A Study on Filter Version of Strongly Central Sets

TL;DR

This paper extends the concept of central sets in semigroup theory by introducing filter-based versions and characterizing strongly central sets dynamically and combinatorially.

Contribution

It introduces filter-based strongly central and very strongly central sets and provides their dynamical and combinatorial characterizations.

Findings

Defined filter version of strongly central sets.

Characterized strongly F- central sets dynamically.

Provided combinatorial characterizations of these sets.

Abstract

Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-\v{C}ech compactification of discrete semigroup. They found that central sets are the members of the minimal idempotents of \b{eta}S, the Stone-\v{C}ech compactification of a semigroup (S, .). Hindman and leader introduced the notion of Central set near zero algebraically. Later dynamical and combinatorial characterization have also been established. For any given filter F in S a set A is said to be a F- central set if it is a member of a minimal idempotent of a closed subsemigroup of \b{eta}S, generated by the filter F. In a recent article Bergelson, Hindman and Strauss introduced strongly central and very strongly central sets in [BHS]. They also dynamically characterized the sets in the same paper. In the present article we will characterize the strongly F- central sets dynamically and combinatorially. Here we introduce the filter version of strongly central sets and very strongly central sets. We also provide dynamical and combinatorial characterization of such sets.