Algebra in Stone-Cech Compactifications and its Combinatorial Applications

摘要主要研究者：Hindman， Neil提案编号：DMS - 0852512机构：Howard大学题目：石-赫紧化代数及其组合应用本奖项由2009年美国复苏与再投资法案（公法111-5）资助。右拓扑半群是一个集合S，它既是半群又是拓扑空间，并且具有S的任意固定元素在右侧的乘法是连续的性质。如果S是一个离散半群，那么它的stone - ech紧化，在自然的情况下，就是一个紧化的Hausdorff右拓扑半群。与任何紧化Hausdorff右拓扑半群一样，这种Stone-Cech紧化具有最小的双边理想，并且这种理想通常具有精细的代数结构。该项目涉及对stone - ech紧化的代数结构及其最小理想的研究，以及对该结构的组合应用的研究，主要是对Ramsey理论的研究。拉姆齐理论是组合学的一部分，它处理的问题是，在特定集合的有限划分（或“着色”）的某个单元中，人们可以期望找到什么样的齐次结构。例如，拉姆齐定理的无限版的最简单的非平凡实例说，每当正整数集合N的两个元素子集是有限颜色时，必然存在N的某个无限子集，它的两个元素子集都是相同的颜色。这位首席研究员在许多年前获得了一些名声，因为他证明了当N是有限着色时，在一种颜色中必然存在一个无限序列及其所有不同项的有限和而不重复。最初的证明很简单，但很复杂。但是在1975年，Fred Galvin和Steven Glazer证明了这个“有限和定理”是一个完全平凡的事实，即N的Stone-Cech紧化可以给出一个扩展普通加法的代数结构，这是一个紧的右拓扑半群，因此具有幂等，即p + p = p的元素。从那时起，发现了Stone-Cech紧化的代数结构在Ramsey理论中的许多其他应用。

ABSTRACTPrincipal Investigator: Hindman, Neil Proposal Number: DMS - 0852512Institution: Howard UniversityTitle: Algebra in Stone-Cech Compactifications and its Combinatorial ApplicationsThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).A 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. If S is a discrete semigroup, then its Stone-Cech compactification is, in a natural way, a compact Hausdorff right topological semigroup. As is true of any compact Hausdorff right topological semigroup, this Stone-Cech compactification has a smallest two sided ideal, and this ideal usually has elaborate algebraic structure. This project involves the study of the algebraic structure of the Stone-Cech compactification and its smallest ideal and investigation of the combinatorial applications of that structure, primarily to Ramsey Theory.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 (or "coloring") of specified sets. For example, the simplest nontrivial instance of the infinite version of Ramsey's Theorem says that whenever the two element subsets of the set N of positive integers are finitely colored, there must be some infinite subset of N all of whose two element subsets are the same color. The principal investigator gained some fame many years ago when he proved that whenever N is finitely colored, there must exist in one color an infinite sequence together with all of its finite sums of distinct terms without repetition. The original proof was elementary, but very complicated. But in 1975, Fred Galvin and Steven Glazer showed that this "Finite Sums Theorem" is a completely trivial consequence of the fact that the Stone-Cech compactification of N can be given an algebraic structure extending ordinary addition which is a compact right topological semigroup, and therefore has idempotents, that is elements such that p + p = p. Since then, numerous other applications of the algebraic structure of Stone-Cech compactifications to Ramsey Theory have been found.