Sets satisfying the Central Sets Theorem

Abstract

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a subset with zero Folner density which satisfies the conclusion of the Central Sets Theorem. We show that this class includes any direct sum of countably many finite abelian groups as well as any subsemigroup of (ℝ,+) which contains ℤ. We also show that if S and T are in this class and either both are left cancellative or T has a left identity, then S×T is in this class. We also extend a theorem proved in (Beiglbock et al. in Topology Appl., to appear), which states that, if p is an idempotent in βℕ whose members have positive density, then every member of p satisfies the Central Sets Theorem. We show that this holds for all commutative semigroups. Finally, we provide a simple elementary proof of the fact that any commutative semigroup satisfies the Strong Folner Condition.