3. Fundamental algebraic structures

3.1 Introduction

A set in itself is of little interest. What can make it richer are the relationships between its elements (for example, an order relationship), and the operations that can be performed on some of them. An operation has an arity: this is the number of operands it brings into play. In a 3-dimensional Euclidean vector space, the mixed product has arity 3, while the vector product has arity 2. The passage to the opposite (noted by the minus sign, known as "minus unary") is of arity 1. In reality, basic operations (sum, product, difference, etc.) are usually of arity 2. It's these binary operations that we're primarily interested in. Of course, an operation cannot be just any application of two variables. We'll see, in the structures we'll be studying (essentially those...