Exhibition of piecewise syndetic and broken IP sets near idempotent

Abstract

Characterizations of ultrafilters belong to the smallest ideal of Stone-Čech 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.

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