Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.

If $\otimes$ is +, what are the possible operators for $\oplus$?

So far I have proven that max and softmax (logsumexp) are solutions. Can we characterize all possible $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.

If $\otimes$ is +, what are the possible operators for $\oplus$?

So far I have proven that max and softmax (logsumexp) are solutions. Can we characterize all possible $\oplus$?

I'd also appreciate references to relevant papers.

Assume that ($\oplus$, $\otimes$) is a semiring over the positive reals.

If $\otimes$ is +, what are the possible operators for $\oplus$?

So far I have proven that max, min, and softmax (logsumexp) are solutions. Can we characterize all possible $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.

If $\otimes$ is +, what are the possible operators for $\oplus$?

So far I have proven that max and softmax (logsumexp) are solutions. Can we characterize all possible $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the positive reals.

If $\otimes$ is +, does that imply that $\oplus$ is logsumexp, thereby making ($\oplus$, $\otimes$) the log semiring?

Clearly $\oplus$ = logsumexp is a solution but I would like to prove or disprove its uniqueness.

Assume that ($\oplus$, $\otimes$) is a semiring over the positive reals.

If $\otimes$ is +, what are the possible operators for $\oplus$?

So far I have proven that max, min, and softmax (logsumexp) are solutions. Can we characterize all possible $\oplus$?