Combinatorial Derivation

Abstract

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 $\Delta :{\mathcal{P}}_G\to {\mathcal{P}}_G$, $A\mapsto \Delta (A)$, is called a combinatorial derivation and can be considered as an analogue of the topological derivation $d:{\mathcal{P}}_X\to {\mathcal{P}}_X$, $A\mapsto A^d$, 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 $d$.