Algebra in the Stone-Čech Compactification by Neil Hindman and Dona Strauss

Abstract

This is an excellent book. This review is an attempt to convince the reader that this verdict is not the prejudice of an enthusiast but a sober, sound judgement. The title might suggest that the subject matter of the book is rather esoteric. The Stone-Cech compactication is a remote object. Its elements are ultralters and the existence of non-trivial ultralters depends on the axiom of choice. This means that an interesting ultralter has never been seen, and although analysts may not be troubled by objects which can only be imagined, a down-to-earth algebraist or number theorist might wonder why time should be wasted reading about them. In fact ultralters have a very desirable property: if the positive integers N are partitioned into a nite number of sets, then any ultralter on N will pick out exactly one set from the partition. From this observation, a really wild speculation would be that maybe ultralters could be used to prove van der Waerden’s Theorem, that given an integer ‘ and a nite partition of N; there is a set in the partition which contains an arithmetic progression of length ‘: A proof of this kind was published in 1989 by V Bergelson, H Furstenberg, Y Katznelson and N Hindman. It is essentially easy. I myself have presented it to a general audience in a one hour lecture, including background theory semigroup theorists already know. The Stone-Cech compactication N { the set of all ultralters on N with a natural topology which happens to be compact { is an essential ingredient in this proof. Whole books have been devoted to this compactication as a topological object since its rst appearance in 1937. But for applications to number theory, the addition of N has to be extended to N: Opinions dier about when this was rst done. M M Day certainly could without diculty have added it to his 1957 paper on amenable semigroups, but he did not. In 1963, P Civin and B Yood mentioned that as a consequence of their study of second dual Banach algebras Z has a natural semigroup structure but they did nothing with it. The real beginnings of the theory may therefore be ascribed to R Ellis [3]. He showed that S is a semigroup for any discrete group S using an ultralter approach, and he did so because he needed that semigroup structure in topological dynamics. There is good reason even for number theorists to consider general semigroups S : the cases (N;+); (N; ); (Z;+) etc can all be dealt with at the same time. Moreover if S is a semigroup under two operations + and simultaneously then both extend to S. That is why the present book is