Some Combinatorial Concepts near an Idempotent

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.