Matrix Semirings

The theory of matrices over semirings is a problem of algebra, and it has important applications in optimization theory and graph theory. The decom-position, indices and periods, generalized inverse of matrices over semirings are studied in this paper. It mainly achieved in the following aspects:1. A matrix over a finite distributive lattice can be decomposed into the sum of some matrices over a finite chain whose number of elements are of the same is proved. As a result, both the decomposition theorem of a matrix over a finite distributive lattice and the decomposition theorem of a matrix over a finite chain are generalized and extended. Further, by the decomposition theorem, the index and period of a matrix over a finite distributive lattice are studied and some interesting results are obtained.2. The permanent of matrices over semirings is studied and the properties of permanent are given. Also, we study the indices and periods of matrices over semirings, as a result, the estimation of periods of some matrices and a sufficient condition for matrices to be nilpotent are given.3. The generalized inverse of matrices over semirings is studied and a nec-essary and sufficient condition for matrices to exist generalized inverse is given. Also, the upper triangular matrices are studied, and sufficient conditions for up-per triangular matrices over semirings to be regular and to exist{2}-inverse are given and a necessary and sufficient condition for 2×2 matrices to be regular is given.