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