Central Sets Theorem on noncommutative semigroups

Abstract

Let D be a directed set without maximal element, S be an infinite semigroup and DS be the collection of all functions from D into S. It is shown that for a commutative semigroup S, A⊆S is a C-set with respect to NS if and only if A is a C-set with respect to DS. We investigate the Central Sets Theorem for arbitrary semigroups. In fact the Central Sets Theorem is stated with respect to SS for arbitrary semigroups.