I haven't read that paper but tropically \\(a+b\\) means the \\(max(a, b)\\) we know and love, and that operation, understood as a semilattice has associated the usual order \\(a \geq b \iff a \wedge b = b\\), as in [here](https://en.wikipedia.org/wiki/Lattice_(order)#Connection_between_the_two_definitions).