Functional equations on discrete sets

Let Y (+) be a group, D ⊆ ℤ2 where ℤ(+, ⩽) denotes the ordered group of all integers, and ℤ2 := ℤ×ℤ. We shall use the notations Dx := {u ∈ ℤ | ∃v ∈ X : (u, v) ∈ D}, Dy := {v ∈ ℤ | ∃u ∈ ℤ : (u, v) ∈ D}, Dx+y := {z ∈ ℤ | ∃(u, v) ∈ D : z = u + v}. 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.