
176 Algebraic Operads: An Algorithmic Companion
1 4
3
2
+
3 4
1
2
+
1 3 2 4
+
1 4 2 3
+
2 4
3
1
+
3 4
2
1
.
For the rewriting using the other divisor T
2
, we obtain:
1 2
3
4
7−→
1 3
2
4
+
2 3
1
4
7−→
1 3
4
2
+
1 3 2 4
+
1 4 2 3
+
2 3
4
1
7−→
1 4
3
2
+
3 4
1
2
+
1 3 2 4
+
1 4 2 3
+
2 4
3
1
+
3 4
2
1
.
(In each of these cases, each arrow represents several rewritings of all non-
reduced monomials in one go.) We see that the results are the same, so the
corresponding S-polynomial can be reduced to zero, and the defining relation
of the operad Lie
f
is the reduced Gröbner basis. A similar computation shows
that in this case for each of the monomials in the defining relation, and any