A Generalized Central Sets Theorem In Partial Semigroups

TL;DR

This paper extends the Central Sets Theorem to all adequate sequences in commutative adequate partial semigroups, broadening the understanding of $C$-sets and their properties.

Contribution

It generalizes the Central Sets Theorem to include all adequate sequences in commutative adequate partial semigroups, providing a new sufficient condition for $C$-sets.

Findings

Extended the Central Sets Theorem to all adequate sequences

Provided a sufficient condition for $C$-sets in this context

Broadened the applicability of the Central Sets Theorem

Abstract

The most powerful formulation of the Central Sets Theorem in an arbitrary semigroup was proved in the work of De, Hindman, and Strauss. The sets which satisfy the conclusion of the above Central Sets Theorem are called $C$-sets. The original Central Sets Theorem was extended by J. McLeod for adequate commutative partial semigroups. In this work, we will extend the Central Sets Theorem obtained by taking all possible adequate sequences in a commutative adequate partial semigroup. We shall also discuss a sufficient condition for being a set $C$-set in our context.