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Various compatibility conditions among replicated copies of operations in agiven algebraic structure have appeared in broad contexts in recent years.Taking an uniform approach, this paper gives an operadic study of compatibilityconditions for nonsymmetric operads with unary and binary operations, andhomogeneous quadratic and cubic relations. This generalizes the previousstudies for binary quadratic operads. We consider three compatibilityconditions, namely the linear compatibility, matching compatibility and totalcompatibility, with increasingly strict restraints among the replicated copies.The linear compatibility is in Koszul dual to the total compatibility, whilethe matching compatibility is self dual. Further, each compatibility can beexpressed in terms of either one or both of the two Manin square products.

Compatible structures on unary binary nonsymmetric operads with quadratic and cubic relations