# Axioms for vector space and affine space under the same system and notation

Background: Here are common axiomatic definition of a vector space and an affine space:

1. A vector space is a set together with two operations satisfying the following eight axioms. For addition: associativity, commutativity, the existence of identity and inverse element; for scalar multiplication: associativity, compatibility, two distributivities, and existence of identity.

2. An affine space satisfies three axioms: two distinct points lie on the unique line, parallel line passing a point is unique, and non-colinear points exists.

Motivation: The easiest way to tell two structures apart is through their axioms. To my knowledge, the two set of axioms for two spaces are from two different systems with different notation, and thus comparing those axioms cannot be obviously and directly done.

At the same time, people seems claim that an affine space is more genenral than a vector space, and a vector space is a special case of an affine space.

Questions: I am looking for the axioms using the same system. That is, a set of axioms defining vector space, but using the notation of (2). Or, a set of axioms definiting affine space, but using the notation of (1).

In other word, I am looking for, for example, a total of four axioms, say $$A1, A2, A3, A4$$. A set is an affine space iff it satisfies $$A1, A2$$, and $$A3$$. A set is a vector space iff it satifies $$A1, A2, A3$$ and $$A4$$.

My try: Here are some axiomatic definition of an affine space that uses similar style as (1): http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/affine+space#two_ternary_operations

• You forgot to compare both of them to Euclidean space. Feb 13, 2022 at 18:53
• Note that one could consider the affine line, in which case there are no non-collinear points. Feb 13, 2022 at 20:42
• For context this is a question related to math.stackexchange.com/a/884671/169085 Feb 13, 2022 at 22:37
• Also related: math.stackexchange.com/q/1515093/169085 and math.stackexchange.com/q/4379190/169085 (the latter in the sense of taking a definition and relativizing it). Feb 13, 2022 at 23:35

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.