Orderings of the Stone–Čech remainder of a discrete semigroup

✍ Scribed by Salvador Garcı́a-Ferreira; Neil Hindman; Dona Strauss

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 214 KB
• Volume: 97
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00073-x

✦ Synopsis

The Rudin-Keisler (and in the case the space S is countable, the Rudin-Frolík) 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.

In this 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.