Continuous right zero homomorphisms of

A mapping π : T → X of a semigroup T into a set X is a right zero homomorphism if π ( p q ) = 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 2 2 ω .