We may as well begin at the beginning and this will involve us in a brief excursion through some of the fundamental concepts essential for any algebraic subject. It will also enable us to become acquainted with the notation used, although the experienced reader could easily skip through this chapter. We will assume that the reader has a knowledge of elementary set theory.

Relations

One of the fundamental concepts in mathematics is that of a relation. It can be introduced in a variety of ways but the most useful one for us is the following abstract approach.

Let A be a non-empty set. A relation, ℛ, on A is a subset ℛ ⊆ A × A. If (a, a′) ∈ A × A and (a, a′) ∈ ℛ we say that a is ℛ-related to a′. Sometimes a natural notation is used in mathematics to express this relationship between two elements of a set, for example if A = ℤ, the set of all integers, then there is a relation ≤ that can be defined on ℤ. We write a ≤ a′ if the number a′ − a is not negative and the set ℛ ⊆ ℤ × ℤ defining this relation consists of all ordered pairs (a, a′)∈ ℤ × ℤ such that a < a≤.

A relation ℛ on the set A is an equivalence relation if:

1. (i)(a, a) ∈ ℛ for all a ∈ A

2. (ii) (a, a′)∈ ℛ ⇒(a′, a) ∈ ℛ for a, a′ ∈ A…

3. […]