functional equations - "Foldable" functions

Suppose $f:2^X\to X$ satisfies $f(x_1,\dots)=f(f(x_1,x_2),x_3,\dots)$. Min, max and sum are three such examples.

1. I've been calling these functions "foldable" because they bear some similarity to that concept from programming, but is there a real name for them?



2. Can anything interesting be said about them?

My motivation is driven by ethics and economics: if $u$ is some utility function, we might regard $u(x,y)=z$ as meaning that the basket $\{x,y\}$ is equivalent to the basket $\{z\}$, so $u$ would be foldable.

Answer

Any associative operator gives rise to (a family of) such functions, like $\sum_{1 \le i \le n} x_i$, $\prod_{1 \le i \le n} x_i$, $\bigcap_{1 \le i \le n} x_i$. Even the maximal common divisor and the minimal common multiple qualify. The cases $\min$ and $\max$ are just the natural extensions of those binary operations to several arguments.