Maps Of Preserving Term Rank Of Matrix Over Nonnegative Semigroup

As one of the main research topics of linear preserving problem, the exploration of the additive mapping of preserving the term rank of matrixs over the nonnegative semigroup is of great significance. In this thesis, the additive preserving of the matrix term rank over the nonnegative semigroup has been investigated through additive maps over nonnegative semigroup rather than preserving of the linear mapping for matrix term rank over nonnegative semiring. Similar conclusions have also been obtained over nonnegative semring in this paper. Furthermore, the propertis of additive mappings on preserving the term rank of symmetric matrix of order n semigroup over nonnegative semigroup have also been discussed. The main contents of this paper are as follows:(1) Over m ×n matrix semigroup over nonnegative semigroup, the additive mappings will be non singular if it preserves two arbitrary given unequal term ranks of the matrix. If the additive mapping preserves term ranks of k and k+2, the additive map is strongly to preserve the term rank of k.(2) Over a nonnegative semigroup, the value difference of term rank between a symmetric symmetric matrix of order n whose elements in its main diagonal are 0 and its image has also been discussed when the additive map T preserves the term ranks of l, preserves k but not l and preserves k but not 2 respectively, in which 2 ≤ l< k≤ n.