For example, let's define the "swap" function $SW(f(x,y))$ as the function that maps $f(x,y) \rightarrow f(y,x)$. I can imagine there are many such functions that have been described. Is there any useful term for such a thing?
EDIT: I'd like to illuminate a particular problem I'm interested in.
I begin with a set of functions that operate on two real numbers. For the sake of simplicity of this example, I'll use only three. For te variables $x,y,z\in \mathbb{R}$:
$$ADD(x,y) = x + y$$ $$MUL(x,y) = x*y$$ $$z = \sin(x)$$
$\sin(x)$ is defined in the traditional way. I'm including it to make my point a bit more clear.
Let's now define an equation that uses only these functions. I'll use a specific example of:
$$x*(y + z) + sin(x) = f(x,y,z)$$
I'm now interested in making the following idea more precise and general:
Define a function EX(f) whose purpose is to distribute multiplication over addition. Then, when applied to $f$ above,
$$EX(f) = g(x,y,z) = x*y + x*z + sin(x)$$
In this case, $f$ and $g$ evaluate to the same value so we might claim that $f=g$ in the numeric sense. However, I would not say they are equal from the perspective of actually computing those values, since a different set of steps must be followed. It is the latter case I'm interested in studying further, in which numeric equality is different from evaluation equality.
I'm interested in defining functions like EX, and determining things like stationary functions. For example, $EX(g) = g$, so $g$ is "stationary" under EX.
Forgive any imprecision. I hope it was enough to explain the type of things I'm looking for.
What you are describing are transformations of expressions or syntax trees. Computer scientists have a lot to do with those things.
While mathematicians are ordinarily interested in "extrinsic" equality, where $f = g$ when $\forall x. f(x) =g(x)$, computer scientists may consider the definition of an "algorithm" that actually presents the value of $f(x)$ when given $x$, and an expression is certainly one reasonable way to define an algorithm. (Particularly, "functional" programming languages will let you define algorithms via expressions and actually run the resulting programs.) In your example, $f$ and $\mathrm{ex}f$ are extrinsically equal, but "intrinsically" distinct.
One approach is to say that $E\,\Sigma$ is some "tree" functor from a set of Latin letters $\Sigma$, and $\mathrm{sw}: \Sigma \to \Sigma$ is a one to one "relabelling" function on those letters, that may be "lifted" to that functor as $E\,\mathrm{sw}: E\,\Sigma \to E\,\Sigma$, while $\mathrm{ex}: E\,\Sigma \to E\,\Sigma$ is a more general transformation of trees.
What you are calling "stationary" is a "fixed point" of a function. These are also used extensively in computer science. For example, if you start with an expression $x$ and apply some number of $log_a$ and $exp_b: x \mapsto b^x$ to it, you will get a family of expressions, and a fixed point of such application is the infinite group generated by $\{\log_a, \exp_b | a, b \in \mathbb{R} \}$. In such way a "language" — a set of expressions — can be defined.
I am merely a student, so in the above there may be errors and omissions. For future study, one entry point is "Algebraic and Coalgebraic Methods in the Mathematics of Program Construction", edited by Roland Backhouse, Roy Crole and Jeremy Gibbons. Not for the faint of heart though.