Is every partially ordered semiring an idempotent?

Question

I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~\forall~x\in R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none idempotent semiring? Or, is every partially ordered semiring an idempotent ?

Answer 1

$\mathbb R$ is a totally ordered semiring that isn't idempotent, for example.

I guess what made you forget about this is that the natural order on idempotent semirings is defined by $a\leq b$ when $a\oplus b=b$, which obviously doesn't occur in the example I gave.

If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.