Moore-Penrose inverse of matrices on idempotent semirings

摘要

We characterize matrices over an idempotent semiring satisfying some additional necessary conditions for which the Moore Penrose inverse exists. The "(max, x) semiring," defined as the set of nonnegative real numbers R+, equipped with the operations a+b = max {a, b} and axb = ab is an example of such a semiring. The "(max, +) semiring", defined as the set of real numbers including, 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 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. [References: 15]