, p-> z <> erz[p[1]]*erz[p[2]]);
- p := Remove(pairs, pos);
-
- rels := EmptyPlist(d^2);
- Add(rels, [ x[p[1]]*x[p[1]]*x[p[2]], x[p[1]]*x[p[1]]*x[p[1]]*x[p[2]] ]);
-
- for q in pairs do
- if z = erz[q[1]]*erz[q[2]] then
- Add(rels, [ x[p[1]]*x[p[1]]*x[p[2]], x[q[1]]*x[q[2]] ]);
- else
- Add(rels, [ x[p[1]]*x[p[2]], x[q[1]]*x[q[2]] ]);
- fi;
- od;
- return f/rels;
-end);
+InstallGlobalFunction(IsomorphicFpSemigroup,
+function(sgrp)
+ local erz, # minimal generating set of
+ d, # minimal number of generators of
+ f, # free semigroup on generators
+ x, # generating set of
+ z, # multiplicative zero of
+ pairs, # list of 2-tuples with entries in [1 .. ]
+ pos, # position of in unmodified
+ p, # a pair that corresponds to a non - zero product in
+ rels, # relations of output semigroup isomorphic to < sgrp >
+ q; # loop counter, element in modified < pairs >
+
+ erz := MinimalGeneratingSet(sgrp);
+ d := Length(erz);
+
+ f := FreeSemigroup(d);
+ x := GeneratorsOfSemigroup(f);
+
+ pairs := Reversed(Tuples([1 .. d], 2));
+
+ z := MultiplicativeZero(sgrp);
+ pos := PositionProperty(pairs, p -> z <> erz[p[1]] * erz[p[2]]);
+ p := Remove(pairs, pos);
+
+ rels := EmptyPlist(d ^ 2);
+ Add(rels,
+ [x[p[1]] * x[p[1]] * x[p[2]], x[p[1]] * x[p[1]] * x[p[1]] * x[p[2]]]);
+
+ for q in pairs do
+ if z = erz[q[1]] * erz[q[2]] then
+ Add(rels, [x[p[1]] * x[p[1]] * x[p[2]], x[q[1]] * x[q[2]]]);
+ else
+ Add(rels, [x[p[1]] * x[p[2]], x[q[1]] * x[q[2]]]);
+ fi;
+ od;
+ return f / rels;
+end);
-#############################################################################
-##
## returns presentations for all semigroups of order and coclass 2
## which have a minimal generating set of size 3 up to equivalence
##
## - implementing list from small coclass paper for n >= 6
## - using the semigroups from the library for n = 5
-##
-InstallGlobalFunction( NilpotentSemigroupsCoclass2Rank3,
-function( n )
- local sgrps, # list of all semigroups
- f, # free semigroup on two elements
- x, # generators of
- k, l, # loop counters
- subsgrps, # list of candidates for V and W
- v, w, # loop counters
- V, W, # subsemigroups of size n-1 and coclass 1
- Vgens, # generators of the subsemigroup V
- Wgens, # generators of the subsemigroup W
- Vrels, # relations of the subsemigroup V
- Wrels, # relations of the subsemigroup W
- rels; # relations of a semigroup
-
- # catch input error
- if not IsPosInt( n ) then
- Error( "the size has to be a positive integer" );
- fi;
- f := FreeSemigroup( 3 );
- x := GeneratorsOfSemigroup( f );
-
- # catch exceptional cases
- if n <= 3 then
- return [ ];
-
- elif n = 4 then # the zero semigroup
- return [ f / [[ x[1]^2, x[1]^3 ], # u^2 = u^3
- [ x[1]^2, x[2]^2 ], # u^2 = v^2
- [ x[1]^2, x[1]*x[2] ], # u^2 = uv
- [ x[1]^2, x[2]*x[1] ], # u^2 = vu
- [ x[1]^2, x[3]^2 ], # u^2 = w^2
- [ x[1]^2, x[1]*x[3] ], # u^2 = uw
- [ x[1]^2, x[3]*x[1] ], # u^2 = wu
- [ x[1]^2, x[2]*x[3] ], # u^2 = vw
- [ x[1]^2, x[3]*x[2] ], # u^2 = wv
- [ x[1]^2, x[3]^2 ] ]]; # u^2 = w^2
-
- elif n = 5 then # just take the semigroups from the library
- return List(AllSmallSemigroups(5,
- Is3GeneratedSemigroup, true,
- NilpotencyDegree, 3),
- IsomorphicFpSemigroup);
- fi;
+InstallGlobalFunction(NilpotentSemigroupsCoclass2Rank3,
+function(n)
+ local sgrps, # list of all semigroups
+ f, # free semigroup on two elements
+ x, # generators of
+ k, l, # loop counters
+ subsgrps, # list of candidates for V and W
+ v, w, # loop counters
+ V, W, # subsemigroups of size n-1 and coclass 1
+ Vgens, # generators of the subsemigroup V
+ Wgens, # generators of the subsemigroup W
+ Vrels, # relations of the subsemigroup V
+ Wrels, # relations of the subsemigroup W
+ rels; # relations of a semigroup
+
+ # catch input error
+ if not IsPosInt(n) then
+ Error("the size < n > has to be a positive integer");
+ fi;
+
+ f := FreeSemigroup(3);
+ x := GeneratorsOfSemigroup(f);
+
+ # catch exceptional cases
+ if n <= 3 then
+ return [];
+ elif n = 4 then
+ # the zero semigroup
+ return [f / [[x[1] ^ 2, x[1] ^ 3], # u ^ 2 = u ^ 3
+ [x[1] ^ 2, x[2] ^ 2], # u ^ 2 = v ^ 2
+ [x[1] ^ 2, x[1] * x[2]], # u ^ 2 = uv
+ [x[1] ^ 2, x[2] * x[1]], # u ^ 2 = vu
+ [x[1] ^ 2, x[3] ^ 2], # u ^ 2 = w ^ 2
+ [x[1] ^ 2, x[1] * x[3]], # u ^ 2 = uw
+ [x[1] ^ 2, x[3] * x[1]], # u ^ 2 = wu
+ [x[1] ^ 2, x[2] * x[3]], # u ^ 2 = vw
+ [x[1] ^ 2, x[3] * x[2]], # u ^ 2 = wv
+ [x[1] ^ 2, x[3] ^ 2]]]; # u ^ 2 = w ^ 2
+ elif n = 5 then
+ # just take the semigroups from the library
+ return List(AllSmallSemigroups(5,
+ Is3GeneratedSemigroup, true,
+ NilpotencyDegree, 3),
+ IsomorphicFpSemigroup);
+ fi;
+
+ if IsEvenInt(n) then
+ sgrps := EmptyPlist((21 * n ^ 2 + 22 * n - 96) / 8);
+ else
+ sgrps := EmptyPlist((21 * n ^ 2 + 36 * n - 81) / 8);
+ fi;
+
+ sgrps := NilpotentSemigroupsCoclassD_NC(n, 2);
+
+ # get semigroups of size n - 1 and coclass 1 .. .
+ subsgrps := NilpotentSemigroupsCoclass1(n - 1);
+ # .. . in which the two generators are not interchangeable
+ Remove(subsgrps, 1);
+
+ for v in [1 .. Length(subsgrps)] do
+ V := subsgrps[v];
+
+ # relations from V translated to new generators
+ Vgens := FreeGeneratorsOfFpSemigroup(V);
+ Vrels := List(RelationsOfFpSemigroup(V),
+ rel -> [MappedWord(rel[1], Vgens, x{[1, 2]}),
+ MappedWord(rel[2], Vgens, x{[1, 2]})]);
+ # u * v = u ^ k
+ k := ExtRepOfObj(Vrels[Length(Vrels) - 1][2])[2];
+
+ for w in [1 .. Length(subsgrps)] do
+ W := subsgrps[w];
+
+ # relations from W translated to new generators
+ Wgens := FreeGeneratorsOfFpSemigroup(W);
+ Wrels := List(RelationsOfFpSemigroup(W),
+ rel -> [MappedWord(rel[1], Wgens, x{[1, 3]}),
+ MappedWord(rel[2], Wgens, x{[1, 3]})]);
+ # first relation is already contained in
+ Remove(Wrels, 1);
+ # u * w = u ^ l
+ l := ExtRepOfObj(Wrels[Length(Wrels) - 1][2])[2];
+
+ rels := Concatenation(Vrels, Wrels);
+
+ # Case 2.d
+ if w >= v and IsCommutative(V) and IsCommutative(W) then
+ if k + l <= n - 2 then
+ # vw = wv
+ Add(rels, [x[2] * x[3], x[3] * x[2]]);
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[2] * x[3], x[1] ^ (k + l - 2)]);
+ Add(sgrps, f / rels);
+ else
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]);
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]);
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]);
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Remove(rels);
+
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]);
+ # vw = u ^ (k + l - 2)
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]);
+ Add(sgrps, f / rels);
+ fi;
+ elif not IsCommutative(V) and IsCommutative(W) then
+ # se 2.b and part of Case 2.a
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Remove(rels);
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+
+ elif v >= w and not IsCommutative(V) and not IsCommutative(W) then
+ # Case 2.a
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Remove(rels);
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+
+ if v = w then
+ Remove(sgrps);
+ fi;
- if IsEvenInt( n ) then
- sgrps := EmptyPlist( (21*n^2 + 22*n - 96) / 8 );
- else
- sgrps := EmptyPlist( (21*n^2 + 36*n - 81) / 8 );
- fi;
+ # take dual of W
+ Wrels[Length(Wrels) - 2][2] := x[1] ^ (n - 3);
+ Wrels[Length(Wrels) - 1][2] := x[1] ^ (n - 2);
- sgrps := NilpotentSemigroupsCoclassD_NC( n, 2 );
-
- # get semigroups of size n-1 and coclass 1 ...
- subsgrps := NilpotentSemigroupsCoclass1( n-1 );
- # ... in which the two generators are not interchangeable
- Remove( subsgrps, 1 );
-
- for v in [ 1..Length(subsgrps) ] do
- V := subsgrps[v];
-
- # relations from V translated to new generators
- Vgens := FreeGeneratorsOfFpSemigroup(V);
- Vrels := List( RelationsOfFpSemigroup(V),
- rel-> [ MappedWord( rel[1], Vgens, x{[1,2]} ),
- MappedWord( rel[2], Vgens, x{[1,2]} ) ]);
- # u*v = u^k
- k := ExtRepOfObj( Vrels[ Length(Vrels)-1 ][2] )[2];
-
- for w in [ 1..Length(subsgrps) ] do
- W := subsgrps[w];
-
- # relations from W translated to new generators
- Wgens := FreeGeneratorsOfFpSemigroup(W);
- Wrels := List( RelationsOfFpSemigroup(W),
- rel-> [ MappedWord( rel[1], Wgens, x{[1,3]} ),
- MappedWord( rel[2], Wgens, x{[1,3]} ) ]);
- # first relation is already contained in
- Remove( Wrels, 1 );
- # u*w = u^l
- l := ExtRepOfObj( Wrels[ Length(Wrels)-1 ][2] )[2];
-
- rels := Concatenation( Vrels, Wrels );
-
- # Case 2.d
- if w >= v and IsCommutative(V) and IsCommutative(W) then
- if k+l <= n-2 then
- Add( rels, [ x[2]*x[3], x[3]*x[2] ] ); # vw = wv
- Add( rels, [ x[2]*x[3], x[1]^(k+l-2) ] ); # vw = u^(k+l-2)
- Add( sgrps, f / rels );
-
- else
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # vw = u^(k+l-2)
- Add( rels, [ x[2]*x[3], x[1]^(n-3) ] ); # vw = u^(k+l-2)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[2]*x[3], x[1]^(n-2) ] ); # vw = u^(k+l-2)
- Add( sgrps, f / rels );
-
- Remove( rels ); Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # vw = u^(k+l-2)
- Add( rels, [ x[2]*x[3], x[1]^(n-2) ] ); # vw = u^(k+l-2)
- Add( sgrps, f / rels );
- fi;
-
- # Case 2.b and part of Case 2.a
- elif not IsCommutative(V) and IsCommutative(W) then
- Add( rels, [ x[2]*x[3], x[1]^(n-3) ] ); # vw = u^(n-3)
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels ); Remove( rels );
- Add( rels, [ x[2]*x[3], x[1]^(n-2) ] ); # vw = u^(n-2)
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
-
- # Case 2.a
- elif v >= w and not IsCommutative(V) and not IsCommutative(W) then
- Add( rels, [ x[2]*x[3], x[1]^(n-3) ] ); # vw = u^(n-3)
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels ); Remove( rels );
- Add( rels, [ x[2]*x[3], x[1]^(n-2) ] ); # vw = u^(n-2)
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
-
- if v = w then
- Remove( sgrps );
- fi;
-
- # take dual of W
- Wrels[ Length(Wrels)-2 ][2] := x[1]^(n-3);
- Wrels[ Length(Wrels)-1 ][2] := x[1]^(n-2);
-
- rels := Concatenation( Vrels, Wrels );
-
- Add( rels, [ x[2]*x[3], x[1]^(n-3) ] ); # vw = u^(n-3)
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels ); Remove( rels );
- Add( rels, [ x[2]*x[3], x[1]^(n-2) ] ); # vw = u^(n-2)
- Add( rels, [ x[3]*x[2], x[1]^(n-2) ] ); # wv = u^(n-2)
- Add( sgrps, f / rels );
-
- Remove( rels );
- Add( rels, [ x[3]*x[2], x[1]^(n-3) ] ); # wv = u^(n-3)
- Add( sgrps, f / rels );
- fi;
- od;
+ rels := Concatenation(Vrels, Wrels);
+
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Remove(rels);
+ Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
+ Add(sgrps, f / rels);
+
+ Remove(rels);
+ Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
+ Add(sgrps, f / rels);
+ fi;
od;
+ od;
- return sgrps;
+ return sgrps;
end);
-#############################################################################
-##
## returns presentations for all semigroups of order and coclass 2
## which have a minimal generating set of size 2 up to equivalence
##
## Implementing list from small coclass paper for n >= 7
-##
-InstallGlobalFunction( NilpotentSemigroupsCoclass2Rank2,
-function( n )
- local sgrps, # list of all semigroups
- f, # free semigroup on two elements
- x, # generators of
- rels, # relations of a semigroup
- k; # loop counter
-
- # catch input error
- if not IsPosInt( n ) then
- Error( "the size has to be a positive integer" );
- fi;
-
- # catch exceptional cases; for n > 4 see end of function
- if n <= 4 then
- return [ ];
- fi;
-
- f := FreeSemigroup( 2 );
- x := GeneratorsOfSemigroup( f );
- sgrps := EmptyPlist( 5*n + Int(n/2) - Int(n/3) - 1 ); # might be 1 too big
-
- # y = v^2 = uv = vu (third semigroup in Case 4, y = v^2)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]*x[2] ], # v^2 = uv
- [ x[2]^3, x[1]^(n-3) ]] ); # v^3 = u^(n-3)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]*x[2] ], # v^2 = uv
- [ x[2]^3, x[1]^(n-2) ]] ); # v^3 = u^(n-2)
-
- # y = uv = vu
- for k in [ 3,4..Int( (n-1)/2 ) ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]^(2*k-4) ], # v^2 = u^(2k-4)
- [ x[1]^2*x[2], x[1]^k ]] ); # u^2v = u^k
- od;
- for k in [ n-Int(n/2)..n-2 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]^(n-4) ], # v^2 = u^(n-4)
- [ x[1]^2*x[2], x[1]^k ]] ); # u^2v = u^k
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]^(n-3) ], # v^2 = u^(n-3)
- [ x[1]^2*x[2], x[1]^k ]] ); # u^2v = u^k
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[2]^2, x[1]^(n-2) ], # v^2 = u^(n-2)
- [ x[1]^2*x[2], x[1]^k ]] ); # u^2v = u^k
- od;
-
- # y = uv = v^2
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[2]^2, x[1]*x[2] ], # v^2 = uv
- [ x[2]*x[1], x[1]^2 ], # vu = u^2
- [ x[1]*x[2]^2, x[1]^3 ]] ); # uv^2 = u^3
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[2]^2, x[1]*x[2] ], # v^2 = uv
- [ x[2]*x[1], x[1]^(n-3) ], # vu = u^(n-3)
- [ x[1]*x[2]^2, x[1]^(n-2) ]] ); # uv^2 = u^(n-2)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[2]^2, x[1]*x[2] ], # v^2 = uv
- [ x[2]*x[1], x[1]^(n-2) ], # vu = u^(n-2)
- [ x[1]*x[2]^2, x[1]^(n-2) ]] ); # uv^2 = u^(n-2)
-
- # y = v^2
- for k in [ 2..n-2 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[1]*x[2], x[1]^k ], # uv = u^k
- [ x[2]^3, x[1]^(3*k-3) ]] ); # v^3 = u^(3k-3)
- od;
- for k in [ n-Int(2*n/3)..n-2 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[2]*x[1]], # uv = vu
- [ x[1]*x[2], x[1]^k ], # uv = u^k
- [ x[2]^3, x[1]^(n-3) ]] ); # v^3 = u^(n-3)
- od;
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^(n-3) ], # uv = u^(n-3)
- [ x[2]*x[1], x[1]^(n-2) ], # vu = u^(n-2)
- [ x[2]^3, x[1]^(n-3) ]] ); # v^3 = u^(n-3)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^(n-3) ], # uv = u^(n-3)
- [ x[2]*x[1], x[1]^(n-2) ], # vu = u^(n-2)
- [ x[2]^3, x[1]^(n-2) ]] ); # v^3 = u^(n-2)
-
- # y = vu
- for k in [ 2..n-Int((n+1)/2)-1 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^k ], # uv = u^k
- [ x[2]^2, x[1]^(2*k-2) ], # v^2 = u^(2k-2)
- [ x[2]*x[1]^2, x[1]^(k+1) ]] ); # vu^2 = u^(k+1)
- od;
- for k in [ n-Int((n+1)/2)..n-2 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^k ], # uv = u^k
- [ x[2]^2, x[1]^(n-2) ], # v^2 = u^(n-2)
- [ x[2]*x[1]^2, x[1]^(k+1) ]] ); # vu^2 = u^(k+1)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^k ], # uv = u^k
- [ x[2]^2, x[1]^(n-3) ], # v^2 = u^(n-3)
- [ x[2]*x[1]^2, x[1]^(k+1) ]] ); # vu^2 = u^(k+1)
- od;
- for k in [ n-3..n-2 ] do
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^(n-4) ], # uv = u^(n-4)
- [ x[2]^2, x[1]^k ], # v^2 = u^k
- [ x[2]*x[1]^2, x[1]^(n-2) ]] ); # vu^2 = u^(n-2)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^(n-3) ], # uv = u^(n-3)
- [ x[2]^2, x[1]^k ], # v^2 = u^k
- [ x[2]*x[1]^2, x[1]^(n-3) ]] ); # vu^2 = u^(n-3)
- Add( sgrps, f / [[ x[1]^(n-1), x[1]^(n-2) ], # u^(n-2) = u^(n-1)
- [ x[1]*x[2], x[1]^(n-2) ], # uv = u^(n-2)
- [ x[2]^2, x[1]^k ], # v^2 = u^k
- [ x[2]*x[1]^2, x[1]^(n-3) ]] ); # vu^2 = u^(n-3)
- od;
-
- if n = 5 then
- sgrps := sgrps{[ 17, 15, 5, 18, 16, 4, 10, 8, 2, 6 ]};
- Add(sgrps, f / [[ x[1]^2, x[1]^2*x[2] ], # u^2 = u^2v
- [ x[2]^2, x[1]^2*x[2] ], # v^2 = u^2v
- [ x[1]*x[2]*x[1], x[1]^2*x[2] ], # uvu = u^2v
- [ x[2]*x[1]*x[2], (x[1]*x[2])^2 ]] ); # vuv = (uv)^2
- fi;
-
- if n = 6 then
- sgrps := sgrps{[ 23, 21, 8, 30, 29, 5, 6, 20, 3, 24, 22, 7, 27, 26, 14,
- 11, 19, 2, 10, 13, 4, 17, 18, 16, 1, 9 ]};
- rels := [[ x[1]^2, x[1]^3 ], # u^2 = u^3
- [ x[1]^2, x[2]^2 ], # u^2 = v^2
- [ x[1]^2, x[1]*x[2]*x[1] ], # u^2 = uvu
- [ x[1]*x[2]*x[1], x[2]*x[1]*x[2] ], # uvu = vuv
- [ x[1]^2, x[2]^3 ], # u^2 = v^3
- [ x[2]^2, x[2]*x[1]*x[2] ], # v^2 = vuv
- [ x[1]^2, x[2]*x[1] ], # u^2 = vu
- [ x[2]*x[1], x[1]*x[2]^2 ], # vu = uv^2
- [ x[2]^2, x[1]*x[2]*x[1] ], # v^2 = uvu
- [ x[1]^2, x[2]*x[1]*x[2] ], # u^2 = vuv
- [ x[2]^2, x[2]*x[1] ], # v^2 = vu
- [ x[1]*x[2], x[2]*x[1] ], # uv = vu
- [ x[1]^2*x[2], x[1]^2 ], # u^2v = u^2
- [ x[1]^3, x[1]*x[2]*x[1] ], # u^3 = uvu
- [ x[1]^3, x[2]^3 ], # u^3 = v^3
- [ x[1]*x[2]^2, x[1]^2 ], # uv^2 = u^2
- [ x[2]*x[1], x[2]*x[1]^2 ], # vu = vu^2
- [ x[1]^2*x[2], x[1]^3 ], # u^2v = u^3
- [ x[1]*x[2], x[1]^2 ], # uv = u^2
- [ x[2]*x[1]^3, x[1]^3 ]]; # vu^3 = u^3
-
- Add(sgrps, f / rels{[1, 2, 3]}); # [6, 42]
- Add(sgrps, f / rels{[1, 2, 4]}); # [6, 141]
- Add(sgrps, f / rels{[3, 5, 6]}); # [6, 338]
- Add(sgrps, f / rels{[1, 5, 7]}); # [6, 413]
- Add(sgrps, f / rels{[1, 5, 8]}); # [6, 414]
- Add(sgrps, f / rels{[5, 9, 10]}); # [6, 481]
- Add(sgrps, f / rels{[1, 5, 11]}); # [6, 583]
- Add(sgrps, f / rels{[5, 6, 9]}); # [6, 659]
- Add(sgrps, f / rels{[1, 5, 12, 13]}); # [6, 880]
- Add(sgrps, f / rels{[2, 10, 14]}); # [6, 1485]
- Add(sgrps, f / rels{[2, 3, 10]}); # [6, 1865]
- Add(sgrps, f / rels{[15, 16, 17]}); # [6, 2416]
- Add(sgrps, f / rels{[7, 15, 16]}); # [6, 2417]
- Add(sgrps, f / rels{[11, 15, 16]}); # [6, 2494]
- Add(sgrps, f / rels{[12, 15, 16, 18]}); # [6, 2556]
- Add(sgrps, f / rels{[11, 19, 20]}); # [6, 2558]
- Add(sgrps, f / rels{[11, 12, 20]}); # [6, 2559]
- fi;
-
- return sgrps;
+InstallGlobalFunction(NilpotentSemigroupsCoclass2Rank2,
+function(n)
+ local sgrps, # list of all semigroups
+ f, # free semigroup on two elements
+ x, # generators of
+ rels, # relations of a semigroup
+ k; # loop counter
+
+ if not IsPosInt(n) then
+ Error("the size < n > has to be a positive integer");
+ fi;
+
+ # catch exceptional cases; for n > 4 see end of function
+ if n <= 4 then
+ return [];
+ fi;
+
+ f := FreeSemigroup(2);
+ x := GeneratorsOfSemigroup(f);
+
+ # might be 1 too big
+ sgrps := EmptyPlist(5 * n + Int(n / 2) - Int(n / 3) - 1);
+
+ # y = v^2 = uv = vu (third semigroup in Case 4, y = v^2)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n - 2) = u ^ (n - 1)
+ [x[1] * x[2], x[2] * x[1]], # uv = vu
+ [x[2] ^ 2, x[1] * x[2]], # v ^ 2 = uv
+ [x[2] ^ 3, x[1] ^ (n - 3)]]); # v ^ 3 = u ^ (n - 3)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n - 2) = u ^ (n - 1)
+ [x[1] * x[2], x[2] * x[1]], # uv = vu
+ [x[2] ^ 2, x[1] * x[2]], # v ^ 2 = uv
+ [x[2] ^ 3, x[1] ^ (n - 2)]]); # v ^ 3 = u ^ (n - 2)
+
+ # y = uv = vu
+ for k in [3, 4 .. Int((n - 1) / 2)] do
+ # u ^ (n - 2) = u ^ (n - 1)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)],
+ [x[1] * x[2], x[2] * x[1]], # uv = vu
+ [x[2] ^ 2, x[1] ^ (2 * k - 4)], # v ^ 2 = u ^ (2k - 4)
+ [x[1] ^ 2 * x[2], x[1] ^ k]]); # u ^ 2v = u ^ k
+ od;
+ for k in [n - Int(n / 2) .. n - 2] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[2] ^ 2, x[1] ^ (n - 4)], # v^2 =u^(n-4)
+ [x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[2] ^ 2, x[1] ^ (n - 3)], # v^2=u^(n-3)
+ [x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n - 1)
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[2] ^ 2, x[1] ^ (n - 2)], # v^2=u^(n-2)
+ [x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
+ od;
+
+ # y = uv = v ^ 2
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[2] ^ 2, x[1] * x[2]], # v^2=uv
+ [x[2] * x[1], x[1] ^ 2], # vu=u^2
+ [x[1] * x[2] ^ 2, x[1] ^ 3]]); # uv^2=u^3
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[2] ^ 2, x[1] * x[2]], # v^2=uv
+ [x[2] * x[1], x[1] ^ (n - 3)], # vu=u^(n-3)
+ [x[1] * x[2] ^ 2, x[1] ^ (n - 2)]]); # uv^2=u^(n-2)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[2] ^ 2, x[1] * x[2]], # v^2=uv
+ [x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
+ [x[1] * x[2] ^ 2, x[1] ^ (n - 2)]]); # uv^2=u^(n-2)
+
+ # y = v ^ 2
+ for k in [2 .. n - 2] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[1] * x[2], x[1] ^ k], # uv=u^k
+ [x[2] ^ 3, x[1] ^ (3 * k - 3)]]); # v^3=u^(3k-3)
+ od;
+ for k in [n - Int(2 * n / 3) .. n - 2] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[1] * x[2], x[1] ^ k], # uv=u^k
+ [x[2] ^ 3, x[1] ^ (n - 3)]]); # v^3=u^(n-3)
+ od;
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
+ [x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
+ [x[2] ^ 3, x[1] ^ (n - 3)]]); # v^3=u^(n-3)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
+ [x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
+ [x[2] ^ 3, x[1] ^ (n - 2)]]); # v^3=u^(n-2)
+
+ # y=vu
+ for k in [2 .. n - Int((n + 1) / 2) - 1] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ k], # uv=u^k
+ [x[2] ^ 2, x[1] ^ (2 * k - 2)], # v^2=u^(2k-2)
+ [x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
+ od;
+ for k in [n - Int((n + 1) / 2) .. n - 2] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ k], # uv=u^k
+ [x[2] ^ 2, x[1] ^ (n - 2)], # v^2=u^(n-2)
+ [x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ k], # uv=u^k
+ [x[2] ^ 2, x[1] ^ (n - 3)], # v^2=u^(n-3)
+ [x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
+ od;
+ for k in [n - 3 .. n - 2] do
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ (n - 4)], # uv=u^(n-4)
+ [x[2] ^ 2, x[1] ^ k], # v^2=u^k
+ [x[2] * x[1] ^ 2, x[1] ^ (n - 2)]]); # vu^2=u^(n-2)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
+ [x[2] ^ 2, x[1] ^ k], # v^2=u^k
+ [x[2] * x[1] ^ 2, x[1] ^ (n - 3)]]); # vu^2=u^(n-3)
+ Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
+ [x[1] * x[2], x[1] ^ (n - 2)], # uv=u^(n-2)
+ [x[2] ^ 2, x[1] ^ k], # v^2=u^k
+ [x[2] * x[1] ^ 2, x[1] ^ (n - 3)]]); # vu^2=u^(n-3)
+ od;
+
+ if n = 5 then
+ sgrps := sgrps{[17, 15, 5, 18, 16, 4, 10, 8, 2, 6]};
+ Add(sgrps, f / [[x[1] ^ 2, x[1] ^ 2 * x[2]], # u^2=u^2v
+ [x[2] ^ 2, x[1] ^ 2 * x[2]], # v^2=u^2v
+ [x[1] * x[2] * x[1], x[1] ^ 2 * x[2]], # uvu=u^2v
+ [x[2] * x[1] * x[2], (x[1] * x[2]) ^ 2]]); # vuv=(uv)^2
+ fi;
+
+ if n = 6 then
+ sgrps := sgrps{[23, 21, 8, 30, 29, 5, 6, 20, 3, 24, 22, 7, 27, 26, 14,
+ 11, 19, 2, 10, 13, 4, 17, 18, 16, 1, 9]};
+ rels := [[x[1] ^ 2, x[1] ^ 3], # u^2=u^3
+ [x[1] ^ 2, x[2] ^ 2], # u^2=v^2
+ [x[1] ^ 2, x[1] * x[2] * x[1]], # u^2=uvu
+ [x[1] * x[2] * x[1], x[2] * x[1] * x[2]], # uvu=vuv
+ [x[1] ^ 2, x[2] ^ 3], # u^2=v^3
+ [x[2] ^ 2, x[2] * x[1] * x[2]], # v^2=vuv
+ [x[1] ^ 2, x[2] * x[1]], # u^2=vu
+ [x[2] * x[1], x[1] * x[2] ^ 2], # vu=uv^2
+ [x[2] ^ 2, x[1] * x[2] * x[1]], # v^2=uvu
+ [x[1] ^ 2, x[2] * x[1] * x[2]], # u^2=vuv
+ [x[2] ^ 2, x[2] * x[1]], # v^2=vu
+ [x[1] * x[2], x[2] * x[1]], # uv=vu
+ [x[1] ^ 2 * x[2], x[1] ^ 2], # u^2v=u^2
+ [x[1] ^ 3, x[1] * x[2] * x[1]], # u^3=uvu
+ [x[1] ^ 3, x[2] ^ 3], # u^3=v^3
+ [x[1] * x[2] ^ 2, x[1] ^ 2], # uv^2=u^2
+ [x[2] * x[1], x[2] * x[1] ^ 2], # vu=vu^2
+ [x[1] ^ 2 * x[2], x[1] ^ 3], # u^2v=u^3
+ [x[1] * x[2], x[1] ^ 2], # uv=u^2
+ [x[2] * x[1] ^ 3, x[1] ^ 3]]; # vu^3=u^3
+
+ Add(sgrps, f / rels{[1, 2, 3]}); # [6, 42]
+ Add(sgrps, f / rels{[1, 2, 4]}); # [6, 141]
+ Add(sgrps, f / rels{[3, 5, 6]}); # [6, 338]
+ Add(sgrps, f / rels{[1, 5, 7]}); # [6, 413]
+ Add(sgrps, f / rels{[1, 5, 8]}); # [6, 414]
+ Add(sgrps, f / rels{[5, 9, 10]}); # [6, 481]
+ Add(sgrps, f / rels{[1, 5, 11]}); # [6, 583]
+ Add(sgrps, f / rels{[5, 6, 9]}); # [6, 659]
+ Add(sgrps, f / rels{[1, 5, 12, 13]}); # [6, 880]
+ Add(sgrps, f / rels{[2, 10, 14]}); # [6, 1485]
+ Add(sgrps, f / rels{[2, 3, 10]}); # [6, 1865]
+ Add(sgrps, f / rels{[15, 16, 17]}); # [6, 2416]
+ Add(sgrps, f / rels{[7, 15, 16]}); # [6, 2417]
+ Add(sgrps, f / rels{[11, 15, 16]}); # [6, 2494]
+ Add(sgrps, f / rels{[12, 15, 16, 18]}); # [6, 2556]
+ Add(sgrps, f / rels{[11, 19, 20]}); # [6, 2558]
+ Add(sgrps, f / rels{[11, 12, 20]}); # [6, 2559]
+ fi;
+ return sgrps;
end);
diff --git a/gap/enums.gd b/gap/enums.gd
index 982d3c9..346c683 100644
--- a/gap/enums.gd
+++ b/gap/enums.gd
@@ -1,1026 +1,918 @@
#############################################################################
##
-#W enums.gd Smallsemi - a GAP library of semigroups
-#Y Copyright (C) 2008-2014 Andreas Distler & James D. Mitchell
+## enums.gd Smallsemi - a GAP library of semigroups
+## Copyright (C) 2008-2024 Andreas Distler & James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
-
-###########################################################################
-##
-## <#GAPDoc Label="AllSmallSemigroups">
-##
-##
-##
-## the number of argument of this function should be odd. The first argument
-## arg[1] should be a positive integer, an
-## enumerator of small semigroups with
-## - a positive integer arg[1] and a set of positive integers less
-## than
-## - a set of positive integers arg[1] and a list of sets of
-## positive integers arg[2] such that x is at most
-##
-## - a list of id numbers, for example, [[7,1], [6,1], [5,1]].
-##
-## The no check version does not check that the arguments are valid and may return unpredictable
-## results.
-##
-## enum:=EnumeratorOfSmallSemigroupsByIds([[7,1],[6,1],[5,1]]);
-##
-## gap> enum:=EnumeratorOfSmallSemigroupsByIds(7, [1..1000]);
-##
-## gap> enum:=EnumeratorOfSmallSemigroupsByIds([2,3], [[1..2],[1..10]]);
-##
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-DeclareOperation("EnumeratorOfSmallSemigroupsByIds", [IsCyclotomicCollection , IsCyclotomicCollColl]);
-
-DeclareOperation("EnumeratorOfSmallSemigroupsByIdsNC", [IsCyclotomicCollection , IsCyclotomicCollColl]);
-
-#############################################################################
-##
-## EnumeratorOfSmallSemigroupsByDiagonals
-##
-## should become obsolete in 0.7 and is therefore undocumented
-##
-
+# <#GAPDoc Label="EnumeratorOfSmallSemigroupsByIds">
+#
+#
+#
+#
+# the argument of this function should be one of the following:
+#
+# -
+# a positive integer arg[1] and a set of positive integers less than
+#
+# -
+# a set of positive integers arg[1] and a list of sets of positive
+# integers arg[2] such that x is at most
+# -
+# a list of id numbers, for example, [[7,1], [6,1], [5,1]].
+#
+#
+# The no check version does not check that the arguments are valid and may
+# return unpredictable results.
+#
+# enum:=EnumeratorOfSmallSemigroupsByIds([[7,1],[6,1],[5,1]]);
+#
+# gap> enum:=EnumeratorOfSmallSemigroupsByIds(7, [1..1000]);
+#
+# gap> enum:=EnumeratorOfSmallSemigroupsByIds([2,3], [[1..2],[1..10]]);
+#
+# ]]>
+#
+#
+# <#/GAPDoc>
+DeclareOperation("EnumeratorOfSmallSemigroupsByIds",
+ [IsCyclotomicCollection, IsCyclotomicCollColl]);
+DeclareOperation("EnumeratorOfSmallSemigroupsByIds",
+ [IsPosInt, IsCyclotomicCollection]);
+DeclareOperation("EnumeratorOfSmallSemigroupsByIds",
+ [IsList]);
+
+DeclareOperation("EnumeratorOfSmallSemigroupsByIdsNC",
+ [IsCyclotomicCollection, IsCyclotomicCollColl]);
+DeclareOperation("EnumeratorOfSmallSemigroupsByIdsNC",
+ [IsPosInt, IsCyclotomicCollection]);
+
+## TODO document
DeclareGlobalFunction("EnumeratorOfSmallSemigroupsByDiagonals");
-###########################################################################
-##
-## <#GAPDoc Label="EnumeratorSortedOfSmallSemigroups">
-##
-##
-##
-## accepts the same arguments and returns the same values as
-##
-## - odd length
-## - arg[1] is a positive integer between 1 and 8, a list of positive integers
-## between 1 and 8, an enumerator of small semigroups or an iterator of small semigroups
-## - arg[2i] (the even indexed arguments) should be functions.
-##
-##
-##
-##
-## <#/GAPDoc>
-
-DeclareGlobalFunction("SMALLSEMI_ARG_OK");
-
-###########################################################################
-##
-## <#GAPDoc Label="SMALLSEMI_CAN_CREATE_ENUM_NC">
-##
-##
-##
-## checks that the argument arg can be used to produce an enumerator.
-## This function does not check
-## - the maximum size of semigroup satisfying arg at most 7; OR
-## - the maximum size of semigroup satisfying arg equal 8 and there exists
-## i such that arg[2i] in
-##
-##
-## The reason for this is that on a 32-bit computer the maximum length of a list is
-## smaller than the number of semigroups with 8 elements. Enumerators use lists of id numbers
-## to specify their elements and so it is not currently possible to create arbitrary
-## enumerators of small semigroups containing semigroups with 8 elements.
-##
-##
-## <#/GAPDoc>
-
+# <#GAPDoc Label="SMALLSEMI_ValidateArgs">
+#
+#
+#
+# checks that the argument arg is valid for any of the functions
+#
+# -
+# odd length;
+#
+# -
+# arg[1] is a positive integer between 1 and 8; a list of positive
+# integers between 1 and 8; an enumerator of small semigroups; or an iterator
+# of small semigroups;
+#
+# -
+# arg[2i] (the even indexed arguments) should be functions.
+#
+#
+#
+#
+# <#/GAPDoc>
+DeclareGlobalFunction("SMALLSEMI_ValidateArgs");
+
+# <#GAPDoc Label="SMALLSEMI_CAN_CREATE_ENUM_NC">
+#
+#
+#
+# checks that the argument arg can be used to produce an enumerator.
+# This function does not check
+# -
+# the maximum size of semigroup satisfying arg at most 7; OR
+#
+# -
+# the maximum size of semigroup satisfying arg equal 8 and there
+# exists i such that arg[2i] in
+#
+# The reason for this is that on a 32-bit computer the maximum length of a list
+# is smaller than the number of semigroups with 8 elements. Enumerators use
+# lists of id numbers to specify their elements and so it is not currently
+# possible to create arbitrary enumerators of small semigroups containing
+# semigroups with 8 elements.
+#
+#
+# <#/GAPDoc>
DeclareGlobalFunction("SMALLSEMI_CAN_CREATE_ENUM_NC");
-###########################################################################
-##
-## <#GAPDoc Label="SMALLSEMI_CONVERT_ARG_NC">
-##
-##
-##
-## arg is assumed to satisfy
-## - source should be a positive integer between 1 and 8, a list of
-## positive integers between 1 and 8, an enumerator of small semigroups, or
-## an iterator of small semigroups;
-## - positions should be the list
-## such that positions[i] is the list of second components of id numbers ## of small semigroups in the enumerator we are creating
-## - names is the list of functions and values being used to create
-## the enumerator. It is not checked if it is possible to create an enumerator ## using Concatenation([source],names) as an argument. See
-##
-##
-## SMALLSEMI_CREATE_ENUM returns the same value as
-##
+# -
+# source should be a positive integer between 1 and 8, a list of
+# positive integers between 1 and 8, an enumerator of small semigroups, or an
+# iterator of small semigroups;
+#
+# -
+# positions should be the list such that positions[i] is the
+# list of second components of id numbers of small semigroups in the
+# enumerator being created;
+#
+# -
+# names is the list of functions and values being used to create the
+# enumerator. It is not checked if it is possible to create an enumerator
+# using Concatenation([source],names) as an argument. See
+#
+# SMALLSEMI_CREATE_ENUM returns the same value as , )
-#O LStarClass(, )
-#O DStarClass(, )
-#O JStarClass(, )
-#O HStarClass(, )
-##
-## <#GAPDoc Label="RStarClassOperation">
-##
-##
-##
-##
-##
-##
-##
-##
-## Creates the X*-class of the element a
-## in the semigroup S where X is one of
-## L, R, D, J, or H.
-## s := SmallSemigroup(7, 280142);
-##
-## gap> elm := AsList(s)[5];;
-## gap> jclass := JStarClass(s, elm);
-## {s5}
-## gap> AsList(jclass);
-## [ s2, s3, s4, s5 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-##
-
+# <#GAPDoc Label="RStarClassOperation">
+#
+#
+#
+#
+#
+#
+#
+#
+# Creates the X*-class of the element a
+# in the semigroup S where X is one of
+# L, R, D, J, or H.
+# s := SmallSemigroup(7, 280142);
+#
+# gap> elm := AsList(s)[5];;
+# gap> jclass := JStarClass(s, elm);
+# {s5}
+# gap> AsList(jclass);
+# [ s2, s3, s4, s5 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
DeclareOperation("HStarClass", [IsSemigroup, IsObject]);
DeclareOperation("RStarClass", [IsSemigroup, IsObject]);
DeclareOperation("LStarClass", [IsSemigroup, IsObject]);
DeclareOperation("DStarClass", [IsSemigroup, IsObject]);
DeclareOperation("JStarClass", [IsSemigroup, IsObject]);
-#############################################################################
-##
-## misc
-##
-
DeclareAttribute("InternalRepStarRelation", IsStarRelation);
DeclareAttribute("CanonicalStarClass", IsStarClass);
-
diff --git a/gap/greensstar.gi b/gap/greensstar.gi
index 4a16d9f..b05ff69 100644
--- a/gap/greensstar.gi
+++ b/gap/greensstar.gi
@@ -1,119 +1,100 @@
#############################################################################
##
-#W greensstar.gi Smallsemi - a GAP library of semigroups
-#Y Copyright (C) 2008-2013 Andreas Distler & James D. Mitchell
+## greensstar.gi Smallsemi - a GAP library of semigroups
+## Copyright (C) 2008-2024 Andreas Distler & James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
-##
-## This file contains the implementations for starred Green's relations of
-## semigroups
-#
-
-#######################
-#######################
-##
-#M RStarRelation()
-#M LStarRelation()
-#M HStarRelation()
-#M DStarRelation()
-#M JStarRelation()
-##
-## returns the appropriate equivalence relation which is stored as an attribute.
-## The relation knows nothing about itself except its source, range, and what
-## type of congruence it is.
+# This file contains the implementations for starred Green's relations of
+# semigroups.
+# Returns the appropriate equivalence relation which is stored as an attribute.
+# The relation knows nothing about itself except its source, range, and what
+# type of congruence it is.
InstallMethod(RStarRelation, "for a small semigroup", [IsSmallSemigroup],
function(X)
- local fam, rel, sc;
-
- fam := GeneralMappingsFamily( ElementsFamily(FamilyObj(X)),
- ElementsFamily(FamilyObj(X)) );
-
- # Create the default type for the elements.
- rel := Objectify(NewType(fam,
- IsEquivalenceRelation and IsEquivalenceRelationDefaultRep
- and IsRStarRelation), rec());
-
- SetSource(rel, X);
- SetRange(rel, X);
- # AD The following causes weird viewing of the relation on the command line
- SetIsLeftSemigroupCongruence(rel,true);
- if HasIsFinite(X) and IsFinite(X) then
- SetIsFiniteSemigroupStarRelation(rel, true);
- fi;
+ local fam, rel;
+
+ fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(X)),
+ ElementsFamily(FamilyObj(X)));
+
+ # Create the default type for the elements.
+ rel := Objectify(NewType(fam, IsEquivalenceRelation
+ and IsEquivalenceRelationDefaultRep
+ and IsRStarRelation), rec());
+ SetSource(rel, X);
+ SetRange(rel, X);
+ # AD The following causes weird viewing of the relation on the command line
+ SetIsLeftSemigroupCongruence(rel, true);
+ if HasIsFinite(X) and IsFinite(X) then
+ SetIsFiniteSemigroupStarRelation(rel, true);
+ fi;
-# AD rel is objectified using IsRStarRelation, so other Green's relations
-# should not be set to the same object; classes might be set at the point
-# where they are actually computed, i.e. RStarClasses
-#
-# if HasIsCommutative(X) and IsCommutative(X) then
-# SetLStarRelation(X, rel);
-# SetDStarRelation(X, rel);
-# SetHStarRelation(X, rel);
-# fi;
+ # AD rel is objectified using IsRStarRelation, so other Green's relations
+ # should not be set to the same object; classes might be set at the point
+ # where they are actually computed, i.e. RStarClasses
- return rel;
+ return rel;
end);
InstallMethod(LStarRelation, "for a small semigroup", [IsSmallSemigroup],
function(X)
- local fam, rel;
-
- fam := GeneralMappingsFamily( ElementsFamily(FamilyObj(X)),
- ElementsFamily(FamilyObj(X)) );
-
- # Create the default type for the elements.
- rel := Objectify(NewType(fam,
- IsEquivalenceRelation and IsEquivalenceRelationDefaultRep
- and IsLStarRelation), rec());
-
- SetSource(rel, X);
- SetRange(rel, X);
- # AD The following causes weird viewing of the relation on the command line
- SetIsRightSemigroupCongruence(rel,true);
- if HasIsFinite(X) and IsFinite(X) then
- SetIsFiniteSemigroupStarRelation(rel, true);
- fi;
+ local fam, rel;
+
+ fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(X)),
+ ElementsFamily(FamilyObj(X)));
+
+ # Create the default type for the elements.
+ rel := Objectify(NewType(fam, IsEquivalenceRelation
+ and IsEquivalenceRelationDefaultRep
+ and IsLStarRelation), rec());
+
+ SetSource(rel, X);
+ SetRange(rel, X);
+ # AD The following causes weird viewing of the relation on the command line
+ SetIsRightSemigroupCongruence(rel, true);
+ if HasIsFinite(X) and IsFinite(X) then
+ SetIsFiniteSemigroupStarRelation(rel, true);
+ fi;
- return rel;
+ return rel;
end);
InstallMethod(JStarRelation, "for a small semigroup", [IsSmallSemigroup],
function(X)
- local fam, rel;
+ local fam, rel;
- fam := GeneralMappingsFamily( ElementsFamily(FamilyObj(X)),
- ElementsFamily(FamilyObj(X)) );
+ fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(X)),
+ ElementsFamily(FamilyObj(X)));
- # Create the default type for the elements.
- rel := Objectify(NewType(fam,
- IsEquivalenceRelation and IsEquivalenceRelationDefaultRep
- and IsJStarRelation), rec());
+ # Create the default type for the elements.
+ rel := Objectify(NewType(fam, IsEquivalenceRelation
+ and IsEquivalenceRelationDefaultRep
+ and IsJStarRelation), rec());
- SetSource(rel, X);
- SetRange(rel, X);
- if HasIsFinite(X) and IsFinite(X) then
- SetIsFiniteSemigroupStarRelation(rel, true);
- fi;
+ SetSource(rel, X);
+ SetRange(rel, X);
+ if HasIsFinite(X) and IsFinite(X) then
+ SetIsFiniteSemigroupStarRelation(rel, true);
+ fi;
- return rel;
+ return rel;
end);
InstallMethod(DStarRelation, "for a small semigroup", [IsSmallSemigroup],
function(X)
local fam, rel;
- fam := GeneralMappingsFamily( ElementsFamily(FamilyObj(X)),
- ElementsFamily(FamilyObj(X)) );
+ fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(X)),
+ ElementsFamily(FamilyObj(X)));
# Create the default type for the elements.
- rel := Objectify(NewType(fam,
- IsEquivalenceRelation and IsEquivalenceRelationDefaultRep
- and IsDStarRelation), rec());
+ rel := Objectify(NewType(fam, IsEquivalenceRelation
+ and IsEquivalenceRelationDefaultRep
+ and IsDStarRelation), rec());
SetSource(rel, X);
SetRange(rel, X);
@@ -127,278 +108,191 @@ end);
InstallMethod(HStarRelation, "for a small semigroup", [IsSmallSemigroup],
function(X)
- local fam, rel;
+ local fam, rel;
- fam := GeneralMappingsFamily( ElementsFamily(FamilyObj(X)),
- ElementsFamily(FamilyObj(X)) );
+ fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(X)),
+ ElementsFamily(FamilyObj(X)));
- # Create the default type for the elements.
- rel := Objectify(NewType(fam,
- IsEquivalenceRelation and IsEquivalenceRelationDefaultRep
- and IsHStarRelation), rec());
+ # Create the default type for the elements.
+ rel := Objectify(NewType(fam, IsEquivalenceRelation
+ and IsEquivalenceRelationDefaultRep
+ and IsHStarRelation), rec());
- SetSource(rel, X);
- SetRange(rel, X);
+ SetSource(rel, X);
+ SetRange(rel, X);
- if HasIsFinite(X) and IsFinite(X) then
- SetIsFiniteSemigroupStarRelation(rel, true);
- fi;
+ if HasIsFinite(X) and IsFinite(X) then
+ SetIsFiniteSemigroupStarRelation(rel, true);
+ fi;
- return rel;
+ return rel;
end);
-
### AD The following methods should probably be installed for ViewString
-# AD This is overwritten by the method for LeftSemigroupCongruence
-InstallMethod( ViewObj, "for RStarRelation", [IsRStarRelation],
- function( obj )
- Print( "< R*-relation on ");
- ViewObj(Source(obj));
- Print(" >");
-end );
-
-# AD This is overwritten by the method for RightSemigroupCongruence
-InstallMethod( ViewObj, "for LStarRelation", [IsLStarRelation],
- function( obj )
- Print( "< L*-relation on ");
- ViewObj(Source(obj));
- Print(" >");
-end );
-
-InstallMethod( ViewObj, "for JStarRelation", [IsJStarRelation],
- function( obj )
- Print( "< J*-relation on ");
- ViewObj(Source(obj));
- Print(" >");
-end );
-
-InstallMethod( ViewObj, "for DStarRelation", [IsDStarRelation],
- function( obj )
- Print( "< D*-relation on ");
- ViewObj(Source(obj));
- Print(" >");
-end );
-
-InstallMethod( ViewObj, "for HStarRelation", [IsHStarRelation],
- function( obj )
- Print( "< H*-relation on ");
- ViewObj(Source(obj));
- Print(" >");
-end );
-
-InstallMethod(\=, "for starred Green's relations", IsIdenticalObj,
-[IsStarRelation and IsEquivalenceRelation,
-IsStarRelation and IsEquivalenceRelation], function(rel1, rel2)
- if not Source(rel1) = Source(rel2) then
- return false;
- else
- # make sure the internal representation is known
- EquivalenceClasses(rel1); EquivalenceClasses(rel2);
- return InternalRepStarRelation(rel1) = InternalRepStarRelation(rel2);
- fi;
+BindGlobal("SMALLSEMI_ViewStarRelation",
+function(obj, type)
+ Print("<", type, "*-relation on ");
+ ViewObj(Source(obj));
+ Print(">");
end);
-#############################################################################
-##
-## The following operations are constructors for starred Green's class with
-## a given element as a representative. The call is for semigroups
-## and an element in the semigroup. This function doesn't check that
-## the element is actually a member of the semigroup.
-##
-#O RStarClass(, )
-#O LStarClass(, )
-#O JStarClass(, )
-#O DStarClass(, )
-#O HStarClass(, )
-##
+# AD This is overwritten by the method for LeftSemigroupCongruence
+InstallMethod(ViewObj, "for Green's R*-relation", [IsRStarRelation],
+16, # to beat IsLeftSemigroupCongruence
+obj -> SMALLSEMI_ViewStarRelation(obj, "R"));
-InstallMethod(RStarClass, "for a small semigroup ", IsCollsElms,
-[IsSmallSemigroup, IsSmallSemigroupElt], function(s,e)
- return EquivalenceClassOfElement( RStarRelation(s), e );
+# AD This is overwritten by the method for RightSemigroupCongruence
+InstallMethod(ViewObj, "for Green's L*-relation", [IsLStarRelation],
+16, # to beat IsRightSemigroupCongruence
+obj -> SMALLSEMI_ViewStarRelation(obj, "L"));
+
+InstallMethod(ViewObj, "for Green's J*-relation", [IsJStarRelation],
+obj -> SMALLSEMI_ViewStarRelation(obj, "J"));
+
+InstallMethod(ViewObj, "for Green's D*-relation", [IsDStarRelation],
+obj -> SMALLSEMI_ViewStarRelation(obj, "D"));
+
+InstallMethod(ViewObj, "for Green's H*-relation", [IsHStarRelation],
+obj -> SMALLSEMI_ViewStarRelation(obj, "H"));
+
+InstallMethod(\=, "for starred Green's relations", IsIdenticalObj,
+[IsStarRelation and IsEquivalenceRelation,
+ IsStarRelation and IsEquivalenceRelation],
+function(rel1, rel2)
+ if Source(rel1) <> Source(rel2) then
+ return false;
+ fi;
+ # make sure the internal representation is known
+ EquivalenceClasses(rel1);
+ EquivalenceClasses(rel2);
+ return InternalRepStarRelation(rel1) = InternalRepStarRelation(rel2);
end);
+# The following operations are constructors for starred Green's class with
+# a given element as a representative. The call is for semigroups
+# and an element in the semigroup. This function doesn't check that
+# the element is actually a member of the semigroup.
+
+InstallMethod(RStarClass, "for a small semigroup ", IsCollsElms,
+[IsSmallSemigroup, IsSmallSemigroupElt],
+{s, e} -> EquivalenceClassOfElement(RStarRelation(s), e));
+
InstallMethod(LStarClass, "for a small semigroup", IsCollsElms,
-[IsSmallSemigroup, IsSmallSemigroupElt], function(s,e)
- return EquivalenceClassOfElement( LStarRelation(s), e );
-end);
+[IsSmallSemigroup, IsSmallSemigroupElt],
+{s, e} -> EquivalenceClassOfElement(LStarRelation(s), e));
InstallMethod(HStarClass, "for a small semigroup", IsCollsElms,
-[IsSmallSemigroup, IsSmallSemigroupElt], function(s,e)
- return EquivalenceClassOfElement( HStarRelation(s), e );
-end);
+[IsSmallSemigroup, IsSmallSemigroupElt],
+{s, e} -> EquivalenceClassOfElement(HStarRelation(s), e));
InstallMethod(DStarClass, "for a small semigroup", IsCollsElms,
-[IsSmallSemigroup, IsSmallSemigroupElt], function(s,e)
- return EquivalenceClassOfElement( DStarRelation(s), e );
-end);
+[IsSmallSemigroup, IsSmallSemigroupElt],
+{s, e} -> EquivalenceClassOfElement(DStarRelation(s), e));
InstallMethod(JStarClass, "for a small semigroup", IsCollsElms,
-[IsSmallSemigroup, IsSmallSemigroupElt], function(s,e)
- return EquivalenceClassOfElement( JStarRelation(s), e );
-end);
-
+[IsSmallSemigroup, IsSmallSemigroupElt],
+{s, e} -> EquivalenceClassOfElement(JStarRelation(s), e));
-#################
-#################
-##
-#M CanonicalStarClass()
-##
-## methods to get any starred Green's class in its canonical form
-##
+# Methods to get any starred Green's class in its canonical form
-InstallMethod(CanonicalStarClass, "for a H*-class", [IsHStarClass],
-cl-> First(HStarClasses(ParentAttr(cl)), y-> Representative(cl) in y));
+InstallMethod(CanonicalStarClass, "for a H*-class", [IsHStarClass],
+cl -> First(HStarClasses(ParentAttr(cl)), y -> Representative(cl) in y));
-InstallMethod(CanonicalStarClass, "for a R*-class", [IsRStarClass],
-cl-> First(RStarClasses(ParentAttr(cl)), y-> Representative(cl) in y));
+InstallMethod(CanonicalStarClass, "for a R * - class", [IsRStarClass],
+cl -> First(RStarClasses(ParentAttr(cl)), y -> Representative(cl) in y));
-InstallMethod(CanonicalStarClass, "for a L*-class", [IsLStarClass],
-cl-> First(LStarClasses(ParentAttr(cl)), y-> Representative(cl) in y));
+InstallMethod(CanonicalStarClass, "for a L*-class", [IsLStarClass],
+cl -> First(LStarClasses(ParentAttr(cl)), y -> Representative(cl) in y));
-InstallMethod(CanonicalStarClass, "for a D*-class", [IsDStarClass],
-cl-> First(DStarClasses(ParentAttr(cl)), y-> Representative(cl) in y));
+InstallMethod(CanonicalStarClass, "for a D*-class", [IsDStarClass],
+cl -> First(DStarClasses(ParentAttr(cl)), y -> Representative(cl) in y));
-InstallMethod(CanonicalStarClass, "for a J*-class", [IsJStarClass],
-cl-> First(JStarClasses(ParentAttr(cl)), y-> Representative(cl) in y));
+InstallMethod(CanonicalStarClass, "for a J*-class", [IsJStarClass],
+cl -> First(JStarClasses(ParentAttr(cl)), y -> Representative(cl) in y));
-#################
-#################
-##
-#M ImagesElm(, )
-##
-## method to find the images under a starred Green's relation of an
-## element of a semigroup.
-##
+# method to find the images under a starred Green's relation of an
+# element of a semigroup.
InstallMethod(ImagesElm, "for a starred Green's relation",
-[IsStarRelation, IsObject], 0,
- function(rel, elm)
- local exp, semi;
-
- semi:=Source(rel);
-
- if IsRStarRelation(rel) then
- exp:=RStarClass(semi, elm);
- elif IsLStarRelation(rel) then
- exp:=LStarClass(semi, elm);
- elif IsHStarRelation(rel) then
- exp:=HStarClass(semi, elm);
- elif IsDStarRelation(rel) then
- exp:=DStarClass(semi, elm);
- elif IsJStarRelation(rel) then
- exp:=JStarClass(semi, elm);
- fi;
+[IsStarRelation, IsObject],
+function(rel, elm)
+ local exp, semi;
- return AsSSortedList(exp);
- end);
+ semi := Source(rel);
-#################
-#################
-##
-#M Successors()
-##
-## returns ImagesElm for one element in each class of
-##
+ if IsRStarRelation(rel) then
+ exp := RStarClass(semi, elm);
+ elif IsLStarRelation(rel) then
+ exp := LStarClass(semi, elm);
+ elif IsHStarRelation(rel) then
+ exp := HStarClass(semi, elm);
+ elif IsDStarRelation(rel) then
+ exp := DStarClass(semi, elm);
+ elif IsJStarRelation(rel) then
+ exp := JStarClass(semi, elm);
+ fi;
-InstallMethod(Successors, "for a starred Green's relation",
-[IsStarRelation], 0,
- function( rel )
- return List(EquivalenceClasses(rel), AsSSortedList);
+ return AsSSortedList(exp);
end);
-#################
-#################
-##
-#M AsSSortedList()
-##
-## returns the elements of the Star class
-##
+# Returns ImagesElm for one element in each class of
-InstallMethod(AsSSortedList, "for a starred Green's class", [IsStarClass],
- x-> AsSSortedList(CanonicalStarClass(x)));
+InstallMethod(Successors, "for a starred Green's relation",
+[IsStarRelation],
+rel -> List(EquivalenceClasses(rel), AsSSortedList));
-#################
-#################
-##
-#M \= (,)
-##
+# Returns the elements of the *-class
+
+InstallMethod(AsSSortedList, "for a starred Green's class", [IsStarClass],
+x -> AsSSortedList(CanonicalStarClass(x)));
-InstallMethod(\=, "for starred Green's classes",
+# Equality of *-classes
+InstallMethod(\=, "for starred Green's classes",
[IsStarClass, IsStarClass],
-function(class1,class2)
- if not ParentAttr(class1)=ParentAttr(class2) then
- return false;
- else
- return Representative(class1) in class2;
- fi;
+function(class1, class2)
+ if ParentAttr(class1) <> ParentAttr(class2) then
+ return false;
+ fi;
+ return Representative(class1) in class2;
end);
-#################
-#################
-##
-#M Size()
-##
-## size of a starred Green's class
-##
-
+# Size of a starred Green's class
InstallMethod(Size, "for a starred Green's class", [IsStarClass],
-function(class)
- return Size(Elements(class));
-end);
+class -> Size(Elements(class)));
-#################
-#################
-##
-#M in
-##
-## membership test for a starred Green's class
-##
-
-InstallMethod(\in, "membership test of starred Green's class",
-[IsObject, IsStarClass], 0,
+# Membership test for a starred Green's class
+InstallMethod(\in, "membership test of starred Green's class",
+[IsObject, IsStarClass],
function(elm, class)
- if elm=Representative(class) then
+ if elm = Representative(class) then
return true;
fi;
return elm in Elements(class);
end);
-#################
-#################
-##
-#M EquivalenceRelationPartition()
-##
-##
-
InstallMethod(EquivalenceRelationPartition, "for a starred Green's equivalence",
[IsEquivalenceRelation and IsStarRelation],
-function(rel)
- return Filtered(Successors(rel), x-> not Length(x)=1);
-end);
+rel -> Filtered(Successors(rel), x -> not Length(x) <> 1));
-#################
-#################
-##
-#M EquivalenceClassOfElementNC(, )
-##
-## new methods required so that what is returned by this function
-## is the appropriate type of starred Green's class
+# New methods required so that what is returned by this function
+# is the appropriate type of starred Green's class.
-InstallOtherMethod(EquivalenceClassOfElementNC,
-"for a starred Green's relation", true,
- [IsEquivalenceRelation and IsStarRelation, IsObject], 0,
+InstallOtherMethod(EquivalenceClassOfElementNC,
+"for a starred Green's relation",
+[IsEquivalenceRelation and IsStarRelation, IsObject],
function(rel, rep)
-
local new;
- new:= Objectify(NewType(CollectionsFamily(FamilyObj(rep)),
- IsEquivalenceClass and IsEquivalenceClassDefaultRep), rec());
+ new := Objectify(NewType(CollectionsFamily(FamilyObj(rep)),
+ IsEquivalenceClass and IsEquivalenceClassDefaultRep),
+ rec());
SetEquivalenceClassRelation(new, rel);
SetRepresentative(new, rep);
SetParent(new, UnderlyingDomainOfBinaryRelation(rel));
SetIsStarClass(new, true);
+
if IsRStarRelation(rel) then
SetIsRStarClass(new, true);
elif IsLStarRelation(rel) then
@@ -414,405 +308,340 @@ function(rel, rep)
return new;
end);
-#################
-#################
-##
-#M EquivalenceClasses()
-##
-
-InstallMethod(EquivalenceClasses, "for a starred Green's R-relation",
-[IsEquivalenceRelation and IsRStarRelation], x-> RStarClasses(Source(x)));
+InstallMethod(EquivalenceClasses, "for a starred Green's R - relation",
+[IsEquivalenceRelation and IsRStarRelation], x -> RStarClasses(Source(x)));
InstallMethod(EquivalenceClasses, "for a starred Green's L-relation",
-[IsEquivalenceRelation and IsLStarRelation], x-> LStarClasses(Source(x)));
+[IsEquivalenceRelation and IsLStarRelation], x -> LStarClasses(Source(x)));
-InstallMethod(EquivalenceClasses, "for a starred Green's H-relation",
-[IsEquivalenceRelation and IsHStarRelation], x-> HStarClasses(Source(x)));
+InstallMethod(EquivalenceClasses, "for a starred Green's H - relation",
+[IsEquivalenceRelation and IsHStarRelation], x -> HStarClasses(Source(x)));
-InstallMethod(EquivalenceClasses, "for a starred Green's D-relation",
-[IsEquivalenceRelation and IsDStarRelation], x-> DStarClasses(Source(x)));
+InstallMethod(EquivalenceClasses, "for a starred Green's D - relation",
+[IsEquivalenceRelation and IsDStarRelation], x -> DStarClasses(Source(x)));
InstallMethod(EquivalenceClasses, "for a starred Green's J-relation",
-[IsEquivalenceRelation and IsJStarRelation], x-> JStarClasses(Source(x)));
+[IsEquivalenceRelation and IsJStarRelation], x -> JStarClasses(Source(x)));
-
-#################
-#################
-##
-#M RStarClass()
-#M LStarClass()
-#M DStarClass()
-#M JStarClass()
-##
-## return the XStarClass containing the smaller class
-##
+# Return the XStarClass containing the smaller class
InstallOtherMethod(RStarClass, "for a starred Green's H-class",
-[IsHStarClass], cl-> RStarClass(ParentAttr(cl), Representative(cl)));
+[IsHStarClass], cl -> RStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(LStarClass, "for a starred Green's H-class",
-[IsHStarClass], cl-> LStarClass(ParentAttr(cl), Representative(cl)));
+[IsHStarClass], cl -> LStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(DStarClass, "for a starred Green's H-class",
-[IsHStarClass], cl-> DStarClass(ParentAttr(cl), Representative(cl)));
+[IsHStarClass], cl -> DStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(JStarClass, "for a starred Green's H-class",
-[IsHStarClass], cl-> JStarClass(ParentAttr(cl), Representative(cl)));
+[IsHStarClass], cl -> JStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(DStarClass, "for a starred Green's R-class",
-[IsRStarClass], cl-> DStarClass(ParentAttr(cl), Representative(cl)));
+[IsRStarClass], cl -> DStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(JStarClass, "for a starred Green's R-class",
-[IsRStarClass], cl-> JStarClass(ParentAttr(cl), Representative(cl)));
+[IsRStarClass], cl -> JStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(DStarClass, "for a starred Green's L-class",
-[IsLStarClass], cl-> DStarClass(ParentAttr(cl), Representative(cl)));
+[IsLStarClass], cl -> DStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(JStarClass, "for a starred Green's L-class",
-[IsLStarClass], cl-> JStarClass(ParentAttr(cl), Representative(cl)));
+[IsLStarClass], cl -> JStarClass(ParentAttr(cl), Representative(cl)));
InstallOtherMethod(JStarClass, "for a starred Green's D-class",
-[IsDStarClass], cl-> JStarClass(ParentAttr(cl), Representative(cl)));
+[IsDStarClass], cl -> JStarClass(ParentAttr(cl), Representative(cl)));
-
-#################
-#################
-##
-#F LStarPartitionByMT()
-#M RStarClasses()
-#M LStarClasses()
-#M JStarClasses()
-#M DStarClasses()
-#M HStarClasses()
-##
-## find all the classes of a particular type
+# Find all the classes of a particular type
InstallGlobalFunction(LStarPartitionByMT, function(table)
+ local mt, # mutable copy of multiplication table
+ n, # size of
+ profiles, # types of rows, sorted
+ partition, # partition of indices into R*-classes
+ k, # element counter
+ pro, # profile of current elements row
+ pos; # position of current profile
+
+ # get a mutable copy of the multiplication table
+ mt := MutableCopyMat(table);
+ n := Length(mt);
+ mt{[1 .. n]}[n + 1] := [1 .. n];
+
+ # initialise sorted list of profiles ...
+ profiles := EmptyPlist(n);
+ # ... and partition into R*-classes
+ partition := EmptyPlist(n);
+
+ # loop over elements
+ for k in [1 .. n] do
+ pro := KernelOfTransformation(TransformationNC(mt[k]));
+ pos := PositionSorted(profiles, pro);
+
+ # either ...
+ if IsBound(profiles[pos]) and profiles[pos] = pro then
+ # add element to part with same profile ...
+ Add(partition[pos], k);
+ else
+ # ... or start a new part
+ Add(profiles, pro, pos);
+ Add(partition, [k], pos);
+ fi;
+ od;
- local mt, # mutable copy of multiplication table
- n, # size of
- profiles, # types of rows, sorted
- partition, # partition of indices into R*-classes
- k, # element counter
- pro, # profile of current elements row
- pos; # position of current profile
-
- ### main function body ###
-
- # get a mutable copy of the multiplication table
- mt := MutableCopyMat(table);
- n := Length(mt);
- mt{[1..n]}[n+1] := [1..n];
-
- # initialise sorted list of profiles ...
- profiles := EmptyPlist(n);
- # ... and partition into R*-classes
- partition := EmptyPlist(n);
-
- # loop over elements
- for k in [1..n] do
-
- # get profile of element
- #pro := FLAT_KERNEL_TRANS(TransformationNC(mt[k]));
- pro := KernelOfTransformation(TransformationNC(mt[k]));
- pos := PositionSorted(profiles, pro);
-
- # either ...
- if IsBound(profiles[pos]) and profiles[pos] = pro then
- # add element to part with same profile ...
- Add(partition[pos], k);
- else
- # ... or start a new part
- Add(profiles, pro, pos);
- Add(partition, [k], pos);
- fi;
- od;
-
- return AsSSortedList(partition);
+ return AsSSortedList(partition);
end);
InstallMethod(RStarClasses, "for a small semigroup", [IsSmallSemigroup],
function(semi)
+ local partition, # partition of indices into R*-classes
+ classes, # the actual classes
+ part, # one part of , used as loop counter
+ rc; # one R*-class
- local partition, # partition of indices into R*-classes
- classes, # the actual classes
- part, # one part of , used as loop counter
- rc; # one R*-class
+ # get partition of index set into R*-classes ...
+ partition := LStarPartitionByMT(TransposedMat(MultiplicationTable(semi)));
- ### main function body ###
+ # ... and use as internal representation
+ SetInternalRepStarRelation(RStarRelation(semi), partition);
- # get partition of index set into R*-classes ...
- partition := LStarPartitionByMT(TransposedMat(MultiplicationTable(semi)));
+ # create acutal classes as GAP objects
+ classes := EmptyPlist(Length(partition));
- # ... and use as internal representation
- SetInternalRepStarRelation(RStarRelation(semi), partition);
-
- # create acutal classes as GAP objects
- classes := EmptyPlist(Length(partition));
-
- for part in partition do
- rc := RStarClass(semi, Elements(semi)[part[1]]);
- Add(classes, rc);
- SetAsSSortedList(rc, Elements(semi){part});
- SetSize(rc, Size(part));
- SetCanonicalStarClass(rc, rc);
- od;
+ for part in partition do
+ rc := RStarClass(semi, Elements(semi)[part[1]]);
+ Add(classes, rc);
+ SetAsSSortedList(rc, Elements(semi){part});
+ SetSize(rc, Size(part));
+ SetCanonicalStarClass(rc, rc);
+ od;
- return classes;
+ return classes;
end);
InstallOtherMethod(RStarClasses, "for a starred Green's D-class",
-[IsDStarClass], cl-> Filtered(RStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
+[IsDStarClass], cl -> Filtered(RStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
InstallOtherMethod(RStarClasses, "for a starred Green's J-class",
-[IsJStarClass], cl-> Filtered(RStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
+[IsJStarClass], cl -> Filtered(RStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
InstallMethod(LStarClasses, "for a small semigroup", [IsSmallSemigroup],
function(semi)
+ local partition, # partition of indices into L*-classes
+ classes, # the actual classes
+ part, # one part of , used as loop counter
+ lc; # one L*-class
- local partition, # partition of indices into L*-classes
- classes, # the actual classes
- part, # one part of , used as loop counter
- lc; # one L*-class
-
- ### main function body ###
-
- # get partition of index set into L*-classes ...
- partition := LStarPartitionByMT(MultiplicationTable(semi));
+ # get partition of index set into L*-classes ...
+ partition := LStarPartitionByMT(MultiplicationTable(semi));
- # ... and use as internal representation
- SetInternalRepStarRelation(LStarRelation(semi), partition);
+ # ... and use as internal representation
+ SetInternalRepStarRelation(LStarRelation(semi), partition);
- # create acutal classes as GAP objects
- classes := EmptyPlist(Length(partition));
+ # create acutal classes as GAP objects
+ classes := EmptyPlist(Length(partition));
- for part in partition do
- lc := LStarClass(semi, Elements(semi)[part[1]]);
- Add(classes, lc);
- SetAsSSortedList(lc, Elements(semi){part});
- SetSize(lc, Size(part));
- SetCanonicalStarClass(lc, lc);
- od;
+ for part in partition do
+ lc := LStarClass(semi, Elements(semi)[part[1]]);
+ Add(classes, lc);
+ SetAsSSortedList(lc, Elements(semi){part});
+ SetSize(lc, Size(part));
+ SetCanonicalStarClass(lc, lc);
+ od;
- return classes;
+ return classes;
end);
InstallOtherMethod(LStarClasses, "for a starred Green's D-class",
-[IsDStarClass], cl-> Filtered(LStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
+[IsDStarClass], cl -> Filtered(LStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
InstallOtherMethod(LStarClasses, "for a starred Green's J-class",
-[IsJStarClass], cl-> Filtered(LStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
+[IsJStarClass], cl -> Filtered(LStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
# AD would it be better to use JoinEquivalenceRelations?
InstallMethod(DStarClasses, "for a small semigroup", [IsSmallSemigroup],
function(semi)
-
- local mt, # multiplication table of
- left, # partition of indices into L*-classes
- right, # partition of indices into R*-classes
- lookupL, # positions of parts in containing each index
- lookupR, # positions of parts in containing each index
- partition, # partition of indices into D*-classes
- classes, # the actual classes
- part, # one part of , used as loop counter
- dc; # one D*-class
-
- ### function body ###
-
- mt := MultiplicationTable(semi);
-
- # get R- and L-classes
- LStarClasses(semi); RStarClasses(semi);
- left := InternalRepStarRelation(LStarRelation(semi));
- right := InternalRepStarRelation(RStarRelation(semi));
-
- lookupL := List([1..Size(mt)], k-> First(left, part-> k in part));
- lookupR := List([1..Size(mt)], k-> First(right, part-> k in part));
-
- while left <> right do
- left := Unique(List(right, part-> Union(List(part, k-> lookupL[k]))));
- right := Unique(List(left, part-> Union(List(part, k-> lookupR[k]))));
- od;
-
- partition := AsSSortedList(left);
-
- SetInternalRepStarRelation(DStarRelation(semi), partition);
- classes := EmptyPlist(Length(partition));
-
- for part in partition do
- dc := DStarClass(semi, Elements(semi)[part[1]]);
- Add(classes, dc);
- SetAsSSortedList(dc, Elements(semi){part});
- SetSize(dc, Size(part));
- SetCanonicalStarClass(dc, dc);
- od;
-
- return classes;
+ local mt, # multiplication table of
+ left, # partition of indices into L*-classes
+ right, # partition of indices into R*-classes
+ lookupL, # positions of parts in containing each index
+ lookupR, # positions of parts in containing each index
+ partition, # partition of indices into D*-classes
+ classes, # the actual classes
+ part, # one part of , used as loop counter
+ dc; # one D*-class
+
+ mt := MultiplicationTable(semi);
+
+ # get R- and L-classes
+ LStarClasses(semi);
+ RStarClasses(semi);
+ left := InternalRepStarRelation(LStarRelation(semi));
+ right := InternalRepStarRelation(RStarRelation(semi));
+
+ lookupL := List([1 .. Size(mt)], k -> First(left, part -> k in part));
+ lookupR := List([1 .. Size(mt)], k -> First(right, part -> k in part));
+
+ while left <> right do
+ left := Unique(List(right, part -> Union(List(part, k -> lookupL[k]))));
+ right := Unique(List(left, part -> Union(List(part, k -> lookupR[k]))));
+ od;
+
+ partition := AsSSortedList(left);
+
+ SetInternalRepStarRelation(DStarRelation(semi), partition);
+ classes := EmptyPlist(Length(partition));
+
+ for part in partition do
+ dc := DStarClass(semi, Elements(semi)[part[1]]);
+ Add(classes, dc);
+ SetAsSSortedList(dc, Elements(semi){part});
+ SetSize(dc, Size(part));
+ SetCanonicalStarClass(dc, dc);
+ od;
+
+ return classes;
end);
InstallOtherMethod(DStarClasses, "for a starred Green's J-class",
-[IsJStarClass], cl-> Filtered(DStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
-
+[IsJStarClass], cl -> Filtered(DStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
InstallMethod(JStarClasses, "for a small semigroup", [IsSmallSemigroup],
function(semi)
-
- local mt, # multiplication table of
- n, # size of
- jideals, # Green's J*-ideals of all elements
- partition, # partition of indices into J*-classes
- set, # a subset of the indices to become a part of
- classes, # list of Green's J*-classes
- rest, # elements not yet sorted into classes
- elm, # element for building next class
- part, # one part of , used as loop counter
- jc, # one J*-class
-
- # AD the ideal computation should certainly become a user function
- # AD but the necessary technicalities are unclear to me
- greensStarJIdealRaw;
-
- greensStarJIdealRaw := function(elm)
- local ideal, # the Green's J*-ideal of
- new, # auxiliary variable for construction of
- trans, # transposed of
- dclasses, # internal representation of Green's D*-classes
-
- greensJIdeal;
-
- ### local local function body ###
-
- # AD this step is rather inefficient in its current form
- # AD it should be replaced by a call to an attribute
- greensJIdeal := function(a)
- local regid; # the regular Greens J-ideal of
-
- # {a} \cup aS
- regid := Union(mt[a], [a]);
-
- # {a} \cup aS \cup aS \cup SaS
- regid := Union(regid, Union(List(regid, x-> trans[x])));
-
- return regid;
- end;
-
- ### local function body ###
-
- trans := TransposedMat(mt);
- DStarClasses(semi);
- dclasses := InternalRepStarRelation(DStarRelation(semi));
- dclasses := List([1..n], x-> First(dclasses, cl-> x in cl));
-
- new := [elm];
-
- repeat
- ideal := new;
- new := Union(List(ideal, greensJIdeal));
- new := Union(List(new, x-> dclasses[x]));
- until new = ideal;
-
- return ideal;
+ local mt, # multiplication table of
+ n, # size of
+ jideals, # Green's J*-ideals of all elements
+ partition, # partition of indices into J*-classes
+ set, # a subset of the indices to become a part of
+ classes, # list of Green's J*-classes
+ rest, # elements not yet sorted into classes
+ elm, # element for building next class
+ part, # one part of , used as loop counter
+ jc, # one J*-class
+ # AD the ideal computation should certainly become a user function
+ # AD but the necessary technicalities are unclear to me
+ greensStarJIdealRaw;
+
+ greensStarJIdealRaw := function(elm)
+ local ideal, # the Green's J*-ideal of
+ new, # auxiliary variable for construction of
+ trans, # transposed of
+ dclasses, # internal representation of Green's D*-classes
+ greensJIdeal;
+
+ # AD this step is rather inefficient in its current form
+ # AD it should be replaced by a call to an attribute
+ greensJIdeal := function(a)
+ local regid; # the regular Greens J-ideal of
+ # {a} \cup aS
+ regid := Union(mt[a], [a]);
+ # {a} \cup aS \cup aS \cup SaS
+ return Union(regid, Union(List(regid, x -> trans[x])));
end;
- ### main function body ###
+ trans := TransposedMat(mt);
+ DStarClasses(semi);
+ dclasses := InternalRepStarRelation(DStarRelation(semi));
+ dclasses := List([1 .. n], x -> First(dclasses, cl -> x in cl));
- mt := MultiplicationTable(semi);
- n := Size(mt);
+ new := [elm];
- jideals := List([1..n], i-> greensStarJIdealRaw(i));
+ repeat
+ ideal := new;
+ new := Union(List(ideal, greensJIdeal));
+ new := Union(List(new, x -> dclasses[x]));
+ until new = ideal;
- partition := EmptyPlist(Size(mt));
- rest := [1..Size(mt)];
+ return ideal;
+ end;
- repeat
+ mt := MultiplicationTable(semi);
+ n := Size(mt);
- # take one of the remaining elements
- elm := rest[1];
+ jideals := List([1 .. n], greensStarJIdealRaw);
- # get remaining elements in the same class
- set := Filtered(Intersection(rest, jideals[elm]),
- x-> elm in jideals[x]);
+ partition := EmptyPlist(Size(mt));
+ rest := [1 .. Size(mt)];
- # remember new class and prepare for next step
- Add(partition, set);
- rest := Difference(rest, set);
+ repeat
+ # take one of the remaining elements
+ elm := rest[1];
- until IsEmpty(rest);
+ # get remaining elements in the same class
+ set := Filtered(Intersection(rest, jideals[elm]),
+ x -> elm in jideals[x]);
- MakeImmutable(partition);
- SetIsSSortedList(partition, true);
+ # remember new class and prepare for next step
+ Add(partition, set);
+ rest := Difference(rest, set);
+ until IsEmpty(rest);
- SetInternalRepStarRelation(JStarRelation(semi), partition);
- classes := EmptyPlist(Length(partition));
+ MakeImmutable(partition);
+ SetIsSSortedList(partition, true);
- for part in partition do
- jc := JStarClass(semi, Elements(semi)[part[1]]);
- Add(classes, jc);
- SetAsSSortedList(jc, Elements(semi){part});
- SetSize(jc, Size(part));
- SetCanonicalStarClass(jc, jc);
- od;
+ SetInternalRepStarRelation(JStarRelation(semi), partition);
+ classes := EmptyPlist(Length(partition));
- SetJStarClasses(semi, classes);
+ for part in partition do
+ jc := JStarClass(semi, Elements(semi)[part[1]]);
+ Add(classes, jc);
+ SetAsSSortedList(jc, Elements(semi){part});
+ SetSize(jc, Size(part));
+ SetCanonicalStarClass(jc, jc);
+ od;
- return classes;
+ SetJStarClasses(semi, classes);
+
+ return classes;
end);
InstallMethod(HStarClasses, "for a small semigroup", [IsSmallSemigroup],
function(semi)
-
- local mt, # multiplication table of
- left, # partition of indices into L*-classes
- lpart, # one part of
- right, # partition of indices into R*-classes
- rpart, # one part of
- partition, # partition of indices into D*-classes
- classes, # the actual classes
- part, # one part of , used as loop counter
- hc; # one H*-class
-
- ### function body ###
-
- mt := MultiplicationTable(semi);
+ local left, # partition of indices into L*-classes
+ lpart, # one part of
+ right, # partition of indices into R*-classes
+ rpart, # one part of
+ partition, # partition of indices into D*-classes
+ classes, # the actual classes
+ part, # one part of , used as loop counter
+ hc; # one H*-class
# get R- and L-classes
- LStarClasses(semi); RStarClasses(semi);
+ LStarClasses(semi);
+ RStarClasses(semi);
+
left := InternalRepStarRelation(LStarRelation(semi));
right := InternalRepStarRelation(RStarRelation(semi));
- partition := [ ];
+ partition := [];
for lpart in left do
- for rpart in right do
- Add(partition, Intersection(lpart, rpart));
- od;
+ for rpart in right do
+ Add(partition, Intersection(lpart, rpart));
+ od;
od;
- partition := AsSSortedList(Filtered(partition, p-> not IsEmpty(p)));
+ partition := AsSSortedList(Filtered(partition, p -> not IsEmpty(p)));
- SetInternalRepStarRelation(HStarRelation(semi), partition);
+ SetInternalRepStarRelation(HStarRelation(semi), partition);
classes := EmptyPlist(Length(partition));
for part in partition do
- hc := HStarClass(semi, Elements(semi)[part[1]]);
- Add(classes, hc);
- SetAsSSortedList(hc, Elements(semi){part});
- SetSize(hc, Size(part));
- SetCanonicalStarClass(hc, hc);
+ hc := HStarClass(semi, Elements(semi)[part[1]]);
+ Add(classes, hc);
+ SetAsSSortedList(hc, Elements(semi){part});
+ SetSize(hc, Size(part));
+ SetCanonicalStarClass(hc, hc);
od;
-
SetHStarClasses(semi, classes);
-
return classes;
end);
-InstallOtherMethod(HStarClasses, "for a starred Green's class",
-[IsStarClass], cl-> Filtered(HStarClasses(ParentAttr(cl)),
- c-> Representative(c) in cl));
-
-
+InstallOtherMethod(HStarClasses, "for a starred Green's class",
+[IsStarClass], cl -> Filtered(HStarClasses(ParentAttr(cl)),
+ c -> Representative(c) in cl));
diff --git a/gap/properties.gd b/gap/properties.gd
index 5abfe9c..9b4b735 100644
--- a/gap/properties.gd
+++ b/gap/properties.gd
@@ -1,546 +1,495 @@
#############################################################################
##
-#W properties.gd Smallsemi - a GAP library of semigroups
-#Y Copyright (C) 2008-2014 Andreas Distler & James D. Mitchell
+## properties.gd Smallsemi - a GAP library of semigroups
+## Copyright (C) 2008-2024 Andreas Distler & James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
-###########################################################################
-##
-## <#GAPDoc Label="Annihilators">
-##
-##
-##
-## returns the set of annihilators of sgrp if sgrp contains a
-## zero element and fail otherwise.
-##
-## An element x in a semigroup with zero z is an
-## annihilator if xy=yx=z for every element y in the
-## semigroup.
-## s := SmallSemigroup(5,6);
-##
-## gap> Annihilators(s);
-## [ s1, s2 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-DeclareAttribute( "Annihilators", IsSemigroup );
-
-###########################################################################
-##
-## <#GAPDoc Label="DiagonalOfMultiplicationTable">
-##
-##
-##
-## returns the diagonal of the multiplication table of the semigroup
-## sgrp.
-## s:=SmallSemigroup(8,10101);;
-## gap> DiagonalOfMultiplicationTable(s);
-## [ 1, 1, 1, 1, 1, 1, 1, 1 ]
-## gap> s:=SmallSemigroup(7,10101);;
-## gap> DiagonalOfMultiplicationTable(s);
-## [ 1, 1, 1, 1, 1, 1, 1 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
+# <#GAPDoc Label="Annihilators">
+#
+#
+#
+# returns the set of annihilators of sgrp if sgrp contains a
+# zero element and fail otherwise.
+#
+# An element x in a semigroup with zero z is an
+# annihilator if xy=yx=z for every element y in the
+# semigroup.
+# s := SmallSemigroup(5,6);
+#
+# gap> Annihilators(s);
+# [ s1, s2 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
+DeclareAttribute("Annihilators", IsSemigroup);
+
+# <#GAPDoc Label="DiagonalOfMultiplicationTable">
+#
+#
+#
+# returns the diagonal of the multiplication table of the semigroup
+# sgrp.
+# s:=SmallSemigroup(8,10101);;
+# gap> DiagonalOfMultiplicationTable(s);
+# [ 1, 1, 1, 1, 1, 1, 1, 1 ]
+# gap> s:=SmallSemigroup(7,10101);;
+# gap> DiagonalOfMultiplicationTable(s);
+# [ 1, 1, 1, 1, 1, 1, 1 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
DeclareAttribute("DiagonalOfMultiplicationTable", IsSemigroup);
-###########################################################################
-##
-## <#GAPDoc Label="DisplaySmallSemigroup">
-##
-##
-##
-## displays all the information about the small semigroup sgrp that
-## is stored in the library and its Green's classes and
-## idempotents.
-## s:=SmallSemigroup(6, 3838);;
-## gap> DisplaySmallSemigroup(s);
-## IsBand: false
-## IsBrandtSemigroup: false
-## IsCommutative: false
-## IsCompletelyRegularSemigroup: false
-## IsFullTransformationSemigroupCopy: false
-## IsGroupAsSemigroup: false
-## IsIdempotentGenerated: false
-## IsInverseSemigroup: false
-## IsMonogenicSemigroup: false
-## IsMonoidAsSemigroup: false
-## IsMultSemigroupOfNearRing: false
-## IsOrthodoxSemigroup: false
-## IsRectangularBand: false
-## IsRegularSemigroup: false
-## IsSelfDualSemigroup: false
-## IsSemigroupWithClosedIdempotents: true
-## IsSimpleSemigroup: false
-## IsSingularSemigroupCopy: false
-## IsZeroSemigroup: false
-## IsZeroSimpleSemigroup: false
-## MinimalGeneratingSet: [ s3, s4, s5, s6 ]
-## Idempotents: [ s1, s5, s6 ]
-## GreensRClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ]
-## GreensLClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ]
-## GreensHClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ]
-## GreensDClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-##
-
+# <#GAPDoc Label="DisplaySmallSemigroup">
+#
+#
+#
+# displays all the information about the small semigroup sgrp that
+# is stored in the library and its Green's classes and
+# idempotents.
+# s:=SmallSemigroup(6, 3838);;
+# gap> DisplaySmallSemigroup(s);
+# IsBand: false
+# IsBrandtSemigroup: false
+# IsCommutative: false
+# IsCompletelyRegularSemigroup: false
+# IsFullTransformationSemigroupCopy: false
+# IsGroupAsSemigroup: false
+# IsIdempotentGenerated: false
+# IsInverseSemigroup: false
+# IsMonogenicSemigroup: false
+# IsMonoidAsSemigroup: false
+# IsMultSemigroupOfNearRing: false
+# IsOrthodoxSemigroup: false
+# IsRectangularBand: false
+# IsRegularSemigroup: false
+# IsSelfDualSemigroup: false
+# IsSemigroupWithClosedIdempotents: true
+# IsSimpleSemigroup: false
+# IsSingularSemigroupCopy: false
+# IsZeroSemigroup: false
+# IsZeroSimpleSemigroup: false
+# MinimalGeneratingSet: [ s3, s4, s5, s6 ]
+# Idempotents: [ s1, s5, s6 ]
+# GreensRClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ]
+# GreensLClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ]
+# GreensHClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ]
+# GreensDClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ]
+# ]]>
+#
+#
+# <#/GAPDoc>
DeclareGlobalFunction("DisplaySmallSemigroup");
-#############################################################################
-##
-## <#GAPDoc Label="IndexPeriod">
-##
-##
-##
-## returns the minimum numbers m, r such that x^{m+r}=x^m; known
-## as the index and period of the small semigroup element x.
-## s:=SmallSemigroup(5,116);
-##
-## gap> x:=Elements(s)[3];
-## s3
-## gap> IndexPeriod(x);
-## [ 2, 1 ]
-## gap> x^3=x^2;
-## true
-## gap> x^2=x^1;
-## false
-## gap> x^3=x^1;
-## false
-## ]]>
-##
-##
-## <#/GAPDoc>
-##
-##
-
+# <#GAPDoc Label="IndexPeriod">
+#
+#
+#
+# returns the minimum numbers m, r such that x^{m+r}=x^m; known
+# as the index and period of the small semigroup element x.
+# s:=SmallSemigroup(5,116);
+#
+# gap> x:=Elements(s)[3];
+# s3
+# gap> IndexPeriod(x);
+# [ 2, 1 ]
+# gap> x^3=x^2;
+# true
+# gap> x^2=x^1;
+# false
+# gap> x^3=x^1;
+# false
+# ]]>
+#
+#
+# <#/GAPDoc>
+#
DeclareAttribute("IndexPeriod", IsSmallSemigroupElt);
-###########################################################################
-##
-## <#GAPDoc Label="IsBand">
-##
-##
-##
-## returns true if the small semigroup sgrp is a band and
-## false otherwise.
-##
-## A semigroup sgrp is a band if every element is an
-## idempotent, that is, x^2=x for all x in sgrp.
-## s:=SmallSemigroup(5,519);;
-## gap> IsBand(s);
-## false
-## gap> s:=OneSmallSemigroup(5, IsBand, true);
-##
-## gap> IsBand(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 5, 1010 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-
-###########################################################################
-##
-## <#GAPDoc Label="IsBrandtSemigroup">
-##
-##
-##
-## returns true if the small semigroup sgrp is a Brandt semigroup
-## and false otherwise.
-##
-## A finite semigroup sgrp is a Brandt semigroup if it is inverse
-## and zero simple.
-## s:=SmallSemigroup(5,519);;
-## gap> IsBrandtSemigroup(s);
-## false
-## gap> s:=OneSmallSemigroup(5, IsBrandtSemigroup, true);
-##
-## gap> IsBrandtSemigroup(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 5, 149 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-
-
-###########################################################################
-##
-## <#GAPDoc Label="IsCliffordSemigroup">
-##
-##
-##
-## returns true if the small semigroup sgrp is a Clifford
-## semigroup and false otherwise.
-##
-## A semigroup sgrp is a Clifford semigroup if it is a
-## regular semigroup whose idempotents are central, that is, for all
-## e,x in sgrp where e^2=e
-## we have that ex=xe.
-## s:=SmallSemigroup(5,519);;
-## gap> IsBand(s);
-## false
-## gap> s:=OneSmallSemigroup(5, IsBand, true);
-##
-## gap> IsBand(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 5, 1010 ]
-## gap> s:=SmallSemigroup(5,519);;
-## gap> IsCliffordSemigroup(s);
-## false
-## gap> s:=OneSmallSemigroup(5, IsCliffordSemigroup, true);
-##
-## gap> IsCliffordSemigroup(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 5, 148 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-
-###########################################################################
-##
-## <#GAPDoc Label="IsCommutativeSemigroup">
-##
-##
-##
-##
-## return true if the small semigroup sgrp is commutative
-## and false otherwise.
-##
-## A semigroup sgrp is commutative if
-## xy=yx for all x,y in sgrp.
-## s:=SmallSemigroup(6,871);;
-## gap> IsCommutativeSemigroup(s);
-## false
-## gap> s:=OneSmallSemigroup(5, IsCommutative, true);
-##
-## gap> IsCommutativeSemigroup(s);
-## true
-## gap> IsCommutative(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 5, 1 ]
-## gap> s:=OneSmallSemigroup(5, IsCommutativeSemigroup, true);
-##
-## gap> IsCommutativeSemigroup(s);
-## true
-## gap> IsCommutative(s);
-## true
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-
-###########################################################################
-##
-## <#GAPDoc Label="IsCompletelyRegularSemigroup">
-##
-##
-##
-## returns true if the semigroup sgrp is completely
-## regular and false otherwise.
-## A semigroup is completely regular
-## if every element is contained in a subgroup.
-## s:=SmallSemigroup(6,13131);
-##
-## gap> IsCompletelyRegularSemigroup(s);
-## false
-## gap> s:=OneSmallSemigroup(6, IsCompletelyRegularSemigroup, true);
-##
-## gap> IsCompletelyRegularSemigroup(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 6, 3164 ]
-## ]]>
-##
-##
-## <#/GAPDoc>
-
-
-###########################################################################
-##
-## <#GAPDoc Label="IsFullTransformationSemigroupCopy">
-##
-##
-##
-## returns true if the semigroup sgrp is isomorphic to a full
-## transformation
-## semigroup and false otherwise.
-##
-## The size of the full transformation semigroup on an n element set is
-## n^n and so there are only two semigroup in the library that have this property.
-## s:=SmallSemigroup(1,1);
-##
-## gap> IsFullTransformationSemigroupCopy(s);
-## true
-## gap> s:=OneSmallSemigroup(4, IsFullTransformationSemigroupCopy, true);
-##
-## gap> IsFullTransformationSemigroupCopy(s);
-## true
-## gap> IdSmallSemigroup(s);
-## [ 4, 96 ]
-## gap> s:=OneSmallSemigroup(6, IsFullTransformationSemigroupCopy, true);
-## fail
-## ]]>
-##
-##
-## <#/GAPDoc>
-
+# <#GAPDoc Label="IsBand">
+#
+#
+#
+# returns true if the small semigroup sgrp is a band and
+# false otherwise.
+#
+# A semigroup sgrp is a band if every element is an idempotent,
+# that is, x^2=x for all x in sgrp.
+# s:=SmallSemigroup(5,519);;
+# gap> IsBand(s);
+# false
+# gap> s:=OneSmallSemigroup(5, IsBand, true);
+#
+# gap> IsBand(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 5, 1010 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
+
+# <#GAPDoc Label="IsBrandtSemigroup">
+#
+#
+#
+# returns true if the small semigroup sgrp is a Brandt semigroup
+# and false otherwise.
+#
+# A finite semigroup sgrp is a Brandt semigroup if it is inverse
+# and zero simple.
+# s:=SmallSemigroup(5,519);;
+# gap> IsBrandtSemigroup(s);
+# false
+# gap> s:=OneSmallSemigroup(5, IsBrandtSemigroup, true);
+#
+# gap> IsBrandtSemigroup(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 5, 149 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
+
+# <#GAPDoc Label="IsCliffordSemigroup">
+#
+#
+#
+# returns true if the small semigroup sgrp is a Clifford
+# semigroup and false otherwise.
+#
+# A semigroup sgrp is a Clifford semigroup if it is a
+# regular semigroup whose idempotents are central, that is, for all
+# e,x in sgrp where e^2=e
+# we have that ex=xe.
+# s:=SmallSemigroup(5,519);;
+# gap> IsBand(s);
+# false
+# gap> s:=OneSmallSemigroup(5, IsBand, true);
+#
+# gap> IsBand(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 5, 1010 ]
+# gap> s:=SmallSemigroup(5,519);;
+# gap> IsCliffordSemigroup(s);
+# false
+# gap> s:=OneSmallSemigroup(5, IsCliffordSemigroup, true);
+#
+# gap> IsCliffordSemigroup(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 5, 148 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
+
+# <#GAPDoc Label="IsCommutativeSemigroup">
+#
+#
+#
+#
+# return true if the small semigroup sgrp is commutative and
+# false otherwise.
+#
+# A semigroup sgrp is commutative if xy=yx for all
+# x,y in sgrp.
+# s:=SmallSemigroup(6,871);;
+# gap> IsCommutativeSemigroup(s);
+# false
+# gap> s:=OneSmallSemigroup(5, IsCommutative, true);
+#
+# gap> IsCommutativeSemigroup(s);
+# true
+# gap> IsCommutative(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 5, 1 ]
+# gap> s:=OneSmallSemigroup(5, IsCommutativeSemigroup, true);
+#
+# gap> IsCommutativeSemigroup(s);
+# true
+# gap> IsCommutative(s);
+# true
+# ]]>
+#
+#
+# <#/GAPDoc>
+
+# <#GAPDoc Label="IsCompletelyRegularSemigroup">
+#
+#
+#
+# returns true if the semigroup sgrp is completely
+# regular and false otherwise.
+# A semigroup is completely regular
+# if every element is contained in a subgroup.
+# s:=SmallSemigroup(6,13131);
+#
+# gap> IsCompletelyRegularSemigroup(s);
+# false
+# gap> s:=OneSmallSemigroup(6, IsCompletelyRegularSemigroup, true);
+#
+# gap> IsCompletelyRegularSemigroup(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 6, 3164 ]
+# ]]>
+#
+#
+# <#/GAPDoc>
+
+# <#GAPDoc Label="IsFullTransformationSemigroupCopy">
+#
+#
+#
+# returns true if the semigroup sgrp is isomorphic to a full
+# transformation
+# semigroup and false otherwise.
+#
+# The size of the full transformation semigroup on an n element set is
+# n^n and so there are only two semigroup in the library that have this
+# property.
+# s:=SmallSemigroup(1,1);
+#
+# gap> IsFullTransformationSemigroupCopy(s);
+# true
+# gap> s:=OneSmallSemigroup(4, IsFullTransformationSemigroupCopy, true);
+#
+# gap> IsFullTransformationSemigroupCopy(s);
+# true
+# gap> IdSmallSemigroup(s);
+# [ 4, 96 ]
+# gap> s:=OneSmallSemigroup(6, IsFullTransformationSemigroupCopy, true);
+# fail
+# ]]>
+#
+#
+# <#/GAPDoc>
DeclareProperty("IsFullTransformationSemigroupCopy", IsSmallSemigroup);
-
-###########################################################################
-##
-## <#GAPDoc Label="IsGroupAsSemigroup">
-##
-##
-##
-## returns true if the small semigroup sgrp
-## is mathematically a group, and returns false otherwise.
-## Note that no small semigroup can lie in the category
-##