# The combinatorial derivation and its inverse mapping

Open Mathematics (2013)

• Volume: 11, Issue: 12, page 2176-2181
• ISSN: 2391-5455

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## Abstract

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Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G 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: P X→ P X, A ↦ A d, where X is a topological space and A d 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 P G such that Δ(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G.

## How to cite

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Igor Protasov. "The combinatorial derivation and its inverse mapping." Open Mathematics 11.12 (2013): 2176-2181. <http://eudml.org/doc/269326>.

@article{IgorProtasov2013,
abstract = {Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G 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: P X→ P X, A ↦ A d, where X is a topological space and A d 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 P G such that Δ(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G.},
author = {Igor Protasov},
journal = {Open Mathematics},
keywords = {Combinatorial derivation; Δ-trajectories; Large, small and thin subsets of groups; Partitions of groups; Stone-Čech compactification of a group; combinatorial derivation; Stone-Čech compactification of groups; partitions of groups; large subsets of groups; small subsets of groups; thin subsets of groups; infinite groups; symmetric subsets; combinatorial conditions on subsets; amenable groups},
language = {eng},
number = {12},
pages = {2176-2181},
title = {The combinatorial derivation and its inverse mapping},
url = {http://eudml.org/doc/269326},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Igor Protasov
TI - The combinatorial derivation and its inverse mapping
JO - Open Mathematics
PY - 2013
VL - 11
IS - 12
SP - 2176
EP - 2181
AB - Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G 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: P X→ P X, A ↦ A d, where X is a topological space and A d 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 P G such that Δ(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G.
LA - eng
KW - Combinatorial derivation; Δ-trajectories; Large, small and thin subsets of groups; Partitions of groups; Stone-Čech compactification of a group; combinatorial derivation; Stone-Čech compactification of groups; partitions of groups; large subsets of groups; small subsets of groups; thin subsets of groups; infinite groups; symmetric subsets; combinatorial conditions on subsets; amenable groups
UR - http://eudml.org/doc/269326
ER -

## References

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13. [13] Protasov I., Zarichnyi M., General Asymptology, Math. Stud. Monogr. Ser., 12, VNTL, L’viv, 2007

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