Semigroup Algebra at Infinity and its Combinatorial Applications

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A compact right topological semigroup is a set S which is a semigroup and a topological space and has the property that multiplication on the right by any fixed element of S is continuous. Compact right topological semigroups are guaranteed to have some well known algebraic properties, principal among which is the existence of a smallest two sided ideal which is the union of (usually infinitely many) pairwise isomorphic groups. Given an infinite semigroup S, such as the set N of positive integers under addition, the largest possible compactification of S, its Stone-Cech compactification, has a natural semigroup operation extending that of S which makes it a compact right topological semigroup. In most reasonable semigroups, including all of the (right or left) cancellative semigroups, the smallest ideal of the Stone-Cech compactification is contained in the Stone-Cech remainder. Also, one usually finds most, or all, of the idempotents of the Stone-Cech compactification lying in this remainder. The algebraic structure of the Stone-Cech remainder is the "algebra at infinity" of the title of this proposal. While much is known about the algebra at infinity of infinite semigroups, many fascinating and very natural questions remain open. Another reason for interest in the algebra at infinity of infinite semigroups is the significant consequences in Ramsey theory that are (usually quite easily) obtainable there. Ramsey Theory is that part of combinatorics that deals with the question of what sort of homogeneous structures one can expect to find in some one cell of a finite partition of specified sets (or sometimes in any suitably "large" subset). Ramsey Theory may be thought of as a generalization of one of the simplest mathematical statements, the "pigeonhole principle". This principle says that if letters are being distributed among pigeonholes and there are more letters than pigeonholes, then some pigeonhole will get more than one letter. The simplest statement in Ramsey Theory says that, if six people are at a party, then either there will be some three, none of whom have met before, or there will be some three, each pair of which have met each other. In spite of (or maybe because of) the simplicity of some of its basic statements, Ramsey Theory has had widespread applications throughout many areas of mathematics: number theory, logic, algebra, and Banach space theory to name a few. The principal investigator and his students study ways to apply the "algebra at infinity" of semigroups to obtain new results in Ramsey Theory. They investigate the algebraic structure of the Stone-Cech compactification of discrete semigroups, deriving new understandings of this structure itself, and obtaining new applications.