Is every axiom in the definition of a vector space necessary?

Question

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms:

1. Commutativity of $+$.
2. Associativity of $+$.
3. Existence of an identity element $\mathbf{0}$ for $+$.
4. Existence of inverses for $+$.
5. Compatibility of $\cdot$ with multiplication in $K$.
6. Distributivity of $\cdot$ over $+$.
7. Distributivity of $\cdot$ over addition in $K$.
8. $1_K$ is a left identity of $\cdot$.

Question: Are all seven of the previous axioms necessary (in the sense that weakening any one of them permits a structure which is not a vector space)? If not, which can be weakened (or removed)?

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EDIT: user7530 has quite cleverly shown that the commutativity of $+$ can be derived from axioms 2-8. Supposing we throw this out, can the remaining axioms all be proven necessary?

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EDIT 2: It was pointed out that axiom 3 cannot simply be thrown out, as the definition of an inverse in axiom 4 depends on the existence of $\mathbf{0}$. What if we tweak the statement of axiom 4 to axiom 4': "For every $x \in V$, there exists $y \in V$ such that $(x+y)+x = x$ and $(y+x)+y = y$"? Is this weakened version equivalent to the original, and if so, does it allow the removal of axiom 3?

Answer 1

I think they are redundant after all! Here's a proof that axiom 1 is redundant. Let $a,b\in V$, and consider $(1+1)\cdot (a+b)$. By axiom 7 and 8, this is equal to $(a+b)+(a+b)$; on the other hand by axiom 6 it is $(1+1)\cdot a + (1+1)\cdot b$, or $(a+a)+(b+b)$ by axiom 7 and 8. We can then use axioms 2, 3, 4 to show that \begin{align*} a^{-1} + (a+b) + (a+b) + b^{-1} &= a^{-1} + (a+a) + (b+b) + b^{-1}\\ b + a &= a + b \end{align*} and $V$ is Abelian.

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Necessity of some of the other axioms:

4: Take $V=[0,\infty)$ under multiplication, and $K=\mathbb{R}$, with $z\cdot x \mapsto \begin{cases} x^z, & x\neq 0\\0, &x=0.\end{cases}$

5: Consider $K=\mathbb{C}$, $V=\mathbb{R}$ with $z\cdot x = \Re(z)x$.

6: Necessary once you toss out commutativity. Take $K=F_3$, and $V$ the Heisenberg group over $F_3$, with $z\cdot x = x^z$. Since all elements of $V$ have order dividing 3, axiom 7 is satisfied, but $$\left(\left[\begin{array}{ccc}1 & 0 &0\\0 & 1 & 1\\0 & 0& 1\end{array}\right]\left[\begin{array}{ccc}1 & 1 & 0\\0 & 1 & 0\\0 & 0 &1\end{array}\right]\right)^2 \neq \left[\begin{array}{ccc}1 & 0 &0\\0 & 1 & 1\\0 & 0& 1\end{array}\right]^2\left[\begin{array}{ccc}1 & 1 & 0\\0 & 1 & 0\\0 & 0 &1\end{array}\right]^2.$$

7: Take $K=\mathbb{C}$, $V=\mathbb{R}$, and $z\cdot x = |z|x$.

8: See comment by Jyrki below.

Answer 2

Selected progress on the definition of Vector Space:

At 1971, Bryant proved that the commutativity of $\oplus$ can be deduced by other axioms$^{(1)}$.

At 1973, Rigby and Wiegold proved that only 6 axioms are needed$^{(2)}$.

In fact, the set of axioms of a vector space may reduce to only 6 as described:

Definition. A vector space over a field $K$ consists of a set $V$ and two binary operations $\oplus: V\times V→V$ and $\odot: K×V→V$ satisfying the following axioms:

1. $(a\oplus b)\oplus c=a\oplus (b\oplus c), \forall a,b,c\in V$,

2. $\lambda\odot(a\oplus b)=(\lambda \odot a)\oplus(\lambda \odot b),\forall a,b\in V, \forall \lambda\in K,$

3. $(\lambda +\mu)\odot a=(\lambda \odot a)\oplus(\mu \odot a),\forall a\in V,\forall\lambda,\mu\in K,$

4. $(\lambda\times\mu)\odot a=\lambda\odot (\mu \odot a), \forall a\in V,\forall \lambda,\mu\in K,$

5. $0\odot a=0\odot b,\forall a,b\in V,$

6. $1\odot a=a, \forall a \in V.$

We have the follow theorems, in which, the Theorem 1. implies the existence of additive inverse and Theorem 2. implies the commutativity axiom.

Theorem 1. If $V$ satisfies axioms, $1, 3, 5, 6,$ then it have additive inverse.

Hints:$$ a =1\odot a=(1+0)\odot a=(1\odot a)\oplus(0\odot a)=a\oplus z$$ $$z=0\odot a=(1+(-1))\odot a=(1\odot a)\oplus(-1)a=a\oplus((-1)\odot a)$$

Denote the element $(-1)\odot a$ as the additive inverse of $a$.

Theorem 2. If $V$ satisfies axioms, $1,2,3,5,6,$ then it is a commutative additive under $\oplus$.

On counterexamples

The example corresponding to additive inverse axiom described in the answer (@user7530) doesn't satisfy the axioms $2, 3$, too. According to Theorem 1., it certainly failed for additive inverse axiom. Therefore, this example cannot prove the necessary of this axiom.

Last but not least

An important rule that intrinsically holds is VERY often to overlook! I call it Rule 0. as follows:

Rule 0. $\lambda\odot a\in V, \forall a\in V, \forall \lambda\in K. $ (Closure under scalar multiplication)

Consider the following counterexample for being a vector space:

$V=\{(a_1,...,a_n):a_i\in \mathbb{R}, i=1,...,n\}, K=\mathbb{C}$ with the operations of coordinatewise addition and multiplication.

It seems satisfy all the axioms listed above. But pay attention to it that $\exists \lambda\in K,$ s.t. $(\lambda\odot a)\notin V$. Now that so, how can we apply axioms $2,3,$ or $4$?

Reference

(1) Bryant, V. (1971). Reducing Classical Axioms. The Mathematical Gazette, 55(391), 38-40. doi:10.2307/3613304

(2) Rigby, J., & Wiegold, J. (1973). Independent Axioms for Vector Spaces. The Mathematical Gazette, 57(399), 56-62. doi:10.2307/3615171

Answer 3

You won't be able to entirely remove axiom 3, since otherwise $V=\emptyset$ would (vacuously) satisfy the other axioms. However, you can remove axiom 4 if you replace axiom 3 with this slightly stronger version (which I will call axiom 3*):

(Axiom 3*) There exists an element $\mathbb{0'} \in V$ such that for all $x \in V$, $0_K \cdot x = \mathbb{0'}$.

(Here, I use the notation $\mathbb{0'}$ to denote that this is a nonstandard definition of $\mathbb{0}$).

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Axiom 3* implies axiom 3 and 4

That this element is an additive identity follows from axioms 6 and 8: we have $$\mathbb{0'} + x = 0_k \cdot x + 1_k \cdot x = (0_k + 1_k)\cdot x = 1_k \cdot x = x.$$

Also, for every $x \in V$, we have $$x + (-1_K)\cdot x = (1_K)\cdot x + (-1_k)\cdot x = (1_K + -1_K)\cdot x = 0_k \cdot x = \mathbb{0'}$$ so each $x \in V$ has as an inverse $(-1_K)\cdot x$.

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Axioms 3 and 4 imply axiom 3*

We have that

$$(0_K)\cdot x + x = (0_k + 1_K) \cdot x = x$$

so, denoting the inverse of $x$ by $-x$,

$$(0_K)\cdot x + x + (-x) = x+(-x)$$ $$(0_K)\cdot x + \mathbb{0} = \mathbb{0}$$ $$(0_k) \cdot x = \mathbb{0}$$

Answer 4

I believe $6$ is indeed indispensible, and may be equivalent to $1$, here's my reasoning:

what $6$ actually says is we have an action of $(F,+)$ upon $(V,+)$, that is, the map $v \mapsto a\cdot v$ (let's call this map $\phi_a$) induces a group homomorphism (the operation being $+$):

$F \to V$ via $a \mapsto \phi_a(v)$ for any fixed $v \in V$.

Indeed, we can relax the vector space axioms to allow $R$ to be a commutative ring with unity, and obtain an $R$-bimodule. Now there is a unique homomorphism $\psi:\Bbb Z \to R$ sending $1 \mapsto 1_R$, and this allows us to define on any $R$-bimodule $M$, a $\Bbb Z$-action by:

$n\cdot m = \psi(n)\cdot m$.

Now the intuitive way to try to impose a $\Bbb Z$-action on a group $G$, is to try to set:

$n\cdot g = g^n$.

However, $g \mapsto g^n$ is an element of $\text{End}(G)$ for all $n \in \Bbb Z$ if and only if $G$ is abelian (the "if" part is obvious, the "only if" can be proved using $n = 2$, which is essentially user7530's argument).

Answer 5

Here is an even more trivial example showing why Axiom 8 is necessary (I wanted to post this as a reply under the remark of Jyrki Lahtonen, but the site did not allow me to do that).

Take any abelian group $V$ with any field $F$, and for any $v \in V$ and $a \in F$ set $av=0$ to be the trivial element of $V$ (written additively).

Answer 6

The answer of user7530 about commutativity axiom removal may not cover all the cases yet.

For, $K$ is any field, so it may also have characteristic $char(K)=2$. In that case we have $1+1=0$, and $(1+1)(\vec u+ \vec v)$ is simply equal to $0(\vec u+ \vec v) = \vec 0$.

In his original work [1] Bryant considered the spaces over the real field $\mathbb R$ only, so that the mentioned issue with the field characteristic did not appear.

[1] Bryant, Victor, Reducing Classical Axioms, The Mathematical Gazette, Vol. 55, No. 391, 38-40

Answer 7

I think 6 is necessary even if 1 is given.

Let $V$ be an abelian group, so it automatically satisfies 1-4.

For any group action $K\setminus{0}\times V\to V$ (where we consider $K\setminus\{0\}$ as a multiplicative group), if we define $\cdot$ to agree with this group action and to send $0\in K$ to the zero map (sending everything to $0\in V$), then this satisfies 5.

Also, whether this satisfies 7 and 8 is determined by whether it's true within each $K$-orbit: Axiom 6 is the only axiom that allows different orbits to talk to each other.

So we can get a counterexample by taking $K=\mathbf{C}, V=\mathbf{C}^2$, and having $K$ act on each 1-dimensional subspace of $V$ either by the usual scalar multiplication or by multiplying by the conjugate (picking randomly for each subspace). (In general, for any field $K$, if we let $P$ denote its prime field (i.e., the smallest subfield), then starting with any usual vector space $V$, one can alter the scalar multiplication on each 1-dimensional subspace by precomposing by any $P$-automorphism $\varphi:K\to K$.

I don't know whether this gives all the examples where all axioms but 6 hold. The other axioms imply $0v=0$ for all $v\in V$, and each $K$-orbit must form a 1-dimensional vector space over $K$, and $P$-automorphisms of $K$ are the same thing as ways of making $K$ act on $K$ as a vector space (given a scalar multiplication $\cdot$, we can get a $P$-automorphism by sending $a\in K$ to $a\cdot 1$), but I don't know whether I can always choose additional automorphisms to ``realign'' $V$ into a usual vector space.)

Answer 8

I think 2 is also necessary.

(Motivation: As in my other answer showing 6 is necessary, we can still get that the $K$-orbits of any vector must be 1-dimensional vector spaces. So $V$ must be a union of $K$-orbits, where the union is disjoint except at the zero vector. But without requiring associativity of vector addition, defining vector addition feels very unrestricted, so we'll start by defining $V$ as a set and making vector addition to make it work.)

Let $K$ be any field. Let $S$ be any set. We will take $V=\left(\coprod_{s\in S}(K\setminus\{0\})S\right)\coprod\{\vec{0}\}$. Scalar multiplication is done in the usual way on each orbit, which gives axioms 5, 7, and 8.

To define vector addition, the addition structure on any $K$-orbit is already determined by being a 1-dimensional vector space, so axioms 3 and 4 are satisfied (unless axiom 4 says that inverses must be unique, which I think cannot be shown without axiom 2).

So we only need to define $+_V$ to satisfy commutativity and axiom 6. This is the same data as a symmetric function $\mathbf{P}((\mathbf{V}\setminus\{0\})^2)\to V$, and there are lots of choices for that. In other words, no associativity means our addition doesn't have any composition rules, so we can pretty much set it pointwise.

For example, if additive inverses (in $V$) are not required to be unqiue, we could use the function 0, which means every pair of vectors not in the same $K$-orbit get sent to 0. Alternatively, if $K\ne\mathbf{F}/2\mathbf{Z}$ and $V$ is identified as a set with a vector space in a way that sends $K$-orbits to $K$-orbits, we could take the addition of two vectors not in the same $K$-orbit to be the given addition multiplied by any fixed scalar (while retaining the usual addition within any $K$-orbit).