The combinatorial derivation and its inverse mapping |
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Authors: | Igor V Protasov |
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Institution: | 1. Department of Cybernetics, Kyiv University, Volodimirska 64, Kyiv, 01033, Ukraine
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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 . |
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