Combinatorial Derivation

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 $d$.

💡 Research Summary

The paper introduces and studies a combinatorial derivation operator (\Delta) on a group (G). For any subset (A\subseteq G) the operator is defined by
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