I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$.
The semi-ring is freely generated from a finite set $\Sigma$, the semi-ring itself would be of infinite rank.
We can also assume $\times$ distributes into $+$ from the left (or right, doesn't matter). There are also the usual identity elements 0 and 1.
How may I construct matrix representations for it? What would be the constraints for those matrices? Can someone give a simple example?