Is there a name for the equations that are related by commutativity?

Question

I know addition and multiplication are both commutative: 2+3 has the same sum as 3+2, and 2x3 has the same product at 3x2. It's the "commutative property" of the two operations.

But is there a word that labels the two equations in that relationship? i.e. "2x3 is the ??? of 3x2" (and vice versa).

I (think?) I've ruled out: "inverse", "complement", and "commutatively flipped lifelong buddy pal".

Answer 1

There isn't a specific term for the relationship between two equations representing the same commutative property. However, you can say: "$2\times3$ is commutative with $3\times2$". However, I don't really see a need to use such terms at all. You can just write "$2\times3$ is equal to $3\times2$".

Answer 2

We can invent a syntax for this. This is also a great introduction to the notion of a group!

If it is just two variables then saying "complement" or "reflection" or "conjugate" would make plenty of sense in the context. It's worth pointing out that a complement of a complement is the original, a reflection of a reflection is the original, a conjugate of a conjugate is the original. This is very important for our next step

In more variables we need to pull in some more complex notions. Each of the different expressions

$$ a \times b \times c = a \times c \times b = b \times c \times a = ... $$

Are permutations of the underlying variables. Every permutation has a standard cycle notation so we can say for example that $a \times c \times b $ is the image of $a \times b \times c$ acted upon by the permutation $(2,3)$. Here the action of $(2,3)$ is to swap the element number $2$ with element number $3$.

Now our earlier "complement", "reflection", "conjugate" choice of words is a bit further clarified. All of these are basically references to the 2 element group $\mathbb{Z}_2$ or the permutation $(1,2)$. Where $1+1=0$ or said another way $(1,2)(1,2) = ()$ or said another way $(-1)\times(-1) = 1$ or said another way "the reflection of a reflection is the original" or said another way the "complement of the complement is the original" etc...

By identifying the underlying group that's hidden in the 2-element case we can now talk about the symmetries of the n-element case.

I'll be the first to admit that this explanation is a bit sophisticated and will require a lot more exposition to make it easy to digest for beginners.