Generalized Inverse Of Matrices Over Quantales (Semirings)

The concept of Quantale was given by Mulvey C J in1986.It was firstly drew into in studying unexchange C*-algebra as a new mathematic model for quantum mechanics.In recent years,Zhao bin?Han Shengwei?Liang Shaohui et al have studied inverse?{1}-inverse and generalized M-P inverse of Quantale matrix,they have obtained some properties.In1936, Von Neumann has given the concept of regular semiring(every element has {1}-inverse) and proved that the necessary and sufficient condition for R is regular ring is the multiplication of Mn(R) is regular.But if R is anti-ring, multiplication of Mn(R)(n?3) is not regular.The conditions for this kind upper triangular matrixes when the order number n satisfied3?n?4in additively-idempotent division semiring exist {1}-inverse have be given by Li pei.This paper is a continuation of the above researches.In Chapter three, an analogy of morphism or ring involution operation, Quantale involution operation and Quantale matrix involution operation are defined. The definition of generalized M-P inverse of Quantale matrix is introduced. Using transposition operation to define generalized M-P inverse of Quantale matrix as a special example of using involution operation to define generalized M-P inverse of Quantale matrix.Using involution operation to define generalized M-P inverse of Quantale matrix is more extensive than using transposition operation to define generalized M-P inverse of Quantale matrix.Under the different involutions for the same Quantale matrix that its generalized M-P inverse are different by examples.In addition,some necessary and sufficient conditions for the uniqueness of generalized M-P inverse of Quantale matrix when it exists are obtained. Some of its properties are studied and some of its expressions are given.In Chapter four, the definition of weighted generalized M-P inverse of Quantale matrix and generalized left(right) cancellable are given.Under these definitions, if Quantale matrix exists weighted generalized M-P inverse,then it is unique. On the basis,by using the method of ring theory,some characterizations for a Quantale matrix exists a weighted generalized M-P inverse are obtained and some of its expressions are given.In Chapter five, The necessary and sufficient condition that reverse order law for the existence of weighted generalized M-P inverse of Quantale matrix is obtained.In Chapter six,the relationship between the entries of this kind upper triangular matrixes are given by mathematical introduction. On this basis,each matrix of this kind upper triangular matrixes is regular matrix and the existence of {2}-generalized inverse is proved.