A Study Of Quotient Semimodules Over Commutative Semirings

In this article,we mainly investigate L-quotient semimodules and quotient transformation.Then we discuss linear and bilinear functions in detail.This article is divided into five parts,the details are as follows:In the first part,we discuss the quotient semimodules over commutative semirings.Based on the properties of L-semimodules,we propose a definition of L-quotient semimodules.First,we propose the definition of the relationship,and discuss the related properties of the relationship,and use the relevant examples to deepen our understanding.Then,through the study of the relationship,it is confirmed that the relation is equivalent,and the classification determined by the equivalence relation is derived,the representative elements are found out,and the common properties of the representative elements are explored.To define this concept of equivalence between two elements in a semimodule is understood by an example.Then,in the (?)-semimodules,the cancellable semimodules,dense subsemimodules and order relations are proposed to further explore the properties of the L-semimodules.Explore the relationship between the equivalence of two elements and the order relation between two elements in the L-semimodules.In addition,the common properties of the elements on the chain over the L-semimodules and its minimal elements are investigated.Based on the exploration of these definitions,properties,theorems and so on,we further know of the structure of Lquotient semimodules.Finally,the natural homomorphism is derived from the epimorphism,and the properties of the rank of L-quotient semimodules are discussed in combination with the natural homomorphism.In the second part,we discuss the quotient transformations on L-quotient semimodules induced by linear transformations over L-semimodules.In this paper,the linear transformations on L-semimodules M and the invariant subsemimodules,range and kernel of linear transformations are proposed,and their properties are discussed by an example of invariant subsemimodules.By discussing the properties of linear transformations over L-semimodules,the quotient transformation on L-quotient semimodules is induced and its concordance is verified.Finally,we explore some properties of the quotient transformations.In the third part,we discuss the linear function on the L-semimodule Vn over commutative semirings.Firstly,we give the definition of linear function.Then the image corresponding to zero vector is deduced by the definition of linear function.Subsequently,we define the equality of linear functions and the addition of linear functions and the scalar multiplication of linear functions,and according to the definition of linear functions,we conclude that the sum of linear functions and the scalar multiplication of linear functions are still linear functions.Moreover,according to the properties of linear functions,the uniqueness of linear functions on L-semimodules Vn is explored.Then we discuss L-semimodule Vn*(L-linear function semimodules)which is composed of linear functions and their operations on the Vn.And we begin to explore the basis on Vn*and the dimension of Vn*.In addition,we study the relation of Vn and Vn*.Finally,we discuss the properties of transition matrix on Vn*by the properties of transition matrix on Vn.According to the properties of Vn and Vn*combined with homomorphism and isomorphism,we discuss the relation between Vn and Vn*.The fourth part,we explore the bilinear function by the properties of the linear function.Firstly we define the bilinear function and we understand the bilinear function by the linear function.Then we explore the properties of bilinear function by the definition of bilinear function.Subsequently,we define the equality,addition and scalar multiplication of bilinear functions and we discuss their properties.Then we give the example of bilinear function and it can be further understood.According to the properties of bilinear function,we discuss the matrix of bilinear function under the basis of vn.Then we induce the contractual relationship between two matrices and discuss its properties,so as to discuss the contract of two matrix and the relation of bilinear functions.