Continuous right zero homomorphisms of U(S)

A mapping π:T→X of a semigroup T into a set X is a right zero homomorphism if π(pq)=π(q) for all p,q∈T. Let S be a discrete cancellative semigroup of cardinality κ⩾ω, let βS be the Stone–Čech compactification of S, and let U(S) denote the ideal of βS consisting of uniform ultrafilters. We show that (a) if κ>ω, then U(S) admits a continuous right zero homomorphism onto U(κ), and (b) if κ=ω, then U(S) admits a continuous right zero homomorphism onto any connected compact metric space and onto a connected compact Hausdorff space of cardinality ω22.