
Symmetric Operads and Shuffle Operads 161
Lemma 5.4.1.5. A tree monomial T = (τ, x, n) is uniquely determined by its
path-permutation data.
Proof. Let us rearrange the words in Path(T ) according to the permuta-
tion σ(T )
−1
; this would list the paths from the root to the leaves according
to the total planar order of leaves. By Lemma 3.4.1.4 we can uniquely recon-
struct a nonsymmetric tree monomial out of that path sequence. It remains to
use σ(T ) to number the leaves of that tree monomial, obtaining the original
shuffle tree monomial.
Definition 5.4.1.6 (Path-permutation extension). Suppose that Ξ is a mono-
mial order on X
∗
.
The path-permutation extension ...