Moore-Penrose inverse of matrices on idempotent semirings

We characterize matrices over an idempotent semiring satisfying some additi onal necessary conditions for which the Moore Penrose inverse exists. The " (max, x) semiring," defined as the set of nonnegative real numbers R+, equi pped with the operations a+b = max {a, b} and axb = ab is an example of suc h a semiring. The "(max, +) semiring", defined as the set of real numbers i ncluding, equipped with the operations a+b = max {a, b} and axb = a + b is another example. Some of our results generalize known results in the case o f the binary boolean algebra ( a trivial idempotent semiring). We give an a lgorithm to compute the Moore Penrose inverse, when it exists. We also make comparisons with similar results over the conventional algebra.