
166 Algebraic Operads: An Algorithmic Companion
T,T
00
2 3
1
=
2 3
1
4
.
One very useful feature of the insertion operations is that they allow us
to give an explicit description of an ideal generated by a given collection S in
the free shuffle operad which is a suitable replacement of the description “the
ideal (S) is the linear span of all elements r
1
sr
2
for all r
1
, r
2
∈ T (X), s ∈ S”
which we had in the associative case.
Proposition 5.4.2.8. Let S ⊂ T
X
(X). The ideal (S) generated by S can be
described explicitly as the linear span of all insertions
T
1
,T
2
(f), where T
1
is
a shuffle tree monomial, T
2
is a divisor of T
1
, and f ∈ S(ar(T
2
)).
Proof. The ideal ...