Consider the following problem:
Take the function $\mathcal{V}:\mathcal{A}\rightarrow\mathbb{R}$ where $\mathcal{A}$ is the set of finite multisets $A=\{a_1,a_2,a_3,...,a_N\}$ for $a_n\in\mathbb{R}$ and $0\le a_n\le1$. $\mathcal{V}$ is commutative $\mathcal{V}(\{a_1,a_2\})=\mathcal{V}(\{a_2,a_1\})$ and incrementative $\mathcal{V}(\{a_1,a_2\})<\mathcal{V}(\{a_1,a_3>a_2\})$. What is the set of operators which can define $\mathcal{V}$?
After playing around with this problem, I'm pretty sure any operator directly proportional to the sum can define $\mathcal{V}$. For example, $\mathcal{V}$ can equal the sum $\sum_{n=1}^{N}a_n$, or the average $\sum_{n=1}^{N}a_nN^{-1}$, but not the product $\prod_{n=1}^{N}a_n$. I would like to be able to prove this though. It is pretty easy to prove a simpler version where $\mathcal{V}$ is resricted to being arithmetic, but I would like a more general proof than that.
More than just having a solution to the above problem, I would like to be able to solve any similar problem; However, I can't figure out which area of mathematics studies this. Which area of mathematics is it, and do you have a recommended textbook?
It's not really a major area of math, but it could be considered a problem in universal algebra.
There are infinitely many $\mathcal{V}$ that satisfy your constraints. To see this, it's useful to imagine selecting the value of $\mathcal{V}(A)$ independently for every multiset $A$. What constraints arise that would restrict this process? Well, both of your constraints only refer to multisets with the same number of elements. So what choices of $\mathcal{V}$ we have for multisets of 3 elements could be some completely independent from the value of $\mathcal{V}$ on multisets of 2 elements. For example, you could have $$ \mathcal{V}(\{x, y\}) = x^2 + y^2 \\ \mathcal{V}(\{x, y, z\}) = x + y + z \\ $$
Also, your "commutative" requirement ($\mathcal{V}(\{a_1,a_2\})=\mathcal{V}(\{a_2,a_1\})$) is technically redundant as it is true for any function on multisets. It would be required if you said $\mathcal{V}$ was a function on lists.
Given the above considerations, here is an exact description of the infinite number of possible choices for $\mathcal{V}$: choose for every $n$ a monotone increasing function $f_n: \mathbb{R}^n \to \mathbb{R}$ for each $n$ independently (there are actually uncountably many choices for each $n$). That is, choose any list of such functions $f_0, f_1, f_2, \ldots$. Then set $$ \mathcal{V}(A) = f_{|A|}(A) $$ where $|A|$ is the size of $A$.