  We introduce notions of ${\mathcal O}$-operators of the Loday algebras
including the dendriform algebras and quadri-algebras as a natural
generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators
give a sufficient and necessary condition on the existence of the $2^{n+1}$
operations on an algebra with the $2^{n}$ operations in an associative cluster.
The analogues of the classical Yang-Baxter equation in these algebras can be
understood as the $\mathcal O$-operators associated to certain dual bimodules.
As a byproduct, the constraint conditions (invariances) of nondegenerate
bilinear forms on these algebras are given.
