# Are there “variables/unknowns” for operations?

We use letters for unknowns/variables:

$x^2=4$

Are there variables/unknowns for operations too?

$8 \star 7$

With the $\star$ being any operation.

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Variables are used for all kinds of things. And I mean pretty much literally all kinds of things. – tomasz Jul 8 '12 at 2:54
I imagined it from the axiom: "If my naive mind could imagine that, then mathematicians mind's should've invented it at least 600 years ago". But I've never read something about it's usage. – Voyska Jul 8 '12 at 2:55
600 years is a bit of an exaggeration. I don't think they used variables in quite the way we do nowadays, not even for numbers. But 100 years is probably a safe bet. – tomasz Jul 8 '12 at 2:57
The nearest thing this request reminds me are the rings and groups, I guess it's a little related to what I'm searching: "A set, 1/2 operations - and these operations could be anything" – Voyska Jul 8 '12 at 2:59
That might be a bit of a long shot if you're not familiar with mathematics, but that general stuff probably belongs to model theory rather than algebra. – tomasz Jul 8 '12 at 3:03

Of course. For example, the Cayley-Hamilton theorem states that, if $a_nx^n+\cdots+a_1x+a_0$ is the characteristic polynomial of a linear operator $M$, then $M$ is a root of $a_nX^n+\cdots+a_1X+a_0$ where $X$ is a variable representing a linear operator (often called a matrix). A less common example (but probably more in the spirit of your question) is the Eckmann-Hilton argument, which shows that any two binary operators $\cdot$ and $\star$ which satisfy certain conditions are equivalent.
Sure, but $\circ$ isn't a good choice; usually it denotes function composition. I would use $\star$, for example, which doesn't have an existing widely-used meaning.