Moore--Penrose Inverse of Matrices on Idempotent Semirings

We characterize matrices over an idempotent semiring satisfying some additional necessary conditions for which the Moore--Penrose inverse exists. The "(max,×) semiring," defined as the set of nonnegative real numbers ${\mathbb R}^+,$ equipped with the operations $a\oplus b= \max\{a,b\}\ \mbox{and}\ a\otimes b= ab$ is an example of such a semiring. The "(max,×) semiring,"', defined as the set of real numbers including $-\ity,$ equipped with the operations $a\oplus b= \max\{a,b\}\ \mbox{and}\ a\otimes b= a+b$ is another example. Some of our results generalize known results in the case of the binary boolean algebra (a trivial idempotent semiring). We give an algorithm to compute the Moore--Penrose inverse, when it exists. We also make comparisons with similar results over the conventional algebra.