  We construct an explicit diagonal \Delta_P on the permutahedra P. Related
diagonals on the multiplihedra J and the associahedra K are induced by Tonks'
projection P --> K and its factorization through J. We introduce the notion of
a permutahedral set Z and lift \Delta_P to a diagonal on Z. We show that the
double cobar construction \Omega^2(C_*(X)) is a permutahedral set; consequently
\Delta_P lifts to a diagonal on \Omega^2(C_*(X)). Finally, we apply the
diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal
generality.
