Dynamics Near An Idempotent

TL;DR

This paper extends the dynamical characterizations of combinatorially rich sets near an idempotent in dense subsemigroups, generalizing previous results near zero and exploring algebraic and combinatorial structures.

Contribution

It provides new dynamical characterizations of quasi-central and C-sets near an idempotent in dense subsemigroups of semitopological semigroups.

Findings

Dynamical characterizations of quasi-central sets near an idempotent.

Dynamical characterizations of C-sets near an idempotent.

Extension of previous results from near zero to near an idempotent.

Abstract

Hindman and Leader first introduced the notion of semigroup of ultrafilters converging to zero for a dense subsemigroups of $((0,\infty),+)$. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Tootkabani and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent $e$ for a dense subsemigroups of a semitopological semigroup $(T, +)$ and they gave the combinatorial proof of central set theorem near $e$. Algebraically one can also define quasi-central sets near $e$ for dense subsemigroups of $(T, +)$. In a dense subsemigroup of $(T,+)$, C-sets near $e$ are the sets, which satisfy the conclusions of the central sets theorem near $e$. S. K. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper we shall prove these dynamical characterizations for these combinatorially rich sets near $e$.