Universal Algebra

Chapter 1

Algebras

This first, short, chapter sets the stage for the entire text. Many of the fundamental notions and several central examples are introduced. This will allow us to provide examples and motivation for the deeper ideas we encounter in the later chapters.

1.1 Operations

Let A be a set and n a positive integer. We define Aⁿ to be the set of all n-tuples of elements of A⁰, and A⁰ = {∅}.

Remark. More generally, for any two sets A and B, A^B denotes the set of all functions from B to A. How do you reconcile this with the definition of Aⁿ just given? In particular, what about A⁰? What does the definition tell us about the case A = ∅?

For any A and n as above, we call a function Aⁿ → A an n-ary operation on A. The natural number ...