Eight times four bialgebras of hypergraphs, cointeractions, and chromatic polynomials

We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show the polynomial associated to a hypergraph is the hypergraph chromatic polynomial. Moreover hypergraphs occurs in quartets: there is a dual, a complement, and a dual complement hypergraph. These correspondences are involutions and give rise to three other double bialgebras, and three more chromatic polynomials. In all we give eight quartets of bialgebras which includes recent bialgebras of M. Aguiar and F. Ardila, and by L. Foissy.