Finiteness Conditions and Index in Semigroups and Monoids

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）沿着一个运算，通常称为乘法，定义在这个集合上（即组合集合中的元素对以从该集合中获得新元素的方法）。对于一个半群，这个运算必须是结合的，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取自然数1，2，3，.的集合。沿着加法+的运算等。当然，如果a，B和c是自然数，那么（a+B）+c = a+（B+c），这就是半群的一个例子。还有比这更复杂、更有趣的半群例子。有一件事确实让这个例子有点有趣，那就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。我们开始从一个叫做字母表的集合A开始，比方说我们让A是包含字母a，B和c的集合。然后，我们考虑我们可以通过将字母表中的字母串在一起来组成的所有单词（注意，这些不是通常意义上的单词，因为它们不需要有任何意义）。在我们的例子中，abc是一个单词，bbcabcbcba也是。如果我们沿着所有可能的词的集合以及词的连接（连接在一起）运算，那么我们就会获得一个半群，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，得到单词abcbcc。在这一阶段，我们进一步得到半群表示的概念。一个半群的表示是由一个字母表给出的，就像我们上面对自由半群所做的那样，沿着一组被称为关系的词对R。R中的成对单词通常用等号分隔。例如，我们可以取A为a、B和c的集合作为我们的字母表，如上所述，设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词转化为另一个词的词。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，我们创建了一组彼此等价的词，在这个意义上，我们可以通过应用R中的规则在它们之间移动。我们现在可以把这些词的集合看作对象，并且以自然的方式，我们可以在这些对象上定义乘法运算。由此产生的结构是一个半群，我们称之为由表示（A，R）定义的半群。如果集合A和R可以被选择为有限的，则半群被称为是有限的。每一个有限半群都是双表示的，但也有许多无限半群也是双表示的。因此，表示是处理无限半群的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信息量表示无限对象（半群）的方法。这个研究项目是围绕通过演示无穷半群的研究。给定一个半群，在该半群内可以找到的任何其他半群都称为子半群。这个研究项目的主要目的之一是考虑无限半群（用表示法表示）和其子半群的性质之间的关系。特别是我的兴趣是在发展的方法来衡量之间的差异大小的半群和其子结构。这种度量应该具有这样的性质，即当半群和子半群被度量为靠近在一起时，它们将共享可能的代数、组合和计算性质。

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.