I am a undergraduate student. I am doing a project on tropical geometry and Max-Plus Algebra. So I started reading about max plus algebra. First thing I came across is that here we don't use conventional algebra. Instead of that we define a new algebra with the operations as follows-
$ a \oplus b = max(a,b)$
$a\otimes b= a+b$
So my question is this why we need to define operations like this only? why can't we have operation like-
$a \oplus b =a+b$
$a\otimes b= max(a,b)$
because here we have new addition same as the old one, only multiplication is different. I am unable to find the mathematical reason behind it where this fails.
Because then the new multiplication does not distribute over addition, $a\otimes(b\oplus c)\ne (a\otimes b)\oplus(a\otimes c)$ in general. For example, $4=\max(2,1+3)\ne \max(2,1)+\max(2,3)=5$ .
On the other hand, there are various versions of tropical rings; you can take, e.g., $a\oplus b:=\max(a,b)$ and $a\otimes b=\min(a,b)$ etc. A useful reference which gives a survey of possible versions and explains what is behind tropical (idempotent) mathematics can be found in http://www.mccme.ru/tropical12/Tropics2012final.pdf