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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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It is very easy to see that this is not feasible. It suffices to try $q=0$, so we'd look for a function $f(\cdot,\cdot)$ such that $x+y+z-1=f(x,y)+f(x,z)$. As we can switch variables, this would yield $f\equiv const$, which is impossible.

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