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 β N does not meet the set of additive idempotents in β N . We also show that the following algebraically defined subsets of β N 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.