Definition:Affine Space/Associativity Axioms

Definition
Let $K$ be a field.

Let $\struct {V, +_V, \circ}$ be a vector space over $K$.

Let $\EE$ be a set on which two mappings are defined:


 * $+ : \EE \times V \to \EE$
 * $- : \EE \times \EE \to V$

satisfying the following associativity conditions:

Then the ordered triple $\struct {\EE, +, -}$ is an affine space.

Notation
Almost invariably the same symbol (usually $+$) is used for the addition $+_V: V \times V \to V$ in the vector space and the addition $+: \EE \times V \to \EE$ in the affine space.

This does not allow any ambiguity as the two mappings have different domains.

For elements $p, q \in \EE$, it is common to write $\vec {p q} = q - p$.

Also see

 * Equivalence of Definitions of Affine Space