Semigroup Algebra at Infinity and its Combinatorial Applications

Howard University, Washington DC

Investigators

Abstract

Abstract for award of Hindman DMS-0243586 The principal investigator and his students will continue the investigation of the algebraic scructure of the Stone-Cech compactification of a discrete semigroup and its applications to Ramsey Theory. Significant progress continues to be made in both areas, but much remains to be done. Among the major algebraic problems which is still unsolved is whether there is a nontrivial continuous homomorphism from the Stone-Cech compactification of the positive integers to its remainder, N*. That question can be stated equivalently as whether there is a finite subsemgroup of N* whose members are not all idempotents. The proposed algebraic studies have significant potential applications to Ramsey Theory -- that part of combinatorics which establishes the existence of highly regular substructures of given structures that are partitioned into finitely many classes (or "finitely colored"). One of the earliest applications of the algebra of the Stone-Cech compactification to Ramsey Theory was a simple proof of the following statement: If the positive integers are colored red and blue, then there is either a sequence all of whose sums (without repetition) are red or there is a sequence all of whose sums are blue. Additional applications continue to be discovered.