By definition: a <= b iff a + b = a
The natural order is a negative partial order iff the semiring is
idempotent. It is trivially monotonic for plus. It is left (resp. right)
monotonic for times iff the semiring is left (resp. right) distributive.
It is a total order iff the semiring has the path property.
"Semiring Framework and Algorithms for Shortest-Distance Problems",
Journal of Automata, Languages and Combinatorics 7(3):321-350, 2002.
We define the strict version of this order below.