Exhibition of piecewise syndetic and broken IP sets near idempotent
Abstract
Characterizations of ultrafilters belong to the smallest ideal of Stone-Cech 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 A set, where A∈\ syndetic, quasi-central, central, strongly central, very strongly central\. Also, a characterization of broken IPn sets using ultrafilters, and the equivalence between the sets that contain a broken IP set and sets that contain a broken IPn are established, n∈ 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.