Question 84
Find all $k \gt 0$ such that there is a function $f : [0,1] \times [0,1] \to [0,1]$ satisfying the following conditions:
a) $f(f(x, y), z) = f(x, f(y, z))$;
b) $f(x, y) = f(y, x)$;
c) $f(x, 1) = x$;
d) $f(z x, z y) = z^{k} f(x, y)$,
for any $x, y, z \in [0,1]$.
a) $f(f(x, y), z) = f(x, f(y, z))$;
b) $f(x, y) = f(y, x)$;
c) $f(x, 1) = x$;
d) $f(z x, z y) = z^{k} f(x, y)$,
for any $x, y, z \in [0,1]$.