The combinatorial derivation and its inverse mapping

Abstract

Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping ∆ :A ∩ A is infinite} is called the combinatorial derivation. The mapping ∆ can be considered as an analogue of the topological derivation : P X → P X , A → A , where X is a topological space and A is the set of all limit points of A. We study the behaviour of subsets of G under action of ∆ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in P G such that ∆(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G . MSC:20A05

Recent progress in subset combinatorics of groups

“…Comments. More information on combinatorial derivation in [26], [27], [28]. In particular, Theorem 6.2 from [26] shows that the trajectory A −→ △(A) −→ △ 2 (A) −→ .…”

Partitions of Groups

“…Comments. The combinatorial derivation was introduced in [33] and studied in [12], [34], [38]. The results of this sections from [5], [38], [39].…”

Around $P$-small subsets of groups

“…We recall [10] that a family F of subsets of a group G is ∆-complete (∇-complete) if ∆(X) ∈ F for each X ∈ F (∆(X) ∈ F implies X ∈ F ). By Theorem 3, the family of all near P-small subsets of a group G need not to be ∆-complete.…”

“…We study the relationships between modified P-small subsets and thin subsets (Theorems 1 and 2), and the behavior of near P-small subsets under the action of the combinatorial derivation and its inverse mapping (Theorems 3 and 4). The combinatorial derivation, the main tool in this note, was introduced in [9] and studied in [5], [10], [11]. Some necessary auxiliary statements on the combinatorial derivation are arranged in section 1.…”