A Further Study On The Properties Of Semi - Lattice Correlation

As a branch subject of algebra, lattice theory is a set which is defined logical operation to study its topological property. In the 19th century, British mathematician George Boole tried to symbolize the traditional logic to let it be more accurate and easy to operate. He introduced the "Boole algebra" which contained a serious symbols and algorithms, which lead to the origin of lattice theory. Lattice theory is developed naturally along with the progress of logic theory, it has important application values in many branch subjects of mathematics. Partial order and lattice theory now play an important role in many disciplines of computer science and engineering. For example, they have applications in vector clocks, global predicate detection of distributed computing, the occurrence nets of concurrency theory, the fixed-point semantics of programming language semantics, and concept analysis of data miming. It also has application values in some mathematics subjects, such as combinatorial mathematics, number theory and group theory.The more general distributive lattices have been investigated whose natural models are systems of sets at the same time the Boole algebra generated. The lattice of all subsets of a set is a complete distribute lattice. The distribute lattice plays an important role in the development history of lattice theory, and it is the motive of the generation of many lattices. The abstract algebra pioneer Dedekind observed that the additive subgroups of a ring and the normal subgroup of a group form lattices in a natural way, and these lattices have a special property, which was referred to as the modular law. The modular law can derived from the distributive law, but vice is not set up, so we can say the modular law is a result deduced from the distributive law. The two laws are generated along with the development of lattice theory, they are used to describe the property of lattice, there are many other laws similar with the two, such as complementation and orthogonality.The generation of orthomodular lattice is based on the complemented lattice, the ortholattice is orthomodular lattice if all the element pairs in the ortholattice are modular pairs. In the recent years, the study of orthomodular lattice focus on distributivity, perspectivity, irreducibility modular pair and center and ideal property. The study of orthomodular lattice have a role in the commutator and quantum mechanics. Maeda,F.and S.Maeda introduced a series of inequalities which could be characters of lattices, these inequalities are equivalent. We reference these inequalities in chapter 3, and introduce another 2 inequalities, and we proof their equivalence.Dedekind discovered the modular lattice through his observation of the lattice of the additive subgroups of a ring and the normal subgroup of a group. The modular law is a result of the distributive law, which means that the former can be deduced by the latter, but vice is not set up. Since we receive the counter-example which satisfy the modular law but not satisfy the distributive law. There are many laws similar with the two, which have relationship each other in lattice theory. The distributive lattice, which is the motivation of the generation of many results, play an important role in the development history of lattice theory, that is, many laws derived from distributive law. I summarize and verify some counter-examples in chapter 4, this job help me learn more about the difference and relation between different laws in lattice theory, which will be of great importance in further study to lattice theory.