We introduce the notion of a matrad M = { Mn,m } whose submodules M*,1 and M1,* are non-Σ operads. We define the free matrad H∞ generated by a singleton θmn in each bidegree (m,n) and realize H∞ as the cellular chains on a new family of polytopes { KKn,m = KKm,n }, called biassociahedra, of which KKn,1 = KK1,n is the associahedron Kn. We construct the universal enveloping functor from matrads to PROPs and define an A∞-bialgebra as an algebra over H∞.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 1, pp.1-57.