Finiteness Conditions and Index in Semigroups and 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.