Semigroup Algebra at Infinity and its Combinatorial Applications

一个紧右拓扑半群是一个集S，它是一个半群和一个拓扑空间，并且具有这样的性质：S的任何固定元素在右边的乘法是连续的。 紧右拓扑半群保证具有一些著名的代数性质，其中主要的是存在一个最小的双边理想，它是（通常是无穷多个）两两同构群的并。给定一个无限半群S，例如正整数的集合N在加法作用下，S的最大可能紧化，即它的Stone-Cech紧化，有一个自然半群运算扩展S的自然半群运算，使它成为紧右拓扑半群。 在大多数合理半群中，包括所有（左或右）可消半群，Stone-Cech紧化的最小理想包含在Stone-Cech剩余中。 此外，人们通常会发现斯通-切赫紧化的大部分或全部幂等元都存在于这个余数中。 Stone-Cech余项的代数结构是本提案标题中的“无穷代数”。虽然我们对无穷半群的无穷代数有很多了解，但仍有许多有趣且非常自然的问题有待解决。另一个原因感兴趣的代数在无穷远的无限半群是重大后果拉姆齐理论是（通常很容易）获得。 拉姆齐理论是组合学的一部分，它处理的问题是，人们可以期望在指定集合的有限划分的某个单元中找到什么样的齐次结构（或者有时在任何适当的“大”子集中）。 拉姆齐理论可以被认为是一个最简单的数学陈述，“鸽子洞原理”的推广。 这个原则是说，如果信件被分配到各个鸽子洞中，并且信件比鸽子洞多，那么一些鸽子洞将得到不止一个字母。 拉姆齐理论中最简单的陈述说，如果有六个人参加聚会，那么要么有三个人以前从未见过面，要么有三个人，每对都见过面。 尽管（或者可能是因为）它的一些基本语句的简单性，拉姆齐理论在数学的许多领域都有广泛的应用：数论，逻辑，代数和Banach空间理论仅举几例。 首席研究员和他的学生研究如何应用“代数在无穷大”的半群，以获得新的成果拉姆齐理论。 他们研究离散半群的斯通-切赫紧化的代数结构，推导出对这种结构本身的新理解，并获得新的应用。

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.