In a computer science class about automata the professor claims that a semiring can not have both additive inverse and canonical partial order at the same time. I do not understand how additive inverse contradicts with canonical partial order?
The canonical partial order is defined as $\forall a,b \in F. [ a \leq b \equiv \exists c \in F. a+c=b ] $.
If the additive inverse exists than a possible $c$ is $c=(-a)+b$ which would result into $a \leq b \equiv a+(-a)+b \leq b = b \leq b$. But this does not look like a valid proof by contradiction.