# 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:

 $(A1)$ $:$ $\displaystyle \forall p, q \in \mathcal E:$ $\displaystyle p + \left({q - p}\right) = q$ $(A2)$ $:$ $\displaystyle \forall p \in \mathcal E: \forall u, v \in V:$ $\displaystyle \left({p + u}\right) + v = p + \left({u +_V v}\right)$ $(A3)$ $:$ $\displaystyle \forall p, q \in \mathcal E: \forall u \in V:$ $\displaystyle \left({p - q}\right) +_V u = \left({p + u}\right) - q$

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 $\vec{p q} = q - p$.