q-product semiring

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?

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