Stop Assuming Commutativity!

The foundation of mathematics is built on axioms – assumptions we declare to be able to prove anything at all. In general, mathematicians try to keep the number of axioms to a minimum. We want to assume as little as possible and prove as much as possible!

The five axioms of Euclidean Geometry (the geometry we know from school) is a classic example. The fifth axiom, so-called Parallel Postulate, is rather complicated compared to the other ones:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Mathematicians didn’t like this axiom. How could such an ugly statement be the last axiom of our geometry? So for 2000 years, many clever minds tried to prove this axiom using Euclid’s first four postulates. No success, however – they all failed.

They all failed for a reason: We now know that it’s impossible to give such a proof. New geometries were constructed that satisfy each of the first four axioms but not the fifth. Such geometries are called Non-Euclidean geometries with examples as Hyperbolic geometry or Spherical geometry.

The moral of the story? Mathematicians usually carefully pick their axioms and want as few as possible.

That, however, seems not to be the case in specific topics of Abstract Algebra. Somehow, the literature likes to assume commutativity.

Commutativity is a strong condition. In everyday life, we are spoiled with commutativity! How do we pay 3€? We could first pay 1€ and then pay another 2€. Or we could start with 2€ and continue with 1€. The idea is simple: 1 + 2 = 2 + 1.

In mathematical structures, however, commutativity is tough to get. Associativity is a much simpler condition than commutativity. There are only very few structures that possess commutativity but no associativity.

In fact, even in everyday life, we are surrounded by non-commutative structures. Do you first put on your shoes or your socks? Obviously the order matters here!

So commutativity should not be taken for granted. Apparently, though, many people do. That happens for basic structures such as vector spaces.

Definition. A vector space over $\mathbb{R}$ is a set $V$ with an operation + on $V$ and another operation $\cdot$ between $K$ and $V$ that satisfies five axioms.

1. The structure $(V, +)$ is an abelian group.
2. For $a \in \mathbb{R}, \ v,w \in V$ we have $a \cdot (v + w) = (a \cdot v) + (a \cdot w).$
3. For $a,b \in \mathbb{R}, \ v \in V$ we have $(a+b) \cdot v = (a \cdot v) + (b \cdot v).$
4. For $a,b \in \mathbb{R}, \ v \in V$ we have $b \cdot (a \cdot v) = (b \cdot a) \cdot v.$
5. For $v \in V$ we have $1 \cdot v = v.$

(Note that the + and $\cdot$ operations are not always the same operations. Some operate on $\mathbb{R}$ and some on $V.$)

These axioms seem harmless, right? Only the last one seems dispensable but one can construct structures that satisfy only the first 4 axioms.

I too naively believed that these axioms are alright… Until a friend of mine asked me

“Did you know that commutativity is redundant?”

In shock, I was speechless: “whaaaaat?”

$(V, +)$ is an abelian group.

Who in their right minds would have guessed that abelian can be left out? I surely did not. And I bet most people don’t. How should we? The literature usually includes commutativity. See for example Linear Algebra Done Right, most Linear Algebra textbooks, Wikipedia, my Linear Algebra lecture notes, most likely your Linear Algebra lecture notes, etc.

Of course, I had trouble believing my friend in the beginning and tried to construct a counterexample. No success, however – I failed. Why? For the opposite reason as in the introductory paragraph. Indeed, I was later able to prove that commutativity can be proven using the other axioms.

I still don’t get why lecturers/books don’t just leave out commutativity. Showing commutativity doesn’t take long (I encourage you to do so!) and can be immediately declared as a theorem.

Axiom. The structure $(V, +)$ is a group.

Theorem. In fact, $(V, +)$ is not only a group but an abelian group!

We do prove “seemingly obvious” facts like $0v = 0,$ so why not also show some love for commutativity?

Vector spaces are not the only structures suffering this fate. In unitary rings (and fields) we also usually assume commutativity. The proof that commutativity does not have to be assumed can be proven analogously as with vector spaces.

Let us not just take commutativity for granted! Where else does the common literature take a shortcut and declare too many axioms?

3 thoughts on “Stop Assuming Commutativity!”

1. “Associativity is a much simpler condition than commutativity.” Are you really, really sure about that?

After playing around with complex numbers and matrices, I wanted to know, what addition and especially multiplication really are. What do we expect from them? Unfortunately, I still have no answers (apart from group structure).

When doing so, I started looking at binary operations on finite sets. Every binary operation can be visualized in a Cayley table with n^2 entries. For each entry you can choose any of the n elements. This gives you

n^(n^2)

possible binary operations on an n-element set. This number gets very big very fast. For n = 5, there are 298 023 223 876 953 125 possible operations, most of which are of no interest to anyone.

Commutativity can be easily spotted in a Cayley table since it must be symmetric to the main diagonal. The total number of commutative operations is thus

n^(n * (n + 1) / 2)

For n = 5 this is still the huge number 30 517 578 125, but that is only 0.000 010% of all possible operations.

Unfortunately, there is no formula to calculate the number of associative operations on an n-element set. http://oeis.org/A023814 gives a list for small n. For n = 5, the number is “only” 183 732. Still more than expected, but much smaller than the number of commutative operations.

I don’t have any proof for all n, but to me it seems that associativity is the much more restrictive property than commutativity. Before looking at these numbers, I also was under the impression that associativity is a run-of-the-mill property, since every half-gorups has it already.

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1. Woah, that was interesting. I too just played around with Cayley tables and must say I’m somewhat surprised about how few associative binary operations there are compared to commutative ones. From these numbers, I guess we can indeed say that associativity seems to be more restrictive.

I was actually quoting my professor. While associativity might be more restrictive, I guess we may be able to say that most of these commutative operations are of no interest to mathematicians. The associative ones are the interesting ones (I recall my professor saying he knew only one structure that was commutative but not associative).

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