Partial ordering on a space of matrices

Question

I am wondering as how to define partial ordering on a space of matrices! The following is what i intuitively constructed: If $M$ is a set all $n×n$ matrices with entries of each matrix are from an idempotent semiring then $M$ is also an idempotent semiring under the matrix addition and multiplication. Now, we can define a natural order on M as $A\leq B$ when $A+B=B$ for all $A, B$ in $M$, which is a partial order relation on $M$. Is this intuition correct?

Answer 1

A semiring $R$ for all $a\in R$, $a + a = a$ can be ordered by $a \leq b$ when $a + b = b$.

In addition $\leq$ produces a ring order. $a \leq b$ implies $a + c \leq a + c$, $ac \leq bc$.

Have you worked out the details?

The same order can be imposed upon M except possibly for $A \leq B$ implies $AC \leq BC$, which I haven't looked into.