# Some Combinatorial Concepts near an Idempotent

## TL;DR

This paper explores the rich combinatorial properties of a small subset of the real line near zero by examining the $wap$-compactification of semitopological semigroups and their ultrafilters near an idempotent.

## Contribution

It introduces and proves new locally combinatorial concepts near a virtual idempotent using $wap$-compactification of semitopological semigroups.

## Key findings

- Ultrafilters near an idempotent form a compact subsemigroup
- Rich combinatorial properties are observed near zero on the real line
- The $wap$-compactification provides a framework for analyzing local combinatorial structures

## Abstract

A small part of real line which is very close to zero has rich combinatorial properties. The aim of this paper is to express and then prove some locally combinatorial concepts near a virtual idempotent by considering the $wap$-compactification of a semitopological semigroup $S$. The $wap-$compactification of a semitopological semigroup $S$, is denoted by $S^w$, the collection of all ultrafilters near an idempotent $\eta\in S^w$ forms a compact subsemigroup of $\beta S_d$, where $S_d$ denotes $S$ as discrete space.