Generalized Inverse Of Matrix. On The Semi-ring

In this paper, the generalized inverses of a matrix over a semiring are stud-ied. The main results are as follow:1. The Moore-Penrose inverse of a matrix and a T-ordering among matri-ces over a semiring are studied. It is shown that the Moore-Penrose inverse is not only the smallest element of A{1,3,4}, but also is the largest element of A{2,3,4}. Some properties of the matrix T-ordering over semirings are given. Some properties of a matrix over a non-negative idempotent partially order semiring are given. It is obtained that a condition such that the Moore-Penrose inverse of a matrix A is equal to the transposed matrix of A.2. The concepts of monic matrices, epic matrices and matrices with Monic-epic factorization over a semiring are introduced. The Moore-Penrose inverse of them are studied. Some necessary and sufficient conditions of the Moore-Penrose inverse and the group inverse are obtianed, also the expressions of them are given. Some relationships between the Moore-Penrose inverse and group inverse are discussed.3. The reverse order law for the Moore-Penrose inverse of the matrices prod-uct is studied. A kind of the characterzation is given that the reverse order law for the Moore-Penrose inverse. This conclusion well be extended to A1A2…An. Finally, an example is given to illustrate that the existence matrices A, B meet the (AB)+=B+A+.