Exhibition of piecewise syndetic and broken IP sets near idempotent

TL;DR

This paper characterizes various special sets in semigroups using ultrafilters and explores their properties near idempotents, providing new insights into recurrence and structure without countability assumptions.

Contribution

It introduces novel characterizations of piecewise syndetic and broken IP sets via ultrafilters and extends results to non-countable semigroups near idempotents.

Findings

Characterization of ultrafilters in the smallest ideal using syndetic and central sets

Representation of piecewise syndetic sets containing broken A sets

Equivalence between sets containing broken IP and broken IP^n sets

Abstract

Characterizations of ultrafilters belong to the smallest ideal of Stone-\v{C}ech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to represent piecewise syndetic sets of a semigroup in terms of the sets that contain a broken $\mathcal{A}$ set, where $\mathcal{A}\in\{$ syndetic, quasi-central, central, strongly central, very strongly central$\}$. Also, a characterization of broken IP$^{n}$ sets using ultrafilters, and the equivalence between the sets that contain a broken IP set and sets that contain a broken IP$^{n}$ are established, $n\in \mathbb{N}$. Without assuming the countability of a semigroup, it is shown that piecewise syndetic sets i.e., sets that contain a broken syndetic set (broken IP set) force uniform recurrence (recurrence respectively) and vice versa. In addition, all the said results are established near idempotent of a semitopological semigroup.