A lemma on extending functions into F -spaces and homomorphisms between Stone–Čech remainders

This papers contains two main results. The first is a theorem about continuous functions from a countably compact Hausdorff space into a compact F -space, which has applications to the algebraic properties of the Stone–Čech compactification βS of a discrete semigroup S . The second main result shows that many continuous homomorphisms from S∗ to G∗ have to arise from homomorphisms mapping S to G , where S is a discrete semigroup and G is a discrete group and S∗ denotes βS\S . The second result is related to the first because it uses it at a crucial point.