Semi-simplicity, Finite Condition And Zero-divisor Graph Of Complete Moular Lattice L

This dissertation,consisting of six chapters, is concerned with the semi-simplicity and finite conditions as well as zero-divisor graph of complete modular lattices.Chapter 1 summarizes the background and main results obtained in this thesis.Chapter 2 briefly introduces some elementary concepts and a few notational oddities that will be used through the thesis.Chapter 3 discusses the semi-simplicity of upper continuous and complete modular lattices. Let L be a complete modular lattice. We first discuss the basic properties of independent subsets of L, from which, some new charaterizations for L to be upper continuous and semi-simple are obtained, furthermore, the equivalent conditions under which the element 1 in L can be expressed as a join of independent atoms are also given if L is upper continuous.Chapter 4 is concerned with the finite conditions of complete modular lat-tices. Let L be a complete modular lattice, as a natural generization of the con-cepts of finitely generated and finitely cogenerated modules, we first introduce the concepts of finitely generated elements and finitely cogenerated elements in L,and give the equivalent charaterizations of the concepts. Second,we discuss the ascending and descending chain conditions, and give a series of equivalent conditions for L being artinian or neotherian, furthermore, the relationships between the finitely generated and finitely cogenerated elements and the chain conditions are established, if L is semisimple,it is proved that the four kinds of finite conditions are equivalent. Finally, we discuss an indecomposable decom-position that complements all maximal direct summands of L, and prove that all decompositions of independent atoms of 1 have the same cardinal in case L is upper continuous and semisimple.Chapter 5 is on the zero-divisor graph of bounded lattices. Let L be a bounded lattice, we introduce the concept of co-zero divisor graph of L, and give an example which shows that there is a bounded lattice L such that its zero-divisor graph is not isomorphic to its co-zero divisor graph. Second, the zero-divisor graph and co-zero divisor graph of boolean algebras are discussed, and an equivalent charaterization for L to be a boolean algebra is given, it is proved that the zero-divisor graph and co-zero divisor graph of a boolean algebra is isomorphic. Finally, we study the zero-divisor graph of an artinian lattice, and give a sufficient and necessary condition for the zero-divisor graph of an artinian lattice to be a finite complete graph, the classification of the zero-divisor of L is also obtained provided that L is artinian and only contains finite number of atoms, it is shown that the number of colors and the clique number of the zero-divisor graph arc both equal to n if the number of atoms contained in L is n,where n≥2.Chapter 6 is Peroration of this dissertation.