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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.

Sourabh
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  • We want multiplication to distribute over addition, as in usual arithmetic. The way the operations are defined makes this true since, in real arithmetic, addition $(\otimes)$ actually distributes over maximum $(\oplus)$ :]

    Also, similar intuitive results carry over and as such we needn't learn a whole new set of theorems :)

     – Musa Al-hassy Jun 28 '14 at 23:09

1 Answers1

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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

Vladimir
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  • I an trying to find a paper which has some real life application of tropical mathematics and max plus algebra. I want to work on those examples. Can you provide me some help? I read this article Tropical Mathematics and found interesting application.If some thing related to this is available then please provide me the link. – Sourabh Jun 24 '14 at 05:59
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    http://www.ams.org/bookstore-getitem/item=CONM/616 – Vladimir Jun 24 '14 at 06:53
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    http://www.springer.com/mathematics/algebra/book/978-0-7923-4509-1 – Vladimir Jun 24 '14 at 06:57
  • I want to ask about something important in the same subject. when we say max-plus algebra we always mean tropical algebra? and is semiring is always tropical semiring? and what about semiring with anegation map is this also tropical semiring so we can apply their properties to the tropical semiring? – Mehrema Jan 27 '19 at 01:19