Reverse Polish notation in (abstract) algebra

In abstract algebra, we don't refer to it as Polish notation, but "prefix notation" and "postfix notation" do seem to be apt names for talking about these two things.

The two notations $(x)f$ and $f(x)$ have both been used historically; however, I think the latter is far more prevalent today. There may be some concrete anthropological reason we prefer the former, but it is probably driven mostly by tradition and history, like a lot of notation.

Each of these leads naturally to more notation as function composition occurs: $((x)f)g)$ and $g(f(x))$. The former can be abbreviated to $(x)fg$ and the latter to $gf(x)$. But really, they define exactly the same function, just written with two different conventions.

Actually, both notations come in handy for noncommutative algebra. When we have a homomorphism $f$ of right $R$ modules, it's conveninent to write $f$ on the left, for then the rule of homogeneity for homomorphisms becomes $f(x\cdot r)=f(x)\cdot r$. If we had used the other convention, then it would look like $(x\cdot r)f=(x)f\cdot r$, which is somewhat less appealing.

But if we are talking about a homomorphism of left $R$ modules, then we would prefer to write $f$ on the right, so that $(r\cdot x)f=r\cdot(x)f$. Arranging things this way is just an aesthetic convenience.

Consider it (and call it, if you need a name) the composition of functions (after all, permutations are functions). Remember:

$$(\phi\circ \psi)(x) = \phi(\psi(x))$$

We work from inside out to determine $(\phi\circ \psi)(x) = \phi(\psi(x))$:

First determine $\psi(x)$, then you can determine $\phi(\psi(x))$.

This is no different than how we approach the notation $(f\circ g)(x) = f(g(x))$, and I've never heard this referred to as "reverse Polish notation."

To me, seems counter-intuitive to interpret $(f\circ g)(x)$ by first finding $f(x)=y$ and then taking $g(y)$.