
Symmetric Operads and Shuffle Operads 181
1 2
u
3
u
−
2 3
1
u
u
−
2 3
1
v
u
−
1 2
v
3
u
−
2
1 3
v
2
u
+
2 3
1
u
v
+
1 2
u
3
v
+
1 3
v
2
v
,
and
1 3
u
2
u
−
2 3
1
u
u
+
2 3
1
v
u
−
1 3
v
2
u
−
2
1 2
v
3
u
+
2 3
1
u
v
+
1 3
u
2
v
+
1 2
v
3
v
.
Let us consider the following monomial order. We set
1 2
v
≺
1 2
u
,
and modify the gpathpermlex order in a way that we first compare the per-
mutations of leaves using the lexicographic order, and then compare the path
sequences. The set of relations is not linearly self-reduced; after making it lin-
early self-reduced, the leading monomials of relations are the tree monomials
1 3
v
2
v
,
1 3
v
2
u
, and
1 3
u
2
u
.
It is easy to see that both S-polynomials corresponding to common multiples of
these tree monomials ...