Is the natural order relation on an idempotent semiring total/linear? We know that on an idempotent semiring $R$, the natural order relation is defined as: for all $x, y\in R$, $x\leq y$ when $x+y=y$, which is clearly a partial order relation. I am unable to point out whether this relation is a total order relation too? i.e., does it satisfy Comparability (trichotomy law)?

A distributive lattice is an idempotent semiring (with addition $\vee$ and multiplication $\wedge$), but most lattices are not totally ordered. 
