A Study On The Number And Structure Of The Convex Sub-lattice Of Lattice

“Lattices” is a special kind of algebraic system.It is an algebraic system integrating “order structure” and “algebraic structure”.Therefore,lattice algebra theory has an extremely wide range of applications in many fields.There have been many achievements in the study of lattice theory by domestic and foreign scholars.It is a very valuable work to further study and enriches the lattice theory on the basis of these valuable achievements."Subalgebraic structure" is an important means to study algebraic systems.For the lattice algebra system,there are three important sub-algebras,which are: ideals,sub-lattices and convex sub-lattices.Since convex sub-lattices is between ideal and sub-lattices under the set inclusion relationship,they should have better characterization potential from the perspective of algebraic structure.It has fairly good properties than the original lattice L.A large part of the study of lattice theory can be transferred to the convex sub-lattice-lattice.This shows a new way for the research work of Lattice Algebras.The convex sub-lattice-lattice theory is very important for broadening the application field of lattice theory.In this thesis,on the basis of previous studies on convex sub-lattice-lattices,the number,structure and characterization of convex sub-lattice-lattices of lattices are studied in depth.This thesis is divided into five parts.Part 1: Introduction.The concept,properties,research background,significance and innovation of this thesis is introduced.Part 2: Preparatory Knowledge.This thesis introduces some basic knowledge used in the full text,including some theorems and lemmas related to lattice theory,convex sub-lattice-lattices and ideal lattice.Part 3: The number and structure of convex sub-lattice-lattices two definitions of convex sub-lattices are given and their equivalence is proved.The sufficient and necessary conditions for convex sub-lattice-lattices to be chain,distributive lattices and power lattices are discussed.The number of elements and the formula of convex sub-lattice-lattices of finite lattice L are proved.The concrete structures of some common convex sub-lattice-lattices are given.The formula for calculating the number of elements in convex sub-lattice-lattices of direct product lattice of finite lattice is given.In this paper,we study the characterization of convex sub-lattice-lattices.The structure of convex sub-lattice-lattices of chain and chain direct product lattice is obtained.Part 4: Isomorphism of convex sub-lattice-lattices.The relationship between convex sub-lattice-lattices and direct product lattice is studied.The properties of convex sub-lattice-lattices isomorphism are discussed,and the degree of describing the original lattice by convex sub-lattice-lattices in the sense of isomorphism is solved.Answer the Problem I.10 raised by George Gratzer in his book "General Lattice Theory".Part 5: Summary and Prospect.