
Nonsymmetric Operads 107
3.7.1 Extensions and normal forms in algebras
Let P = T (X)/I be a nonsymmetric operad, and let A be an algebra
over P. The following construction is an explicit adaptation for the nonsym-
metric case of the construction of the enveloping operad for the pair (P, A),
see [17] and references therein.
Definition 3.7.1.1 (Extension of an operad by an algebra). The extension
of an operad P by its algebra A, denoted P n A, is the operad
T (X ⊕ A)/(I ⊕ I
A
),
where A is the same vector space as A but viewed as a collection of operations
of arity 0 (for each element a ∈ A, we denote the corresponding element of A
by a), and I
A
consists of ...