Confluence laws and Hopf-Borel type theorem for operads

In 2008, Loday shed light on the existence of Hopf-Boreltheorems for operads. Using the vocabulary of category theory, Livernet,Mesablishvili and Wisbauer extended such theorems to monads. In bothcases, the reasoning was to start from a mixed distributive law andthen to prove that it induces an isomorphism of species to finally geta rigidity theorem. Our reasoning goes here backward: we prove thatfrom an isomorphism of species one can get what we called a confluencelaw, which generalises mixed distributive laws, and that it is enough toobtain a rigidity theorem. This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements. Our reasoning permits us to generatemany new examples, while recovering the known ones by consideringdual relations.