Matrix Representation Of Finite Distributive Lattices

This paper mainly considers the Boolean matrices on binary Boolean algebras,mainly including the following aspects:matrix representation of finite distributive lattices,the properties of matrix representations of congruence relations of finite distributive lattices,distributive pseudocomplemented algebras,Heying algebras and their corresponding congruence matrices and principal congruence matrices,conditions for column space lattices to become Stone algebras,matrix characterization of algebraic properties of subdirectly irreducible of various algebras.This paper consists of six main chapters:Chapter 1,introduction.It briefly introduces the background of this research topic,the status of domestic and foreign research and the significance of the research,and briefly outlines the main work and innovation points of this paper.Chapter 2,preliminary knowledge.The concepts of finite distributive lattices,distributive pseudocomplemented algebras,Stone algebras and Heying algebras and their related properties are introduced.It also introduces the concept of Boolean matrices and their related properties.Chapter 3,matrix representation of finite distributive lattices.Firstly,we define the preserving incidence relation mapping between the elements of a matrix and study that any finite distributive lattice can be represented as the column space lattice of some reduced partial order relation matrix.Secondly,we define the down sets of the partial order relation matrix,and give the relationship between the down sets of the matrix and the preserving incidence relation mapping.Finally,the preserving incidence relation mappings between reduced partial order relation matrices and the homomorphisms between their column space lattices are established.Chapter 4,matrix characterization of congruence relations for finite distributive lattices.Firstly,this chapter establishes a new operation f(α)=(?)~TA,g(β)=A(?)~T,by using the row vectors and column vectors of the partial order relation matrix A to form the row and column space lattice B(A),and obtains the matrix characterization of the congruence relations of the finite distributive lattice.Then,by using the difference and complement relation between matrix column vectors,we obtain the condition of principal congruence relation of matrix induced finite distributive lattice.Secondly,we further explore how to express the P-congruence relations and P-principal congruence relations in matrix language under the condition of distributive pseudocomplemented algebra,and give some matrix properties.Finally,the H-congruence matrix and the H-principal congruence matrix of Heying algebra are studied in the context of the reduced partial order relation matrix.Chapter 5,when is a column space lattice C(A)a Stone algebra?First,this chapter explore the distributive pseudocomplemented algebra in more depth:the Stone algebra.The Stone algebra is characterized using inverse preserving incidence relation mapping,and three equivalent propositions are given for characterizing the Stone algebra.Next,this chapter uses the inverse preserving incidence relation mapping to represent its congruence relations.Based on Chapter 4,the matrix representation of the S-principal congruence relation of the Stone algebra is characterized.Chapter 6,subdirectly irreducible.After studying the matrix characterization of congruence relations and principal congruence relations of various algebraic categories,this chapter further explores matrix characterization of algebraic properties of subdirectly irreducible.