Associahedra, cellular W-construction and products of $A_{\mathrm{\infty }}$-algebras

Our aim is to construct a functorial tensor product of $A_{\mathrm{\infty }}$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_{\mathrm{\infty }}$-algebras.