Algebra where operators are assumed as variables. Does there exists any form of Algebra where operators can be assumed as variables?
For example:
$$
1+2\times3=7
$$
can be considered as:
$$
1\:(\mathrm{\,X})\:2\:(\mathrm{Y})\:3=7
$$
?
 A: I would say the answer is most definitely yes, if you construe the operators to be functions.  For example, we could represent the "operator" $+$ (on $\mathbb{Z}$, the integers, for example) to be the function $p: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ given by $p(z_1, z_2) = z_1 + z_2$ for $(z_1, z_2) \in  \mathbb{Z} \times \mathbb{Z}$; likewise the "operator" * would be a function $t: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ such that $t(z_1, z_2) = z_1*z_2$; with these conventions, the equation $1 + 2*3 = 7$ would be written $p(1, t(2, 3)) = 7$.  Now observe that $p$ and $t$ are "merely" elements of the set $\mathcal{F}$ of functions from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Z}$; letting $f_1, f_2$ denote variables ranging over the set $\mathcal{F}$, we see that our expression $p(1, t(2, 3))$ in fact corresponds to letting $f_1 = p$ and $f_2 = t$; clearly many other such choices are possible, though most of them would yield an $f_1(1, f_2(2, 3))$ which is more difficult to compute!  But in fact a similar paradigm occurs over and over again in certain areas of mathematics.
One problem which might be encountered in this approach is that of parsing an expression like $1(X)2(Y)3$ into appropriate functional form, how to group the elements and so forth.  But I'll leave that discussion for future research and questions!
Hope this helps!  Cheers, Bob Lewis.
A: Absolutely! In abstract algebra, you study general operations, besides $+$ and $\times$, that satisfy certain identities. An example of an identity is your equation $1\mathop X (2\mathop{Y}3)=7$. This equation constrains the variables $X$ and $Y$, subject to the constants 1, 2, 3, and 7. So you could say that $X={+}$ and $Y={\times}$ is a "solution" to the equation, whereas $X={\times}$ and $Y={+}$ is not a solution. However, unless you're working with differential equations, I don't think that terminology is very common. Usually you would give a name to a kind of operation that satisfies a given identity. For example, an operation $\circ$ is called commutative if $a\circ b = b\circ a$ for all $a$ and $b$. Then you would say that addition is commutative, but subtraction is not commutative.
In universal algebra, you go even further and treat the identities themselves as variables! From this viewpoint, you can consider all possible sets of identities and their interrelationships. I'm horribly rusty, so please don't ask me to expand on this idea. :)
