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This question come to my mind with this example:

Let's say $a\oplus b = \min(a,b)$ and $a\otimes b = a+b$. We will work in $S = \mathbb{R} \cup \{\infty \}$. They do tropical geometry with this. But some sources do same thing with this definitions: $a\oplus b = \max(a,b)$ and $a\otimes b = a+b $ in $S = \mathbb{R} \cup \{ -\infty \}$. And, if you look the topic, you can see that main objects have same structure in both situation. We use $\infty$ or $-\infty$ in the sense of identity element of $\oplus$ and they do (maybe a little) nothing else in the work of tropical geometry. So, we can use both structures to do tropical geometry. This is nearly obvious but if I want to be sure, how should I prove such a thing?

When I say "structure", I mean that a system which has operations and a set. How can we show that two structures are equivalent? First of all, what is equivalent mean? For example, we can focus on polynomials in this structure (I know, some of these structures may have not a polynomial definition) and prove the equivalence of structures for just that aspect. Still, how can we write such a proof formally?

dfeuer
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    Paging Lord_Farin.... Darn, I wish that worked. There are multiple layers to this question. The simple step is to demonstrate an "isomorphism" between the structures: in this case the function $x\to -x$ will do the trick. The second and trickier trick is to show that within the formal logic you wish to work in, any two isomorphic structures are indistinuishable. I haven't studied logic and/or category theory, so I don't know the details. – dfeuer Aug 12 '13 at 22:33
  • About equivalence, there's the strong concept of isomorphism, and then there's the weaker concept of elementary equivalence. In order to "focus on polynomials", perhaps you're more interested in equivalence? I guess it's moot, since the two structures in question are indeed isomorphic, and it's easy to prove it. – Chris Culter Aug 12 '13 at 23:49

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For clarity, I use the following notation:

$$a\boxplus b:=\min\{a,b\}$$ $$a\oplus b:=\max\{a,b\}$$ $$\Bbb R_\uparrow:=\Bbb R\cup\{\infty\}$$ $$\Bbb R_\downarrow:=\Bbb R\cup\{-\infty\}.$$

Note: I assume that $\pm\infty$ are additively absorptive elements in their respective structures--that is, that for any $a\in\Bbb R_\uparrow,$ we have $a+\infty=\infty=\infty+a,$ and for any $a\in\Bbb R_\downarrow,$ we have $a+-\infty=-\infty=-\infty+a.$

(This assumption isn't necessary, really. More of a side note. The answer below only requires that +, $\boxplus,$ and $\oplus$ are well-defined on their relevant sets.)

Showing that the structures $\Bbb R_\uparrow$ and $\Bbb R_\downarrow$ are equivalent means demonstrating a bijection $f:\Bbb R_\uparrow\to\Bbb R_\downarrow$ such that for all $a,b\in\Bbb R_\uparrow$ we have that $$f(a+b)=f(a)+f(b)$$ and $$f(a\boxplus b)=f(a)\oplus f(b).$$ (Alternately, you could work in the other direction. It makes no difference.)

Your polynomials in these structures will be formal (that is, we needn't be able to evaluate them), and so the bijection above will induce a bijection between the polynomials in these structures. If you'd like more details on that, let me know, and tell me exactly what you mean by polynomials. I'll be glad to add to my answer.

Cameron Buie
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