Research On Upper And Lower Sets Of Lattices And Some Counterexamples

Lattice theory,as a branch of abstract algebra,is a logical operation defined on a set system.Its molding mark was the "Lattice Theory" by American mathematician Garrett Birkhoff in 1937-1939.After nearly 80 years of development,lattice theory has developed into an independent mathematics discipline,with extensive applications in many fields including abstract algebra,projective geometry,topology,point set,number,logic and probability theory.There are many types in lattice theory,such as distributive lattice,modular lattice,semimodular lattice,atomic lattice,algebraic lattices,there are inclusion relations and even equivalence relations,while those lattices of types that do have no equivalence relationship need to enumerate the corresponding counter-examples,through the study of counter-examples can deeply learn to understand lattice theory.In this paper,we introduce the upper and lower sets of lattice to characterize lattice,and study some counter-examples in lattice theory.The first chapter briefly describes the research background and significance of counter-examples,research status in China and abroad and major research results.The second chapter introduces the definitions and theorems of modular laws,atomic lattice,complementarity and ideal and their related conditions to be used in this study.The third chapter introduces a new type of special subsets—upper sets and lower sets of lattices.By studying,this special subset is found to have a fairly good characterization of the lattices,so we give the characterization theorems of some lattice classes based on the upper and lower sets.At the same time,these characterization theorems also have a good role in understanding the lattice theory,and discuss the construction of the upper and lower sets themselves.The fourth chapter discusses some representative counter-examples of atomic lattice,module laws,ideal and complementarity in lattice theory,and combines the relevant theories to add appropriate characterization to make them equivalent and give proof,while simplifying the known counter-examples and giving minimization proof.