Image Partition near an Idempotent

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 .