  If M is a monoid (e.g. the lattice Z^D), and G is a finite (nonabelian)
group, then G^M is a compact group; a `multiplicative cellular automaton' (MCA)
is a continuous transformation F:G^M-->G^M which commutes with all shift maps,
and where nearby coordinates are combined using the multiplication operation of
G.
  We characterize when MCA are group endomorphisms of G^M, and show that MCA on
G^M inherit a natural structure theory from the structure of G. We apply this
structure theory to compute the measurable entropy of MCA, and to study
convergence of initial measures to Haar measure.
