| Topology Atlas Document # zaaa-48 | Topology Atlas Invited Contributions vol. 1 issue 2 (1996) p. 6 |
Howard University, Washington, DC 20059-0001, U.S.A.
Homepage: http://members.aol.com/nhindman/
Given a discrete semigroup (S, ×), the operation × can be extended to the Stone-Cech compactification bS of S in such a way that (bS, ×) becomes a right topological semigroup with dense topological center. This fact was implicitly established by M. Day [Amenable Semigroups, Ill. J. Math. 1(1959), 509-544] using methods of R. Arens. The first explicit statement of this fact seems to have been made by P. Civin and B. Yood [The second conjugate space of a Banach algebra as an algebra, Pac. J. Math. 11(1961), 847-870]. The algebraic structure of bS has now been extensively studied. It has also been a very powerful tool in the branch of combinatorics known as Ramsey Theory, especially in the case S is the semigroup (bN, +).
See the book [Algebra in the Stone-Cech Compactification, de Gruyter, Berlin, 1988], by Dona Strauss and this writer, for an elementary introduction to the algebraic structure of bS and its applications to Ramsey Theory. In addition to the elementary introduction, this book surveys the major developments in the area through December of 1997. There is an associated bibliography which is an updated (and expanded) version of the bibliography in the book.
Among the more exciting relatively recent results is E. Zelenuk's proof that (bN, +) contains no nontrivial finite groups. By contrast, it has been known for some time that (bN, +) contains copies of the free group on 2c generators.
A related problem is probably the most significant unanswered question about the algebraic structure of (bN, +):
Are there any nontrivial continuous homomorphisms from (bN, +) into its Stone-Cech remainder? Equivalently, do there exist distinct p and q in the Stone-Cech remainder such that p + p = q + p = p + q = q + q = q?See [ Algebra in the Stone-Cech compactification, Corollary 10.20] for this equivalence.
Recieved: December 11, 1995. Revised: December 2000.