Central Sets Theorem along filters and some combinatorial consequences

TL;DR

This paper extends the Central Sets Theorem to closed subsemigroups of the Stone-Cech compactification using filters, exploring algebraic properties and combinatorial consequences of largeness notions along filters.

Contribution

It introduces a method to derive the Central Sets Theorem for closed subsemigroups of βS via filters and examines the preservation of largeness notions under certain homomorphisms.

Findings

Central Sets Theorem extended to closed subsemigroups of βS.

Largeness notions along filters are preserved under specific homomorphisms.

Derived combinatorial consequences from the filter-based approach.

Abstract

The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions and extensions of this theorem. All of these theorems deal with central sets, and its origin from the algebra of Stone-Cech compactification of arbitrary semigroup, say $\beta S$. It can be proved that every closed subsemigroup of $\beta S$ is generated by a filter. We will show that, under some restrictions, one can derive the Central Sets Theorem for any closed subsemigroup of $\beta S$ . We will derive this theorem using the corresponding filter and its algebra. Later we will also deal with how the notions of largeness along filters are preserved under some well behaved homomorphisms and give some consequences.