Orderings of the Stone–Čech remainder of a discrete semigroup

The Rudin–Keisler (and in the case the space S is countable, the Rudin–Frolı́k) order of the Stone–Čech remainder βS\S of the discrete space S has often been studied, yielding much useful information about βS. More recently, the Comfort order has been introduced. If (S,·) is a semigroup, then the operation · extends naturally to βS, and the study of the semigroup (βS,·) is both fascinating in its own right and useful in terms of applications to Ramsey Theory. s paper, we study the Rudin–Keisler and Comfort orders on βS\S when S is a semigroup. We show, for example, that the set of Comfort predecessors of a given point p∈βS\S is always a subsemigroup of βS, while if S is cancellative, the set of Rudin–Keisler predecessors of a point p is never a subsemigroup.