D-sets in arbitrary semigroup

Abstract

We define the notion of D-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many D-sets is a D-set. A similar partial result has been proved for Cartesian product of infinitely many D-sets. Finally, in a commutative semigroup we deduce that D-sets (with respect to a Følner net) are C-sets.