Informally, we can say that a vector space (also called a linear space) is an commutative group with a scalar multiplication, while an affine space is a commutative heap with a scalar multiplication. Another possibility is to define an affine space as a set of points with a semigroup on its set of bipoints (=pairs of points) together with a scalar multiplication. All these options are systematically explored below.
For ease of comparison, I will start with usual definitions of a group and a vector space.
Definition 0.
A group is a non-empty set $G$ equipped with a binary operation $$\mathbin{\star} : G\times G \to G, (a, b) \mapsto a\star b $$ that satisfies the following conditions:
- G$_1$ (associativity) for all $a, b, c \in G$, one has
$ (a \star b) \star c = a \star (b \star c);$
- G$_2$ (identity element) there exists an element $e \in G$ such that for all $a\in G$ one has $ a \star e = a = e \star a ;$
- G$_3$ (inverse elements) for each $a \in G$, there exists an element $b \in G$ such that
$a \star a' = e = a' \star a,$ where $e$ is the identity element.
A group is said to be commutative (or abelian) if in addition the following condition is satisfied:
- G$_4$ (commutativity) for all $a, b \in G$, one has $ a\star b = b\star a$.
Definition 1. Suppose $\mathbb{K}$ is a field.
A vector space over $\mathbb{K}$ is an additive abelian group $X$ equipped with a scalar multiplication $$ \cdot :\mathbb{K} \times X \to X, (a,v) \mapsto a \cdot v$$ that satisfies the following conditions:
- V$_1$ (distributivity of scalar multiplication over group addition) for all $a\in \mathbb{K}$ and for all $u, v \in X$, one has $a\cdot(u+v)=a\cdot u+a\cdot v$,
- V$_2$ (distributivity of scalar multiplication over field addition) for all $a, b\in \mathbb{K}$ and for all $v \in X$, one has $(a+b)\cdot v =a\cdot v+ b\cdot v$,
- V$_3$ (compatibility of scalar and field multiplication) for all $a, b\in \mathbb{K}$ and for all $v \in X$, one has $(ab)\cdot v =a\cdot ( b\cdot v)$,
- V$_4$ (identity element of scalar multiplication) for all $v \in X$, one has $1_\mathbb{K} \cdot v = v$.
Definition 2.
A heap is a non-empty set $H$ equipped with a mapping
\begin{align*}
- \mathbin {\star}(-,-) : H \times H^2 &\to H,\\
(a, (b,c)) &\mapsto a \star (c,d)
\end{align*}
that satisfies the following conditions:
- H$_1$ (associativity) for all $a,b,c,d,e \in H$, one has $$(a\star(b, c)) \star (d, e) = a \star(b, c \star (d, e));$$
- H$_2$ (identity) for all $a, b \in H$, one has $a\star(b,b)=a$;
- H$_3$ (transitivity) for all $a, b\in H$, one has $a\star (a, b) =b$.
A heap is said to be laterally commutative if in addition the following condition is satisfied:
- H$_4$ (lateral commutativity) for all $a, b, c \in H$, one has $ a\star (b, c) = c\star (b, a).$
A heap operation is usually written as $[a,b,c]$. I have selected a notation $a \star (b,c)$ to highlight that $H^2$ can be given a semigroup structure and can be thought as acting on the set $H$. I will call elements of $H^2$ bipoints. This is made precise by a following proposition. Note, that I overload $\star$. This should not cause any issue ass both operations are used with different arguments.
Proposition 1.
Suppose $H$ is a heap. Then the binary operation on the set of bipoints $H^2$ defined by
$$ (-,-)\mathbin{{\star}}(-,-) : H^2 \times H^2 \to H^2, ((a, b), (c,d)) \mapsto (a, b \star (c,d))$$
satisfies following properties:
- H$^\prime_1$ (associativity) for all $(a, b), (c, d), (e, f) \in H^2$, one has $$((a,b) \mathbin{{\star}} (c,d)) \mathbin{{\star}} (e, f) = (a,b) \mathbin{{\star}} ((c,d) \mathbin{{\star}} (e,f));$$
- H$^\prime_2$ (right identity elements) for all $(a, b), (c,c) \in H$, one has $ (a,b) \mathbin{{\star}} (c,c) = (a,b)$
- H$^\prime_3$ (Chasles' relation) for all $a, b, c \in H$, one has $(a,b)\mathbin{{\star}} (b,c) = (a,c)$;
If in addition $H$ is commutative then
- H$^\prime_4$ (commutativity of terminal points) for all $a, b, c, d \in H$, one has $(a,b)\mathbin{{\star}}(c,d) = (a,d)\mathbin{{\star}}(c,b)$;
- H$^\prime_5$ (inverse elements) each element $(a, b) \in H^2$ has a unique inverse $(b,a) \in H^2$ in the sense that $$(a,b) = (a,b)\mathbin{{\star}}(b,a)\mathbin{{\star}}(a,b) \text{ and } (b,a) = (b,a)\mathbin{{\star}}(a,b) \mathbin{{\star}}(b,a).$$
By virtue of H$^\prime_1$ and H$^\prime_2$, a heap structure on $H$ induces a semigroup structure on $H^2$ where each bipoint $(c,c)\in H^2$ is an right identity element. This justifies using a group action notation for the heap operation.
Definition 3.
Suppose $\mathbb{K}$ is a field.
An affine space over $\mathbb{K}$ is an additive laterally commutative heap $X$ equipped with a scalar multiplication
$$ -\cdot (-, -) : \mathbb{K}\times X^2 \to X, (a, (x, y)) \mapsto a \cdot (x, y)
$$
that satisfies the following conditions:
- A$_0$ (compatibility of scalar multiplication and heap structure) for all $a\in \mathbb{K}$ and for all $x, y \in X$, one has $ a\cdot(x,y) + (x,z) = a\cdot (z, y + (x, z)) $,
- A$_1$ (distributivity of scalar multiplication over heap operation) for all $a\in\mathbb{K}$ and for all $x, y, z \in X$, one has $a\cdot (x, y + (x, z))=a\cdot(x,y) + (x, a\cdot (x, z))$
- A$_2$ (distributivity of scalar multiplication over field addition) for all $a, b\in \mathbb{K}$ and for all $x, y \in X$, one has $(a+b)\cdot (x, y) =a\cdot (x,y) + (x, b \cdot (x, y))$,
- A$_3$ (compatibility of scalar and field multiplication) for all $a, b\in \mathbb{K}$ and for all $x, y \in X$, one has $ab\cdot (x,y) =a\cdot ( x, b\cdot (x,y))$,
- A$_4$ (scalar multiplication by field identity element) for all $x, y \in X$, one has $1_\mathbb{K} \cdot (x, y) = y$.
A definition of an affine space along the definition 3 can be found in Heaps of modules and affine spaces by Simion Breaz, Tomasz Brzeziński, Bernard Rybołowicz, Paolo Saracco (available here) where axioms of a commutative heap are given a more concise form. They in turn refer to Der baryzentrische Kalkül als axiomatische Grundlage der affinen
Geometrie by Fritz Ostermann, Jürgen Schmidt (available here) who called the heap operation with a different order of arguments a parallelogram operation.
As with the heap operation, the scalar multiplication can be extended to bipoints. I use $\odot$ instead of overloading $\cdot$ to stress that the result of multiplication is a bipoint, not a point.
Proposition 2.
Suppose $X$ is an affine space over $\mathbb{K}$ as per definition 3. Then the scalar multiplication on the set of bipoints $X^2$ defined by
$$ - \mathbin{\odot} (-, -) : \mathbb{K}\times X^2 \to X^2, (a, (x, y)) \mapsto (x, a \cdot (x, y))
$$
satisfies following properties:
- A$^\prime_0$ (compatibility of scalar multiplication and addition of bipoints) for all $a\in\mathbb{K}$ and for all $x, y, z \in X$, one has $a\mathbin{\odot} (x, y) \mathbin{+} (x, z)=(x, z) \mathbin{+} a\mathbin{\odot} ((z,y) \mathbin{+} (x, z))$
- A$^\prime_1$ (distributivity of scalar multiplication over addition of bipoints) for all $a\in\mathbb{K}$ and for all $x, y, z \in X$, one has $a\mathbin{\odot} ((x, y) \mathbin{+} (x, z))=a\mathbin{\odot}(x,y) \mathbin{+} a\mathbin{\odot} (x, z)$
- A$^\prime_2$ (distributivity of scalar multiplication over field addition) for all $a, b\in\mathbb{K}$ and for all $x, y \in X$, one has $(a+b)\mathbin{\odot} (x, y) =a\mathbin{\odot} (x,y) \mathbin{+} b \mathbin{\odot} (x, y)$,
- A$^\prime_3$ (compatibility of scalar and field multiplication) for all $a, b\in\mathbb{K}$ and for all $x, y \in X$, one has $ab\mathbin{\odot} (x,y) =a\mathbin{\odot} (b\mathbin{\odot} (x,y))$
- A$^\prime_4$
(scalar multiplication by identity element of field) for all $x, y \in X$, one has $1_\mathbb{K}\mathbin{\odot}(x, y) = (x,y)$.
Proposition 1 and 2 can be turned into stand alone definitions.
Definition 4
Suppose $H$ is a non-empty set.
A semigroup of bipoints is the set $H^2$ equipped with a binary operation
$$ (-,-)\mathbin{{\star}}(-,-) : H^2 \times H^2 \to H^2, ((a, b), (c,d)) \mapsto (a, b) \star (c,d)$$
that satisfies following properties:
- SG$_0$ (uniqueness of terminal points) for all $(a, b), (a', b), (c, d)\in H^2$, one has $$(a,b) \mathbin{{\star}} (c,d) =(a,e) \text{ and } (a',b) \mathbin{{\star}} (c,d) =(a',e)$$ for some unique $e\in H$;
- SG$_1$ (associativity) for all $(a, b), (c, d), (e, f) \in H^2$, one has
$$((a,b) \mathbin{{\star}} (c,d)) \mathbin{{\star}} (e, f) = (a,b) \mathbin{{\star}} ((c,d) \mathbin{{\star}} (e,f));$$
- SG$_2$ (right identity elements) for all $(a, b), (c,c) \in H$, one has $ (a,b) \mathbin{{\star}} (c,c) = (a,b)$
- SG$_3$ (Chasles' relation) for all $a, b, c \in H$, one has $(a,b)\mathbin{{\star}} (b,c) = (a,c)$;
A semigroup of bipoints is said to be terminally commutative if in addition the following condition is satisfied:
- SG$_4$ (commutativity of terminal points) for all $(a, b), (c, d) \in H^2$, one has $(a,b)\mathbin{{\star}}(c,d) = (a,d)\mathbin{{\star}}(c,b)$;
A semigroup of bipoints is said to be inverse if in addition the following condition is satisfied:
- SG$_5$ (inverse elements) each element $(a, b) \in H^2$ has a unique inverse $(b,a) \in H^2$ in the sense that $$(a,b) = (a,b)\mathbin{{\star}}(b,a)\mathbin{{\star}}(a,b) \text{ and } (b,a) = (b,a)\mathbin{{\star}}(a,b) \mathbin{{\star}}(b,a).$$
An additive commutative inverse semigroup of bipoints formalizes the addition of bound vectors usually taught in schools.
Definition 5.
Suppose $\mathbb{K}$ is a field.
An affine space over $\mathbb{K}$ is a non-empty set $X$ with an additive terminally commutative inverse semigroup of bipoints on $X^2$ equipped with a scalar multiplication
$$ -\cdot (-, -) : \mathbb{K}\times X^2 \to X, (a, (x, y)) \mapsto a \cdot (x, y)
$$
satisfies following properties:
- A$^\prime_0$ (compatibility of scalar multiplication and addition of bipoints) for all $a\in\mathbb{K}$ and for all $x, y, z \in X$, one has $a\mathbin{\cdot} (x, y) \mathbin{+} (x, z)=(x, z) \mathbin{+} a\mathbin{\cdot} ((z,y) \mathbin{+} (x, z))$
- A$^\prime_1$ (distributivity of scalar multiplication over addition of bipoints) for all $a\in\mathbb{K}$ and for all $x, y, z \in X$, one has $a\mathbin{\cdot} ((x, y) \mathbin{+} (x, z))=a\mathbin{\cdot}(x,y) \mathbin{+} a\mathbin{\cdot} (x, z)$
- A$^\prime_2$ (distributivity of scalar multiplication over field addition) for all $a, b\in\mathbb{K}$ and for all $x, y \in X$, one has $(a+b)\mathbin{\cdot} (x, y) =a\mathbin{\cdot} (x,y) \mathbin{+} b \mathbin{\cdot} (x, y)$,
- A$^\prime_3$ (compatibility of scalar and field multiplication) for all $a, b\in\mathbb{K}$ and for all $x, y \in X$, one has $ab\mathbin{\cdot} (x,y) =a\mathbin{\cdot} (b\mathbin{\cdot} (x,y))$
- A$^\prime_4$
(scalar multiplication by identity element of field) for all $x, y \in X$, one has $1_\mathbb{K}\mathbin{\cdot}(x, y) = (x,y)$.
I haven't seen the definition 4 and 5 anywhere but I would not be surprised to find out I'm not the first to define affine space like this. The terminology 'semigroup of bipoints' 'terminally commutative' is my own invention.
The axioms A$_1$-A$_4$ (respectively A$^\prime_1$-A$^\prime_4$)in the definition 3 (respectively 5) ensure that $X$ is a vector space relative to any base point $x\in X$. The axiom A$_0$ (respectively A$^\prime_0$) ensures that these vector spaces are isomorphic. The following proposition makes this observation precise.
Proposition 3
Suppose $X$ is an affine space over a field $\mathbb{K}$ as per definition 3 or 5. Then for any base point $x\in X$, $\mathbf{T}_{x}X=\{x\}\times X$ equipped with the addition given by
\begin{align*}
\mathbin{+} : \mathbf{T}_{x}X \times \mathbf{T}_{x}X & \to \mathbf{T}_{x}X,\\
((x,u), (x, v)) &\mapsto (x,u+(x,v)) \text{ or } (x,u)+(x,v)
\end{align*}
and the scalar multiplication given by
\begin{align*}
\mathbin{\cdot} : \mathbb{K} \times \mathbf{T}_{x}X & \to \mathbf{T}_{x}X,\\
((a, (x, v)) &\mapsto (x,a\cdot (x,v)) \text{ or } a \cdot (x,v)
\end{align*}
is a vector space over $\mathbb{K}$. For any $x, y\in X$ the mapping
\begin{align*}
\Gamma_{x,y}:\mathbf{T}_{x}X&\to\mathbf{T}_{y}X,\\
(x,v)&\mapsto(y, y+(x,v)) \text{ or } (y,y)+(x,v)
\end{align*}
is a linear isomorphism such that for any $x,y,z \in X$, one has $\Gamma_{y,z} \circ \Gamma_{x,y} = \Gamma_{x,z}$.
A usual affine space can be retrieved from either of the definitions by defining a group of translations $\mathbf{T}(X)$ and turning it into a vector space.
Proposition 4
Suppose $X$ is an affine space over a field $\mathbb{K}$ as per definition 3 or 5.
A translation by $(u,v)$ is a mapping \begin{align*}
[u,v] : X &\to X,\\ x&\mapsto x + (u,v) \text{ or } (x,x)+(u,v).
\end{align*}
Then the set of all translations $\mathbf{T}(X)$ equipped with the addition given by
\begin{align*}
\mathbb{+} : \mathbf{T}(X) \times \mathbf{T}(X) & \to \mathbf{T}(X), \\
([u, v], [u', v']) & \mapsto [u,v]+[u',v']=[u, v + (u',v')] \text{ or } =[(u,v)+(u',v')].
\end{align*}
and the scalar multiplication given by
\begin{align*}
\mathbin{\cdot} :\mathbb{K} \times \mathbf{T}(X) & \to \mathbf{T}(X), \\
(a, [u, v]) & \mapsto a\cdot [u,v]= [u, a \cdot (u, v)] \text{ or } [a\cdot (u,v)].
\end{align*}
is a vector space over $\mathbb{K}$.
The following mapping
\begin{align*}
\mathbb{+}: X \times \mathbf{T}(X) &\to X, \\
(x, [u,v]) & \mapsto x + [u, v] = x + (u, v) \text{ or } (x,x)+(u,v)
\end{align*}
is a well-defined action of the additive group $\mathbf{T}(X)$ on $X$ that is free and transitive.
