# The combinatorial derivation and its inverse mapping

Open Mathematics (2013)

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

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topIgor 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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