There are in widespread use operators "$\sum$" & "$\prod$" that generalize binary "$+$" and "$\times$" to accept a set of operands. It seems obvious that we can generalize any binary operation this way, to make it act upon a (finite) set of operands if it's associative & commutative or upon a list of operands if it is not. How do I call such generalization and would the "polyary" operators so produced possess any "nice" properties?
Indeed, As Rahul points out, "if it's not associative, there isn't a unique way to apply it to a list". I say, there are as many ways to apply an operation "$+$" as there are ways to label "$+$" signs with natural numbers, which I believe would be $(n - 1)!$ for a list of length $n$.† (For example: $w +^3 x +^1 y +^2 z$ is one of $6$ ways to apply "$+$" to a list of length $4$.) As there are only so many ways to make a polyary operation from a binary one, we can safely label them with natural numbers and speak of things like $\sum^1$ (be it $\mathtt{foldl1}(+)$) to $\sum^{(n-1)!}$ (that would be $\mathtt{foldr1}(+)$).
My question then would be, whether with such tooling we could safely replace all the occurrences of repetitive application of binary operations in any algebra with their polyary counterparts, thus transforming an expression represented as a binary tree to an equivalent expression represented as a rose tree.
For example, $\sqrt[2]{\sqrt[3]{\sqrt[5]{7}}} \to \mathtt{foldr1}(\sqrt[]{})[2, 3, 5, 7]$ while $\sqrt [{ \sqrt [{ \sqrt[2] 3 }] {5} }] 7 \to \mathtt{foldl1}(\sqrt[]{})[2, 3, 5, 7]$
† As people in the comments correct me, there are as many ways to apply an operation "$+$" to a list of length $n$ as there are rooted planar binary trees with $n$ leaves, which is less than $(n-1)!$ — so (to my understanding) since it does not matter which of any two "$+$" operations we evaluate first if one of them is not going to use the results of the evaluation of the other, and therefore the corresponding interchange of labels does not increase the count of distinct ways of making an operator polyary.