  Tootkaboni and Vahed introduced the notion of some large sets near idempotent
along with some combinatorial properties. We characterize when the finite
Cartesian product of central sets near idempotent is central near idempotent.
Moreover, we provide a partial characterization for the infinite Cartesian
product of the same. We then study the abundance of some large sets near
idempotent. We also investigate the effect of tensor product near zero.
Finally, as an application we provide a characterization of members of
polynomials (with constant term 0) evaluated at idempotents in a near zero
semigroup.
