Combinatorial Derivation

TL;DR

This paper introduces and explores a combinatorial derivation on groups, analyzing its properties, special subsets, and relationships with topological derivations, providing new insights into group subset structures.

Contribution

It defines the combinatorial derivation $ riangle$, investigates its properties, and establishes connections with topological derivations, offering novel methods for studying group subsets.

Findings

Properties of the combinatorial derivation $ riangle$

Characterization of thin and almost thin subsets

Analysis of $ riangle$-trajectories and their behaviors

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 $\Delta:\mathcal{P}_G\rightarrow\mathcal{P}_G$, $A\mapsto\Delta(A)$, 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 $d$.