  A notion of vertex bialgebra and a notion of module nonlocal vertex algebra
for a vertex bialgebra are studied and then a smash product construction of
nonlocal vertex algebras is presented. For every nonlocal vertex algebra $V$
satisfying a suitable condition, a canonical bialgebra $B(V)$ is constructed
such that primitive elements of $B(V)$ are essentially pseudo derivations and
group-like elements are essentially pseudo endomorphisms. Furthermore, vertex
algebras associated with Heisenberg Lie algebras as well as those associated
with nondegenerate even lattices are reconstructed through smash products.
