Let $R$ be a commutative semiring with unit.
Assume that addition is cancellative so that $R$ is included in its ring completion.
Fix ${a, b \in R}$. A simple calculation shows that ${a - b}$ is invertible in the ring completion of $R$ if and only if there are ${x, y \in R}$ such that ${a x + b y = 1 + a y + b x}$.
I am looking for (nontrivial, needless to say) general sufficient conditions on $R$ guaranteeing that, for ${a, b, x, y}$ as above, ${a \leq b}$ or ${b\leq a}$ in $R$.
Do you know some?