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When we are discussing a binary operation $*:X \times X \to X$, we typically say that $*$ is commutative if $*(x,y) = *(y,x)$ for all $x,y \in X.$ However, when discussing a function $F: X \times X \to Y$ (where $X \neq Y$) such that $F(x,y) = F(y,x)$ for all $x,y \in X,$ I often hear $F$ called "symmetric" - as in the case of a metric or an inner product.

What exactly is the difference between the words "commutative" and "symmetric" in describing a function? Is the above the full story?

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2 Answers 2

I wouldn't use the term commutative to describe a binary operation unless it were also associative. In that case, commutativity can be rephrased in a particular way that doesn't at all apply to functions $X \times X \to Y$. But first I need to describe how associativity can be rephrased in this way.

First, there's no reason to privilege binary operations. In practice we don't only, say, multiply two numbers; we often multiply three or more numbers. To say that a binary operation $a \ast b$ is associative is precisely to say that it comes from a compatible family of $n$-ary operations $a_1 \ast a_2 \ast ... \ast a_n$ in a particular way; compatible means, in particular, that

$$(a_1 \ast ... \ast a_n) \ast (a_{n+1} \ast ... \ast a_m) = (a_1 \ast ... \ast a_m).$$

Said another way, associativity really means that I have a nice way to eat a list of elements and spit out an element in a way which is compatible with concatenation of lists.

An operation with the above property which is in addition commutative can be thought of as eating, not a list, but a multiset of elements and spitting out another element; that is, order doesn't matter. Being able to interpret commutativity in this way requires that I be able to write down $n$-fold compositions so it doesn't apply to functions $X \times X \to Y$.

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Ré: your first sentence: we have been talking about commutative Jordan álgebras for almost a century now. –  Mariano Suárez-Alvarez May 21 at 2:17
    
I appreciate this detail, but I am still curious about when the word "symmetric" should be used (esp. as opposed to "commutative.") –  Optional May 21 at 2:40
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It's exactly the thing you said, with an annoying caveat involving symmetric monoidal categories (these should really be called "commutative" for consistency but they just aren't; some kind of historical accident). –  Qiaochu Yuan May 21 at 2:40

The only difference I can see between the two terms is that commutativity is a property of internal products $X\times X\to X$ while symmetry is a property of general maps $X\times X\to Y$ in which $Y$ might differ from $X$.

I'd say that describes the difference in most common usage, and that there is really nothing more to it. There are exceptions, due to history. For example, if $A$ is an algebra, then we sometimes sat that a bilinear map $f:A\times A\to k$ to the base field is associative if $f(xy,z)=f(x,yz)$, and $f$ is not an internal product...

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