Central Sets and Their Combinatorial Characterization

Central sets in semigroups are known to have very rich combinatorial structure, described by the “Central Sets Theorem”. It has been unknown whether the Central Sets Theorem in fact characterizes central sets, and if not whether some other combinatorial characterization could be found. We derive here a combinatorial characterization of central sets and of the weaker notion of quasi-central sets. We show further that in (N, +) these notions are different and strictly stronger than the characterization provided by the Central Sets Theorem. In addition, we derive an algebraic characterization of sets satisfying the conclusion of the Central Sets Theorem and use this characterization to show that the conclusion of the Central Sets Theorem is a partition regular property in any commutative semigroup.