Image Partition near an Idempotent

TL;DR

This paper explores the concept of image partition regularity near an idempotent in Hausdorff semitopological semigroups, extending classical Ramsey Theory results with new combinatorial applications.

Contribution

It introduces the study of image partition regularity near an idempotent in semitopological semigroups and derives applications based on the Central Sets Theorem.

Findings

Extended classical results to semigroups near idempotents

Derived combinatorial applications for finite and infinite matrices

Connected image partition regularity with the Central Sets Theorem

Abstract

Some of the classical results of Ramsey Theory can be naturally stated in terms of image partition regularity of matrices. There have been many significant results of image partition regular matrices as well as image partition regular matrices near zero. Here, we are investigating image partition regularity near an idempotent of an arbitrary Hausdorff semitopological semigroup (T, +) and a dense subsemigroup S of T . We describe some combinatorial applications on finite as well as infinite image partition regular matrices based on the Central Sets Theorem near an idempotent of T .