An $A_{\infty }$-bialgebra is a DGM $H$ equipped with structurally compatible operations $\left\{ \omega ^{j,i}:H^{\otimes i}\rightarrow H^{\otimes j}\right\} $ such that $\left( H,\omega ^{1,i}\right) $ is an $% A_{\infty }$-algebra and $\left( H,\omega ^{j,1}\right) $ is an $A_{\infty }$% -coalgebra. Structural compatibility is controlled by the biderivative operator $Bd$, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP $U=End\left( TH\right) $.
Homology, Homotopy and Applications, Vol. 7(2005), No. 2, pp. 161-177