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:

\((\text A 1)\)	$:$			\(\ds \forall p, q \in \EE:\)	\(\ds p + \paren {q - p} = q \)
\((\text A 2)\)	$:$			\(\ds \forall p \in \EE: \forall u, v \in V:\)	\(\ds \paren {p + u} + v = p + \paren {u +_V v} \)
\((\text A 3)\)	$:$			\(\ds \forall p, q \in \EE: \forall u \in V:\)	\(\ds \paren {p - q} +_V u = \paren {p + u} - q \)

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