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This operad governs associative algebras. ASSOC_n is given by k[Sn] (the algebra of the symmetric group Sn) for any n ≥ 0, whereas ASSOC₀ := {0}. The right action of Sn on ASSOC_n is given by linear extension of right multiplication:

[1.70]

Let σ ∈ ASSOC_k and . The partial compositions are given for any i = 1,…,k by:

[1.71]

with the notations of equation [1.68]. The reader is invited to check the two associativity axioms, as well as the equivariance axiom which reads:

[1.72]

for any σ, σ′ ∈ ASSOC_k and . Let us denote by ek the unit element in the symmetric group Sk. We obviously have ek ∘_i el = e_{k + l – 1} for any i = 1,…, k. In particular,

[1.73]

Now let V be an algebra over the operad ASSOC, and let Φ : ASSOC → Endop(V) be the corresponding morphism of operads. Let μ : V ⊗ V → V be the binary operation Φ(e2). In view of equation [1.73] we have:

[1.74]

In other words, μ is associative. As ek can be obtained, for any k ≥ 3, by iteratively composing k – 2 times the element e₂, we see that any element of ASSOC_k can be obtained from e₂, partial compositions, symmetric group actions and linear combinations. As a result, any k-ary operation on V, which is in the image of Φ, can be obtained in terms of the associative product μ, partial compositions, symmetric group actions and linear combinations. Summing up, an algebra over the operad ASSOC is nothing but an associative algebra. In view of equation [1.69], the free ASSOC-algebra over a vector space W is the (non-unital) tensor algebra .

In the same line of thoughts, the operad governing unital associative algebras is defined similarly, except that the space of 0-ary operations is k.e₀, with ek ∘_i e₀ = e_{k – 1} for any i = 1,…,k. The unit element u : k → V of the algebra V is given by u = Φ(e₀). The free unital algebra over a vector space W is the full tensor algebra .