Compatible associative bialgebras

We introduce a non-symmetric operad $\mathcal{N}$, whose dimension in degree $n$ is given by the Catalan number $c_{n-1}$. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a $\mathcal{N}$-algebra. The data $({\rm As},{\rm As}^2, \mathcal{N})$ is a good triple of operads, in J.-L. Loday's sense. Our construction induces another triple of operads $({\rm As},{\rm As}_2,{\rm As})$, where ${\rm As}_2$ is the operad of matching dialgebras. Motivated by A. Goncharov's Hopf algebra of paths $P(S)$, we introduce the notion of bi-matching dialgebras and show that the Hopf algebra $P(S)$ is a bi-matching dialgebras.