Is there an algebraic structure for affine spaces

Question

So in normal Vector spaces, you have vectors and scalars, where the vectors form a field, and scalars form an abelian group.

But in affine spaces you have the notion of point as opposed to vector, and while it makes sense to add vectors to each other, there is no such notion of adding points to each other, however you can subtract points to get vectors. And while you can multiply scalars with vectors, it doesn't make sense to multiple scalars with points.

I wonder if there is a algebraic structure that generalizes this notion, because I feel like it would be very useful after reading this article.

Answer 1

EDIT: I think the list of axioms in this answer is slightly incomplete and I don't know how to fix it. But I'll leave the answer up since I think it's mostly right, and informative about the right way to think about things.

In a vector space the most general kind of operation you can do is a linear combination, i.e. if you have a list of scalars $(a_1,\dots,a_n)$ and a list of vectors $(v_1,\dots,v_n)$ then you can form the vector

$$\sum_{i=1}^na_iv_i$$

In an affine space linear combinations don't make sense. For example $0.5v$ means "the point halfway between the origin and $v$", but there is no origin in the affine space. But if the coefficients $(a_i)_1^n$ sum to $1$ the linear combinations do make sense. For example $0.75v+0.25w$ means the point a quater of the way from $v$ to $w$, which is in the affine space. Linear combinations where the coefficients sum to $1$ are called "affine combinations".

Affine spaces can be defined abstractly using affine combinations with two arguments. We define an affine space over $\mathbb R$ to be a set $A$ equipped with a function $c_p:A\times A\to A$ for each scalar $p\in\mathbb R$ such that the following axioms hold

$$\forall x,y \quad c_0(x,y)=x$$ $$\forall x,p \quad c_p(x,x)=x$$ $$\forall x,y,p\quad c_p(x,y)=c_{1-p}(y,x)$$ $$\forall x,y,z,p,q \quad \text{either} \quad c_p(c_q(x,y),z)=c_{pq}(x,c_{\frac{p-pq}{1-pq}}(y,z)) \quad \text{or} \quad1-pq=0$$

The point is that $c_p(x,y)$ is supposed to be interpreted as $(1-p)x+py$. Indeed if $A$ is an affine subspace of a vector space then $A$ can be given the above abstract structure by letting $c_p(x,y)=(1-p)x+py$. Conversely, you can check that if you pick any point $o\in A$ then $A$ can be made into a vector space with $o$ as the origin by taking the scalar multiple of $x$ by $p$ to be $c_p(o,x)$ and by taking the sum of $x$ and $y$ to be $c_2(o,c_{\frac 12}(x,y))$.

Answer 2

A related answer. In geometry, there is a construction of scalars for a geometry.

You start with an $n$-dimensional affine geometry, specified by geometric axioms, with no mention of coordinates. Then you construct from it a field $K$ so that the geometry is identified with $K^n$.

An interesting tidbit: if the dimension $n$ is ${}\ge 3$ the field $K$ is commutative as expected. But if $n=2$ you could get a non-commutaitive "skew field" or "division ring". Unless you add Desargues's Theorem as an axiom, which then insures that $K$ is commutative.