Compatible associative bialgebras

We introduce a non-symmetric operad 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 𝒩-algebra. The data (As,As²,𝒩) is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads (As,As₂,As), where As₂ 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.