
Nonsymmetric Operads 87
Lemma 3.4.2.15. For all elements f, g ∈ T (X) such that r
g
(f) is defined,
we have
r
g
(f) = 0 or lm(r
g
(f)) ≺ lm(f ).
Proof. Indeed, by construction we have
lt(f) = lt
lc(f)
lc(g)
T
1
,T
2
(g)
.
One can view a reduction as one step of a version of the long division
algorithm. We make it more precise as follows.
Algorithm 3.4.2.16 (Long division for nonsymmetric operads).
Input: An element f ∈ T (X), and a finite set S ⊂ T (X).
Output: An element
˜
f, reduced with respect to S, for which
lt(
˜
f) lt(f) such that f + (S) =
˜
f + (S).
• If f = 0, return f.
• Replace S by its linear self-reduction (Proposition 1.2.1.6).
• If D := {s ∈ S: lm(f) is