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 $Δ(A)=\{g\in G:|gA\cap A|=\infty\}$. The mapping $Δ:\mathcal{P}_G\rightarrow\mathcal{P}_G$, $A\mapstoΔ(A)$, 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 $Δ$-trajectories, $Δ$ and $d$.