Nilpotent Matrix And Idempotent Matrix On The Distributive Lattice

From the aspect of mathematical structure,mathematics has three basic structures:order,algebra and topology.Lattice is an important combination of ordered structure and algebraic structure,which is closely related to fuzzy mathematics,topology and other modern mathematics.The concept of lattice appear in various fields of mathematics.It is widely used in secrecy,logic,combinatorial science and computer science.The distributive lattice occupies a very important position in the study of lattice theory,which promotes the development of the general lattice theory.Matrix is an important tool of mathematical research and application,so the matrix on the distributive lattice is particularly important.The matrix on the distributive lattice is derived from practical problems.It has be widely used in many fields.Such as automation theory,finite graph theory and computer switch design.Therefore,intensive study of the matrix on the distributive lattice will certainly play a very good role in promoting the solution of the actual problems.Adjoint matrix on the distributive lattice inherits many properties of the original matrix.It is the main tool for studying the operation of lattice matrices.Nilpotent matrix and idempotent matrix on the distributive lattice are important types in matrices.Many scholars have used(?)or(?)to define the operation of the matrix on the lattice for many years.Nilpotent matrix and idempotent matrix on the distributive lattice are studied by using the adjoint matrix and its sequential principal subexpression.And norm operation is the basic structure in the Fuzzy set theory,which summarizes the various operations in the Fuzzy set theory and has good properties.Therefore,it is necessary to study the nilpotent matrix and idempotent matrix with the help of adjoint matrix on the distributive lattice by extendeing norm operation to the lattice and using triangular norm to define operation of the lattice matrix.On the basis of previous studies,this paper makes a further study about adjoint matrix,nilpotent matrix and idempotent matrix on the distribution lattice and obtains some important conclusions by using new method and improved method.The article is divided into three parts:The first part:preparatory knowledge.This part introduces the significance,function,research status and innovation points of adjoint matrix,nilpotent matrix and idempotent matrix on the distributive lattice and gives the basic concepts,reasoning and results used in the study of adjoint matrix,nilpotent matrix and idempotent matrix based on triangular norms on the distributive lattice,including:lattice,distributive lattice,S norm,T norm,adjoint matrix,S-nilpotent(idempotent)matrix,T-nilpotent(idempotent)matrix and so on.The second part:adjoint matrix on the distributive lattice.This part expounds the relationship between the adjoint matrix and the original matrix on the distributive lattice and gives the properties of the invertible matrix and positive definite matrix on the distributive lattice.And rows(columns)orthogonal and 1 decomposition of the matrix on the distributive littice equivalent to invertible also be pointed.The third part:nilpotent matrix and idempotent matrix based on triangular norm on the distributive lattice.For nilpotent matrices based on triangular norm over distributive lattice,sufficient conditions of irreflexive matrix becomes a nilpotent matrix after doing(?)-T operation are given.By using adjoint matrix,we obtain the necessary and sufficient conditions for the irreflexive matrix to be a S-nilpotent matrix on the distributive lattice after doing(?)-S operation.With the help of principal submatrix,the sufficient condition of irreflexive matrix becomes a S-nilpotent matrix after doing(?)-S operation is obtained.In addition,It is proved that the result of after doing(?)-S operation between S-nilpotent matrix and irreflexive matrix is still S-nilpotent matrix.For idempotent matrices based on triangular norm over distributive lattice,conditions of T-idempotent matrix and S-idempotent matrix are obtained by means of adjoint matrix.The irreflexivity,reflexivity and idempotence of T-idempotent matrix and S-idempotent matrix on the distributive lattice were introduced.A necessary and sufficient condition of irreflexive matrix becomes a S-idempotent matrix were given.