  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
F{\o}lner net) are $C$-sets.
