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# This function checks the compatibility condition
# in a skew brace:
# a(b+c)=ab-a+ac for all a,b,c
# It works for braces and skew braces!
InstallGlobalFunction(IS_BRACE, function(obj)
local a,b,c;
for a in obj do
for b in obj do
for c in obj do
if a*(b+c) <> a*b-a+a*c then
return false;
fi;
od;
od;
od;
return true;
end);
InstallGlobalFunction(IS_CYCLESET, function(m)
local x, y, z, xy, xz, yx, yz, size;
size := Size(m);
### Check bijectivity
if fail in List([1..size], i->PermList(m[i])) then
return false;
fi;
### Check (xy)(xz)=(yx)(yz)
for x in [1..size] do
for y in [1..size] do
xy := m[x][y];
yx := m[y][x];
for z in [1..size] do
xz := m[x][z];
yz := m[y][z];
if m[xy][xz] <> m[yx][yz] then
return false;
fi;
od;
od;
od;
return true;
end);
InstallGlobalFunction(IS_RACK,
function( m )
local i, j, k;
for i in [1..Size(m)] do
for j in [1..Size(m)] do
for k in [1..Size(m)] do
if m[i][m[j][k]] <> m[m[i][j]][m[i][k]] then
return false;
fi;
od;
od;
od;
for i in [1..Size(m)] do
if PermList(m[i]) = fail then
return false;
fi;
od;
return true;
end);
InstallGlobalFunction(IS_YB, function(l, r)
local x, y, z, v;
if Size(r) <> Size(l) then
return false;
fi;
for x in [1..Size(l)] do
for y in [1..Size(l)] do
for z in [1..Size(l)] do
v := [x,y,z];
if YB_ij(l, r, YB_ij(l, r, YB_ij(l, r, v, 2, 3), 1, 2), 2, 3) \
<> YB_ij(l, r, YB_ij(l, r, YB_ij(l, r, v, 1, 2), 2, 3), 1, 2) then
return false;
fi;
od;
od;
od;
return true;
end);
### This function returns the value of <obj> acting on the coordinates (<i>,<j>) of <v>
InstallGlobalFunction(YB_ij,
function(l, r, v, i, j)
local w;
w := ShallowCopy(v);
w[i] := l[v[i]][v[j]];
w[j] := r[v[j]][v[i]];
return w;
end);
[ Dauer der Verarbeitung: 0.28 Sekunden
(vorverarbeitet)
]
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