The combinatorial derivation and its inverse mapping

Let G be a group and PG be the Boolean algebra of all subsets of G. A mapping Δ: PG → PG defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: PX→ PX, A ↦ Ad, where X is a topological space and Ad is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in PG such that Δ(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = PG.