If a continuous monotone function $f:[0,1]^2\to[0,1]$ aggregates nicely, is it "multiplication in disguise"? Suppose we have a continuous function $f:[0,1]^2\to [0,1]$ such that:

*

*$f(a,b)=f(b,a)$

*$b\ge c \implies f(a,b) \ge f(a,c)$

*$f(a,f(b,c))= f(f(a,b),c)$

*$f(0,0)=0, f(1,1)=1$
In other words, $f$ is monotone increasing and surjective onto $[0,1]$, and the result of turning a multiset of real numbers into a single real number by repeatedly replacing $a,b$ with $f(a,b)$ is independent of the order in which this aggregation is performed.
Obviously, $f(a,b)=ab$ works. In general, if $g:[0,1]\to[0,1]$ is a strictly montone function (either increasing or decreasing) whose image is the whole interval, then $f(a,b)=g^{-1}(g(a)\cdot g(b))$ also works.
Equivalently, if $h:[0,1]\to\mathbb{R}_{\ge0}\cup\{\infty\}$ is a strictly monotone function whose image is the full set of nonnegative real numbers along with $\infty$, then taking $f(a,b) = h^{-1}(h(a)+h(b))$ works (where we take $x+\infty=\infty$ for all $x$). This is just a transformation of the above example with $h(x)=-\log(g(x))$.
Are there any other examples? In other words, do all such nicely-aggregating functions necessarily behave like a relabeling of addition/multiplication under some bijective map?
One corollary of this that might either be provable without showing the full result or (if false) suggestive of a potential counterexample would be that $f(0,1)$ must equal either $0$ or $1$.
 A: $f(a, b) = \min(a, b)$ or $f(a, b) = \max(a, b)$ also satisfy your conditions but are not isomorphic to multiplication. Any disguised version of multiplication in your sense has the property that $f(a, a) = a$ iff $a = 0, 1$, but for $\min$ and $\max$ this is true for all $a$.
However, $\min$ and $\max$ can be obtained as limits of disguised versions of multiplication. It's a little simpler to discuss this example by taking logarithms so we can work on $\mathbb{R}_{\ge 0} \cup \{ \infty \}$ instead, and talking about disguised versions of addition. Then we can consider for $p \neq 0$ the "$p$-norms"
$$f_p(a, b) = \sqrt[p]{a^p + b^p}$$
which are disguised versions of addition, and which have the property that $\lim_{p \to \infty} f_p(a, b) = \max(a, b)$ and similarly $\lim_{p \to -\infty} f_p(a, b) = \min(a, b)$. ($\min$ and $\max$ have the nice property that they are invariant under strictly increasing maps, so translating this example back to $[0, 1]$ we just get $\min$ and $\max$ again.)
This is closely related to but not quite the same as the study of generalized means. You can also check out tropical geometry.
Also I don't know if these are the only counterexamples!
