MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

q-product is defined as

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


  • $(+,\otimes_\infty)$ 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?

share|cite|improve this question
The paper you link has a discussion of generalized distributivity in section 4.2, but it requires a modification of the addition operation. – S. Carnahan Apr 24 at 12:06

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.

share|cite|improve this answer
You can also set $z=y$ to find that $2f(x,y) = x + 2y - 1$, which is not symmetric in $x$ and $y$. – S. Carnahan Apr 24 at 11:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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