  Let A be an arbitrary set. For any transformation T (self-map of A) let
T(f)(x):=f(T(x)) (for all x in A) be the usual shift operator. A function g is
called periodic, i.e., invariant mod T, if Tg=g (=Ig, where I is the identity
operator).
  As a natural generalization of various earlier investigations in different
function spaces, we study the following problem. Let T_j (j=1,...,n) be
arbitrary commuting mappings -- transformations -- from A into A. Under what
conditions can we state that a function f from A to A is the sum of "periodic",
that is, T_j-invariant functions f_j?
  An obvious necessary condition is that the corresponding multiple difference
operator annihilates f, i.e., D_1 ... D_n f= 0, where D_j:=T_j-I. However, in
general this condition is not sufficient, and our goal is to complement this
basic condition with others, so that the set of conditions will be both
necessary and sufficient.
