About A Half-ring Of Generalized Inverse Matrix

Generalized inverse of matrice and linear preserver problem (LPP for short) are two active areas in matrix theory, It has wide applications in differential graph theory, geomatics, economics and so on. This dissertation studies the generalized Moore-Penrose inverses of matrices over a semiring and invertible linear operators which preserve Moore-Penrose inverse of matrices over semirings.The main results are as follows:1. The generalized Moore-Penrose inverses of matrices over a semiring are introduced and studied. The properties of eliminating and symmetry of matrices over a semiring are used to analyse the existence and uniqueness of the gen-eralized Moore-Penrose inverses. Also some charaterization of the generalized Moore-Penrose inverses are given. The obtained results generalize some conclu-sions of Moore-Penrose inverses of matrices over a semiring.2. The weighted group inverses of matrices over asemiring are studied. The existence and uniqueness of weighted group inverses over a semiring are anal-ysed. Some necessary and sufficient conditions of product of three matrices the weighted group inverses over a semiring are obtianed, also the expressions of are given. The obtained results generalize some conclusions of group inverses of square matrices over a semiring.3. It is studied that invertible linear operators which preserve Moore-Penrose inverse of matrices over ai-semirings. Also it is completely charaterized that invertble linear operators which preserve Moore-Penrose inverse of matrices over special classes of semirings, including chain semiring, bounded distributive lattice, completely distributive lattice and general Boolean algebras.