Characterization of elements of polynomials in βS

Given the discrete space of natural numbers, we characterize the elements of polynomials evaluated on the points of βℕ. We establish these results by proving the characterization in a far more general setting. Let S be a discrete set which is a semigroup under two operations ⋅ and +. Let g(z 1,z 2,…,z k ) be any polynomial and p 1,p 2,…,p k be elements of βS. We provide a sufficient condition that a set A⊆S is a member of g(p 1,p 2,…,p k ) and use it to characterize the members of g(p 1,p 2,…,p k ) if each p i is an idempotent in (βS,+).