Interacting Hopf algebras

We introduce the theory IHRof interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IHRare derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoidâcomonoid pairs. This construction is instrumental in showing that IHRis isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R