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When tropicalizing a subtraction-free expression (see Do all subtraction-free identities tropicalize?), is it more common to replace addition by max or by min?

Related issues:

Is there a name for the duality (for piecewise-linear functions involving addition, subtraction, max, and min) that switches max with min?

Is there a name for the duality (for rational functions involving multiplication, division, addition, and "parallel addition" as defined in Name and notation for a binary operation) that switches addition with parallel addition?

Background: I'm writing an article that defines certain rational functions involving multiplication, division, addition, and parallel addition, as well as their piecewise-linear tropicalizations, and I want to get the tropicalization correspondence right.

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It seems that the duality that switches $\max$ and $\min$ is called "negation", am I wrong? –  Theo Johnson-Freyd Jul 12 '13 at 4:47
    
@Theo: It's related to negation. You negate the variables in the expression and then negate the expression as a whole, e.g., min(x,y) = -max(-x,-y). –  James Propp Jul 12 '13 at 14:49
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