Algebra in Stone-Cech Compactifications and its Combinatorial Applications

主要研究者：Hindman，Neil 提案编号：DMS -0852512机构：霍华德大学题目：Stone-Cech紧化中的代数及其组合应用该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。右拓扑半群是一个集合S，它是一个半群和拓扑空间，并且具有S的任何固定元素在右边的乘法是连续的性质。 如果S是一个离散半群，那么它的Stone-Cech紧化自然地是一个紧Hausdorff右拓扑半群。与任何紧Hausdorff右拓扑半群一样，这个Stone-Cech紧化有一个最小的双边理想，并且这个理想通常具有复杂的代数结构。该项目涉及研究斯通-切赫紧化的代数结构及其最小理想，并调查该结构的组合应用，主要是拉姆齐理论。拉姆齐理论是组合学的一部分，它处理的问题是什么样的齐次结构，人们可以期望在指定集合的有限划分（或“着色”）的某个单元中找到。例如，拉姆齐定理的无限版本的最简单的非平凡例子说，每当正整数集合N的两个元素子集是双着色的时，必须存在N的某个无限子集，其所有两个元素子集都是相同的颜色。首席研究员获得了一些名声多年前，当他证明，每当N是彩色的，必须存在于一个颜色的无限序列连同其所有的有限和不同的条款没有重复。最初的证明是基本的，但非常复杂。但是在1975年，Fred Galvin和Steven Glazer证明了这个“有限和定理”是一个完全平凡的结果，因为N的Stone-Cech紧化可以被赋予一个代数结构，该代数结构扩展了作为紧右拓扑半群的普通加法，因此具有幂等元，即使得p + p = p的元素。Stone-Cech紧化的代数结构在Ramsey理论中的许多其他应用也被发现。

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.