Finiteness Conditions and Index in Semigroups and Monoids

半群和幺半群中的有限性条件和索引

• 批准号: EP/E043194/1
• 负责人: Robert Gray
• 金额: $ 25.87万
• 依托单位: University of St Andrews
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE043194%2F1
• 关键词: Finiteness; Conditions; Index; Semigroups; Monoids

A semigroup is one of the most simple, and fundamental, of mathematical objects. The ingredients of a semigroup are a set (i.e. a collection of symbols) along with an operation, often called multiplication, defined on this set (i.e. a method for combining pairs of elements from the set to get new elements from that set). For a semigroup this operation must be associative, which means that when we multiply a string of elements from the set together it does not matter how the terms are bracketed. A very easy example is to take the set of natural numbers 1, 2, 3, ... etc. along with the operation of addition +. Of course, if a, b and c are natural numbers then (a+b)+c = a+(b+c) and so this gives an example of a semigroup. Far more complicated and interesting examples of semigroup exist than this one. One thing that does make this example slightly interesting is the fact that it is an infinite semigroup. A more interesting example of an infinite semigroup is a so called free semigroup . We begin with a set A called an alphabet, say for example we let A be the set containing the letters a,b and c. We then consider all words we can make by stringing together letters of the alphabet (note that these are not words in the usual sense, since they do not need to have any meaning). In our example abc is a word, as is bbcabcbcba. If we take the set of all possible words along with the operation of concatenation (joining together) of words then we obtain a semigroup, called the free semigroup over the alphabet A. So for example we can multiply the word abc with the word bcc to obtain the word abcbcc. Taking this one stage further we come to the concept of a semigroup presentation . A semigroup presentation is given by an alphabet, like we had for the free semigroup above, along with a set of pairs of words R called relations. The pairs of words in R are usually written with an equals sign separating them. For example we could take A to be the set with a,b and c as our alphabet, as above, and let R be the set of relations abc = a and bca = a. These relations may now be applied to words transforming one word into another. For example, we can apply the relation abc = a to the word cabcabcccbc to obtain the word cabcaccbc (we replaced abc which appears in the middle of the first word by the word a since abc = a is one of our relations). In this way we create sets of words that are equivalent to one another in the sense that we can move between them by applying the rules from R. We can now consider these sets of words as objects and, in the natural way, we can define an operation of multiplication on these objects. The resulting structure is a semigroup and we call it the semigroup defined by the presentation (A,R). If the sets A and R may be chosen to be finite then the semigroup is said to be finitely presented . Every finite semigroup is finitely presented but there are also many infinite semigroups that are also finitely presented. As a result presentations are a very useful tool for working with infinite semigroups because, in many situations, they give us a way of representing an infinite object, the semigroup, using a finite amount of information, the presentation. This research project is centred around the study of infinite semigroups via presentations. Given a semigroup, any other semigroup that can be found inside that semigroup is called a subsemigroup. One of the main aims of this research project is to consider the relationship between the properties of infinite semigroups (represented using presentations) and those of its subsemigroups. In particular my interest is in developing methods for measuring the difference in size between a semigroup and its substructures. This measurement should have the property that when the semigroup and subsemigroup are measured to be close together they will share may algebraic, combinatorial and computational properties.

半群是数学对象的最简单，最基本的之一。半群的成分是一组（即符号集合）以及在此集合上定义的操作（通常称为乘法）（即一种组合集合中元素对的方法，从该集合中获取新元素）。对于半群，此操作必须是关联的，这意味着，当我们将集合中的一串元素乘在一起时，术语的括号如何都无关紧要。一个非常简单的例子是将自然数量1、2、3，...等以及添加 +的操作加上。当然，如果a，b和c是自然数，则（a+b）+c = a+（b+c），因此给出了一个半群的示例。比这更复杂且有趣的示例。确实使这个例子稍微有趣的一件事是它是一个无限的半群。无限半群的一个更有趣的例子是所谓的免费半群。我们从一个称为字母的集合开始，例如，我们让A是包含字母A，B和C的集合。然后，我们通过将字母的字母串在一起来考虑我们可以制作的所有单词（请注意，这些单词在通常的意义上不是单词，因为它们不需要任何含义）。在我们的示例中，ABC是一个单词，bbcabcbcba也是如此。如果我们将所有可能单词的集合以及单词的串联操作（一起）进行操作，那么我们获得了一个半群，称为Alphabet A上的自由半群。进一步迈出了一个阶段，我们来到了半群演讲的概念。像上面的自由半群一样，字母表以及一组称为关系的单词r。 r中的一对单词通常用平等的符号将它们分开。例如，我们可以将A作为字母作为字母（如上所述）作为a，b和c的集合，让r为一组abc = a和bca = a。这些关系现在可以应用于将一个单词转换为另一个单词的单词。例如，我们可以将关系abc = a应用于cabcabccccbc一词以获取cabcaccbc一词（我们替换了abc在第一个单词中间出现的abc a a a a abc = a是我们的关系之一）。通过这种方式，我们创建了一组等同于彼此的单词，从而可以通过应用R的规则在它们之间移动。我们现在可以将这些单词集视为对象，并且以自然的方式可以定义对这些对象的乘法操作。最终的结构是一个半群，我们称其为由演示文稿定义（a，r）定义的半群。如果可以选择A和R的集合为有限，则据说该半群是有限的。每个有限的半群都有限地呈现，但也有许多无限的半群也有限地呈现。结果，演示是一种非常有用的工具，可用于使用无限的半群，因为在许多情况下，它们为我们提供了一种使用有限的信息（演示文稿）来表示无限对象，即半群。该研究项目围绕通过演示进行无限半群的研究。给定一个半群，可以在该半群中发现的任何其他半群都称为subsemigroup。该研究项目的主要目的之一是考虑无限半群（使用演示文稿表示）的性质与其子群的性质之间的关系。特别是我的兴趣是开发用于测量半群及其子结构之间大小差异的方法。该测量应具有以下特性，即当测量半群和亚群时，他们将共享可能具有代数，组合和计算属性。

项目成果

On maximal subgroups of free idempotent generated semigroups

• DOI: 10.1007/s11856-011-0154-x
• 发表时间: 2011-09
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: R. Gray;N. Ruškuc

Locally-finite connected-homogeneous digraphs

局部有限连通齐次有向图

• DOI: 10.1016/j.disc.2010.12.017
• 发表时间: 2011
• 期刊: Discrete Mathematics
• 影响因子: 0.8
• 作者: Gray R

Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

可数代数闭图的自同构群和随机图的自同态

• DOI: 10.1017/s030500411500078x
• 发表时间: 2016
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: DOLINKA I

Groups acting on semimetric spaces and quasi-isometries of monoids

作用于半群空间和幺半群拟等距的群

• DOI: 10.1090/s0002-9947-2012-05868-5
• 发表时间: 2012
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Gray R

IDEALS AND FINITENESS CONDITIONS FOR SUBSEMIGROUPS

子半群的理想和有限条件

• DOI: 10.1017/s0017089513000086
• 发表时间: 2013
• 期刊: Glasgow Mathematical Journal
• 影响因子: 0.5
• 作者: GRAY R