The combinatorial derivation and its inverse mapping

• Autorzy: Igor Protasov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Combinatorial derivation; Δ-trajectories; Large, small and thin subsets of groups; Partitions of groups; Stone-Čech compactification of a group

Kategorie tematyczne

• 06E15: Stone spaces (Boolean spaces) and related structures
• 20A05: Axiomatics and elementary properties
• 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.)
• 20F99: None of the above, but in this section
• 22A15: Structure of topological semigroups

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 12
• Strony: 2176-2181

Daty

wydano

2013-12-01

online

2013-10-08

Twórcy

autor

Igor Protasov

• Kyiv University

Bibliografia

• [1] Banakh T., Lyaskovska N., On thin-complete ideals of subsets of groups, Ukrainian Math. J., 2011, 63(6), 865–879 http://dx.doi.org/10.1007/s11253-011-0549-1
• [2] Brown T.K., On locally finite semigroups, Ukrainian Math. J., 1968, 20(6), 631–636 http://dx.doi.org/10.1007/BF01085231
• [3] Erde J., A note on combinatorial derivation, preprint available at http://arxiv.org/abs/1210.7622
• [4] Hindman N., Strauss D., Algebra in the Stone-Čech Compactification, de Gruyter Exp. Math., 27, Walter de Gruyter, Berlin, 1998 http://dx.doi.org/10.1515/9783110809220
• [5] Lutsenko Ie., Thin systems of generators of groups, Algebra Discrete Math., 2010, 9(2), 108–114
• [6] Lutsenko Ie., Protasov I.V., Sparse, thin and other subsets of groups, Internat. J. Algebra Comput., 2009, 19(4), 491–510 http://dx.doi.org/10.1142/S0218196709005135
• [7] Lutsenko Ie., Protasov I., Thin subsets of balleans, Appl. Gen. Topology, 2010, 11(2), 89–93
• [8] Lutsenko Ie., Protasov I.V., Relatively thin and sparse subsets of groups, Ukrainian Math. J., 2011, 63(2), 254–265 http://dx.doi.org/10.1007/s11253-011-0502-3
• [9] Protasov I.V., Counting Ω-ideals, Algebra Universalis, 2010, 62(4), 339–343 http://dx.doi.org/10.1007/s00012-010-0032-0
• [10] Protasov I.V., Selective survey on subset combinatorics of groups, J. Math. Sci. (N.Y.), 2011, 174(4), 486–514 http://dx.doi.org/10.1007/s10958-011-0314-x
• [11] Protasov I.V., Combinatorial derivation, preprint available at http://arxiv.org/abs/1210.0696
• [12] Protasov I., Banakh T., Ball Structures and Colorings of Graphs and Groups, Math. Stud. Monogr. Ser., 11, VNTL, L’viv, 2003
• [13] Protasov I., Zarichnyi M., General Asymptology, Math. Stud. Monogr. Ser., 12, VNTL, L’viv, 2007

Identyfikatory

DOI

10.2478/s11533-013-0313-x