Functional equations on discrete sets.

Let Y (+) be a group, D ... ℤ² where ℤ (+, ≤) denotes the ordered group of all integers, and ℤ² : = ℤxℤ. We shall use the notations Dx: = ... . The main purpose of the article is to find sets D ... ℤ² that the general solution of the functional equation f(x+y) = g(x)+h(y) for all (x, y) ∈ D with unknown functions f: Dx+y → Y, g : Dx → Y, h : Dy → Y is in the form of f(u) = a(u) + C1 + C2 for all u ∈ Dx+y, g(v) = a(v) + C1 for all v ∈ Dx, h(z) = a(z)+C2 for all z ∈ Dy where a: ℤ → Y is an additive function, C1, C2 ∈ Y are constants. [ABSTRACT FROM AUTHOR]