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.

半群是最简单、最基本的数学对象之一。半群的成分是一个集合（即符号的集合）以及在此集合上定义的通常称为乘法的运算（即组合集合中的元素对以从该集合中获取新元素的方法）。对于半群，此运算必须是关联的，这意味着当我们将集合中的一串元素相乘时，这些项如何括起来并不重要。一个非常简单的例子是采用自然数集合 1, 2, 3, ... 等以及加法 + 运算。当然，如果 a、b 和 c 是自然数，则 (a+b)+c = a+(b+c)，因此这给出了半群的示例。存在比这个更复杂、更有趣的半群例子。确实使这个例子有点有趣的一件事是它是一个无限半群。无限半群的一个更有趣的例子是所谓的自由半群。我们从称为字母表的集合 A 开始，例如我们让 A 为包含字母 a、b 和 c 的集合。然后，我们考虑通过将字母表中的字母串在一起可以组成的所有单词（请注意，这些不是通常意义上的单词，因为它们不需要具有任何含义）。在我们的示例中，abc 是一个单词，bbcabcbcba 也是一个单词。如果我们采用所有可能单词的集合以及单词的串联（连接在一起）操作，那么我们将获得一个半群，称为字母表 A 上的自由半群。例如，我们可以将单词 abc 与单词 bcc 相乘以获得单词 abcbcc。进一步推进这一阶段，我们提出了半群演示的概念。半群表示由字母表给出，就像我们上面的自由半群一样，以及一组称为关系的词对 R 。 R 中的单词对通常用等号分隔。例如，如上所述，我们可以将 A 视为以 a、b 和 c 作为字母表的集合，并令 R 为关系 abc = a 和 bca = a 的集合。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 应用于单词 cabcabcccbc 以获得单词 cabcaccbc （我们将出现在第一个单词中间的 abc 替换为单词 a，因为 abc = a 是我们的关系之一）。通过这种方式，我们创建了彼此等价的单词集，因为我们可以通过应用 R 中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，我们可以定义这些对象的乘法运算。由此产生的结构是一个半群，我们将其称为由表示（A，R）定义的半群。如果集合 A 和 R 可以被选择为有限的，则半群被称为有限呈现。每个有限半群都是有限表示的，但也有许多无限半群也是有限表示的。因此，表示对于处理无限半群来说是一个非常有用的工具，因为在许多情况下，它们为我们提供了一种使用有限数量的信息（即表示）来表示无限对象（半群）的方法。该研究项目的重点是通过演示研究无限半群。给定一个半群，可以在该半群内找到的任何其他半群称为子半群。该研究项目的主要目标之一是考虑无限半群的性质（使用演示表示）与其子半群的性质之间的关系。我特别感兴趣的是开发测量半群及其子结构之间大小差异的方法。这种测量应该具有以下性质：当半群和子半群被测量为接近时，它们将共享可能的代数、组合和计算性质。

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