Composing PROPs

A PROP is a way of encoding structure borne by an object of a symmetric monoidal category. We descibe a notion of distributive law for PROPs, based on Beck's distributive law for monads. A distributive law between PROPs allows them to be composed, and an algebra for the composite PROP consists of a single object with an algebra structure for each of the original PROPs, subject to compatability conditions encoded by the distributive law. An example is the PROP for bialgebras, which is a composite of the PROP for coalgebras and that for algebras.