The Study Of Generated Subsemimodules Over Commutative Semirings

In the article,we mainly investigate the bases and dimension of generated subsemimod-ules over semiring(R+,+,·,0,1)and a cancellative and yoked semiring,and discuss the linear transformation of L-semimodule M in detail.This article is divided into three parts,the details are as follows:In the first part,we first discuss some properties of linear relation of vector groups in semimodule Vn over commutative zerosumfree semirings,and give some conditions that each maximum linear independence group has the same number of elements.Especially,we discuss the generated subsemimodule over semiring(R+,+,·,0,1),and present conditions that each basis has the same number of elements.Finally,we study the dimension of generated subsemimodule over(R+,+,·,0,1)and obtain that there exists a subsemimodule with the dimension more than n in L-semimodule Vn.In the second part,we study the free bases and dimension in generated subsemimodule over a cancellative and yoked semiring C.First,we prove that each basis of W has the same number of elements under certain conditions,and present the structure of the bases.In particular,we obtain that there exists a linearly independent set of vectors with the number of vectors more than n,and there exists a subsemimodule W such that dimW>n.Then we discuss that the row[resp.column]vectors of the matrix are free when a matrix is invertible and give conditions that make vector groups free.In the third part,we mainly study some properties of range and nuclear of linear trans-formation A in semimodule M over commutative semirings.First,we discuss some operation laws and a series of related properties of linear transformation A,and give some condition-s for the existence of dimension in finitely generated subsemimodules.Then we give the concept of invertible linear transformation,and prove that if ?(L)= 1 in finitely generated free L-semimodule M,then the matrix of invertible transformation A under a set of bases is invertible.Based on this,we discuss the construction of range AM and nuclear A-1{0}in detail,and present some conditions that the formula dimAM+dimA-1{0} = dimM in classical linear algebra holds.