2
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ The paper you link has a discussion of generalized distributivity in section 4.2, but it requires a modification of the addition operation. $\endgroup$
    – S. Carnahan
    Apr 24 '16 at 12:06
1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ You can also set $z=y$ to find that $2f(x,y) = x + 2y - 1$, which is not symmetric in $x$ and $y$. $\endgroup$
    – S. Carnahan
    Apr 24 '16 at 11:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.