
Symmetric Operads and Shuffle Operads 171
an S-polynomial of g
1
and g
2
; the common term cancels, since both g
1
and g
2
are monic.
Example 5.5.1.2. Let us consider the shuffle associative operad Ass
f
dis-
cussed in Example 5.3.4.3, and the following two of its six defining relations:
g
1
=
1 3
a
2
a
−
2 3
1
b
a
and g
2
=
1 3
b
2
a
−
1 2
a
3
b
.
Let us consider, for the purpose of this example, the following ordering: to
compare two shuffle tree monomials T and T
0
of the same arity, we compare
the number of internal vertices labelled by b, then if those numbers are the
same, compare T and T
0
using the order gpathpermlex for b ≺ a. In this case,
the leading monomials of the elements above