Some Studies Of Semilinear Spaces Over Pseudorings

In recent years,the study of linear algebra over semiring has attracted the attention of many scholars.In this thesis,the linear transformation of semilinear space Vn(D)over pseudoring is studied,the rank and maximum column rank of the matrix over the pseudoring are discussed.The invertible matrix over the pseudoring is studied,and the main results are as followsFirst,by discussing the basis of Vn(D),it is proved that for each a ∈Vn(D)is unique represented by the basis of Vn(D).the intimate connection between the linear transformation of Vn(D)and matrix over pseudoring is revealed when the basis is determined.At the same time,the equivalent conditions for judging the invertible linear transformation of Vn(D)are given.By defining the similar transformation of Vn(D),the similar linear transformation of Vn(D)is studiedSecond,the relationship between the rank and maximum column rank of a matrix over pseudoring is discussed.The condition that the rank of the matrix over pseudoring is equal to the maximum column rank of matrix over pseudoring is given.Also,the linear operator that preserve the rank of the matrix and maximum column rank of matrix over the pseudoring is describedThird,according to the characteristic of invertible matrices over pseudor-ing,the relationship between an invertible matrix over pseudoring and n-th symmetry group Sn is studied.Invertible matrices over pseudoring are classified by means of n-th symmetry group Sn.Also,the properties of invertible matrix over pseudoring are discussed.Give the method that the invertible matrix over pseudoring become the invertible diagonal matrix.