I'm a deep learning researcher, and these days studying algebraic geometry for my research and for personal interest.
I'm noob to this field, and I found a research area called tropical geometry (TG).
In TG, multiplication and summation is defined as:
$$x\oplus y = \max\{x,y\}$$ $$x\odot y= x + y$$
I don't get the motivation of these operations.
Up to my opinion, these operations makes sense if we define
$$x\oplus y = \lim_{t\to\infty}\log_t(t^x+t^y)=\max\{x,y\}$$
$$x\odot y = \lim_{t\to\infty}\log_t(t^x\cdot t^y)=(x+y)\lim_{t\to\infty}\log_tt=x+y$$
But I can't get why operations should defined like this.
Is there any motivation or motivating example?