The combinatorial derivation


  • Igor V. Protasov Kyiv University



Combinatorial derivation, $\Delta$-trajectories, large, small and thin subsets of groups, partitions of groups, Stone-\v{C}ech compactification of a group


Let $G$ be a group, $\mathcal{P}_G$ be the family of all subsets of $G$. For a subset $A\subseteq G$, we put
$\Delta(A)=\{g\in G:|gA\cap A|=\infty\}$. The mapping $\Delta:\mathcal{P}_G\rightarrow\mathcal{P}_G$, $A\mapsto\Delta(A)$, is called a combinatorial derivation and can be considered as an analogue of the topological derivation $d:\mathcal{P}_X\rightarrow\mathcal{P}_X$, $A\mapsto A^d$, where $X$ is a topological space and $A^d$ is the set of all limit points of $A$. Content: elementary properties, thin and almost thin subsets, partitions, inverse construction and $\Delta$-trajectories,  $\Delta$ and $d$.


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T. Banakh and N. Lyaskovska, Weakly P-small not P-small subsets in groups, Intern. J. Algebra Computations, 18 (2008), 1-6.

M. Filali and I. Protasov, Ultrafilters and Topologies on Groups, Math. Stud. Monorg. Ser., Vol. 13, VNTL Publishers, Lviv, 2010.

V. Gavrylkiv, Algebraic-topological structure on superextensions, Dissertation, Lviv, 2009.

N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification, Walter de Grueter, Berlin, New York, 1998.

The Kourovka Notebook, Novosibirsk, Institute of Math., 1995.

K. Kuratowski, Topology, Vol. 1, Academic Press, New York and London, PWN, Warszawa, 1969.

Ie. Lutsenko, Thin systems of generators of groups, Algebra and Discrete Math., 9 (2010), 108-114.

Ie. Lutsenko and I. V. Protasov, Sparse, thin and other subsets of groups, Intern. J. Algebra Computation, 19 (2009), 491-510.

I. V. Protasov, Selective survey on Subset Combinatorics of Groups, J. Math. Sciences, 174 (2011), 486-514.

I. Protasov and T. Banakh, Ball Structure and Colorings of Groups and Graphs, Math. Stud. Monorg. Ser., Vol. 11, VNTL Publishers, Lviv, 2003.

T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, 2006.




How to Cite

I. V. Protasov, “The combinatorial derivation”, Appl. Gen. Topol., vol. 14, no. 2, pp. 171–178, Sep. 2013.