Definition:Affine Space/Associativity Axioms

Definition
Let $K$ be a field.

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

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


 * $+ : \mathcal E \times V \to \mathcal E$
 * $- : \mathcal E \times \mathcal E \to V$

satisfying the following associativity conditions:

Then the ordered triple $\tuple{\mathcal E, +, -}$ 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 $+: \mathcal E \times V \to \mathcal E$ in the affine space.

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

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

Also see

 * Equivalence of Definitions of Affine Space