q-product is defined as

$x \otimes _q y = (x^{1-q}+y^{1-q}-1)^{1/(1-q)}$

Observation:

- $(+,\otimes_\infty)$ is min-plus tropical semiring on the segment $[0,1]$
- $(+,\otimes_1)$ is R
- $(+,\otimes_{-\infty})$ is max-plus tropical semiring on $[0,\infty]$

Consider the following "generalized distributivity" law:

$x \otimes _q (y+z) = (x \otimes _p y)+(x \otimes _p z)$

I'm looking for closed form expression for $p$ in terms of $q$, or this is not feasible?