  Let S be the set of scalings 1, 2,3,4, ... and consider the corresponding set
of scaled lattices in the plane. In this paper averaging operators are defined
for plaquette functions on a lattice to plaquette functions on a coarser
lattice for all scale factors and their coherence is proved. This generalizes
the averaging operators introduced by Balaban and Federbush. There are such
coherent families of averaging operators for any dimension D and not only for
D=2. Finally there are uniqueness theorems saying that in a sense, besides a
form of straightforward averaging, the weights used are the only ones that give
coherent families of averaging operators.
