Motivational distinctions between max and min conventions in tropical geometry

Question

I am aware that algebraically, there is no real distinction between the tropical semirings

1. $A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \infty, +, 0)$
2. $B = (\mathbb{R} \cup \{ - \infty \}, \text{max}, -\infty, +, 0)$
3. $C = (\mathbb{R}_{> 0} \cup \{ \infty \}, \text{min}, \infty, \times, 1)$
4. $D = (\mathbb{R}_{\ge 0}, \text{max}, 0, \times, 1)$

as made clear in this post.

But I wonder if there are motivational distinctions. What reasons might there be to prefer using one over the other in a certain field of mathematics. In the applications of tropical geometry, does one ever get tied to a specific one?

For example, if we define

$$x \oplus_{t,\mathbb{R}} y = \log_t(\log_t^{-1}x+_{\mathbb{R}} \log_t^{-1}y) = \log_t(t^x+_\mathbb{R}t^y)$$

and $$x \odot_{t,\mathbb{R}} y = \log_t(t^x \cdot_{\mathbb{R}} t^y)= x+_\mathbb{R}y.$$

Then $$\begin{cases} x \oplus y = \lim_{t \to \infty} (x \oplus_{t,\mathbb{R}} y) = \max \{x,y\} \\ x \odot y = \lim_{t \to \infty} (x \odot_{t,\mathbb{R}} y) = x +_\mathbb{R} y.\end{cases}$$

So in this case, the max convention arises if we're looking at the limit of semirings under this $\log_t$ map. I've seen this in algebraic geometry as a motivation, where we map varieties under a similar map to get amoebas, and see the tropical structure emerge as we take the limit. But maybe there's an analogous map leading to the min convention and varieties under this map look the same.

Either way, I wonder if the lack of distinction between these conventions extends to their motivation.

Answer 1

The $p$-adic valuation is usually defined to send a rational number $a/b$ to the greatest power of $p$ dividing $a$ minus the greatest power of $p$ dividing $b$, and this indeed leads to the opposite convention (since the $p$-adic valuation is just minus the logarithm of the $p$-adic absolute value). So one could perhaps argue that the opposite convention is natural from a $p$-adic perspective.