Bases In Semilinear Spaces Over Zerosumfree Semirings And Drazin Inverse M-matrices

This paper investigates the cardinality of basis in semilinear spaces of n-dimensional vectors over zerosumfree semirings and Drazin inverse M-matrices which are closed under the operation of Hadamard multiplication. First, it introduces the notion of an irredundant decomposition of element in zerosumfree semirings, discusses the cardinality of basis, and gives some equivalent conditions that the cardinality of each basis is n in Vn which give an answer to an open problem raised by Di Nola et al. in their work [Algebraic analysis of fuzzy systems, Fuzzy Sets and Systems158(2007)1-22]. Then, it generalizes the inverse M-matrix to Drazin inverse M-matrices and proves that Drazin inverse M-matrices are closed under the operation of Hadamard multiplication. In the end, it gives a question about Drazin inverse M-matrix, i.e., if A is Drazin inverse then so is A(r) for any real number r>1.