2) Consider that E is generated by an n-ary operation [mu] of degree 1.
and more generally an element of F(E)(k(n - 1) + 1) by a rooted planar n-ary tree with levels and (k(n - 1) + 1)) leaves.
The relations that we consider will be quadratic in the sense that we compose two n-ary multiplications.
To compute the dual operad of the operad associated to n-ary algebras we need some differential graded operad.
Now if we consider n-ary algebras, we have seen that we can still define the notion of quadratic operad, that is we consider a generating multiplication which is an n-ary multiplication n, that is E = < [mu] > [subset] K [[[summation].
This follows directly from the definition of the dual operad of a quadratic n-ary operad.