All results mentioned in this abstract assume Martin's Axiom. (Some of them are known to not be derivable in ZFC.) It is known that if S is the free semigroup on countably many generators, then there exists an idempotent p∈βS such that if q,r∈βS and qr=p, then q=r=p. We show that the same conclusion holds for the semigroups (N,⋅) and (F,∪) where F is the set of finite nonempty subsets of N. Such a strong conclusion is not possible if S is the free group on countably many generators or is the free semigroup on finitely many (but more than one) generators, since then any idempotent can be written as a product involving elements of S. But we show that in these cases we can produce p such that if q,r∈βS and qr=p, then either q=r=p or q and r satisfy one of the trivial exceptions that must exist. Finally, we show that for the free semigroup on countably many generators, the conclusion can be derived from a set theoretical assumption that is at least potentially weaker than what had previously been required.

The first author acknowledges support received from the National Science Foundation via Grant DMS-1160566. The second author acknowledges support received from the Simons Foundation via Grant 210296