Topological properties of some algebraically defined subsets of βN

Let S be a discrete semigroup and let the Stone-Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. We show that the closure of the set of multiplicative idempotents in βN does not meet the set of additive idempotents in βN. We also show that the following algebraically defined subsets of βN are not Borel: the set of idempotents; the smallest ideal; any semiprincipal right ideal of N⁎; the set of idempotents in any left ideal; and N⁎+N⁎. We extend these results to βS, where S is an infinite countable semigroup algebraically embeddable in a compact topological group.