Definition:Affine Space/Associativity Axioms

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

Also see