Semigroup Algebra at Infinity and its Combinatorial Applications

紧右拓扑半群是一个集S，它是一个半群，是一个拓扑空间，并且具有右乘于S的任何固定元是连续的性质。紧右拓扑半群被保证具有一些众所周知的代数性质，其中的主要性质是存在一个最小的双边理想，它是(通常无穷多个)成对同构群的并。给定无限半群S，如加法下的正整数集N，S的最大可能紧化，即它的Stone-Cech紧化，有一个自然的半群运算，推广了S的运算，使其成为紧的右拓扑半群.在大多数有理半群中，包括所有(右或左)可消半群，Stone-Cech紧化的最小理想包含在Stone-Cech余项中。此外，人们通常会发现大部分或全部的Stone-Cech紧化的幂等元都在这个余项中。Stone-Cech余数的代数结构是这个提议标题的“无穷大代数”。虽然关于无限半群在无穷远处的代数已知很多，但许多有趣且非常自然的问题仍然悬而未决。对无限半群的无穷远处的代数感兴趣的另一个原因是Ramsey理论中的重要结果，这些结果在那里(通常很容易)可以得到。拉姆齐理论是组合学的一部分，它处理的问题是，人们可以在特定集合的有限划分的某个单元中(有时在任何适当的“大”子集中)找到什么样的同质结构。拉姆齐理论可以被认为是最简单的数学表述之一--“鸽子原理”的推广。这一原则是说，如果信件被分配到文件夹中，并且信件比文件夹多，那么一个文件夹将得到不止一个字母。拉姆齐理论中最简单的说法是，如果一个派对上有六个人，那么要么有三个人，他们以前都没有见过面，要么有三个人，每个人都见过面。尽管它的一些基本陈述很简单(或者可能是因为)，拉姆齐理论已经在许多数学领域得到了广泛的应用：数论、逻辑、代数和Banach空间理论等等。主要研究人员和他的学生研究如何应用半群的“无穷代数”在Ramsey理论中获得新的结果。他们研究了离散半群的Stone-Cech紧化的代数结构，对这种结构本身有了新的理解，并得到了新的应用。

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.