Relative Convex Sublattice As Well As Its Relative Ideal And Relative Filter

In abstract algebra,lattices are as important as groups?rings and fields.They're the most basic algebraic structures and are widely used in logic,algebra,topology,combinatorics and other mathematical fields.In terms of mathematical structure,there are three basic structures in mathematics:order?algebra and topology.A lattice is an important combination of ordered structure and algebraic structure.And its geometric features is the basis of topology on the lattice.As a subclass of lattices,the convex sublattice is used in vector space,mode theory and root theory.In physics,the convex sublattice is also used in the study of antiferromagnetism of insulators and complex mechanical systems.Therefore,it is necessary to study convex sublattice.As a special form of convex sublattice,the relative convex sublattice has attracted much attention in recent years.It provides a more convenient and effective way for research in related fields.In recent years,many scholars have studied the relative convex sublattice,such as:Professor Ping-hua Ming studied the property of union(intersection)of non-empty relative convex sublattice in distributive lattice;Professor Ai-ping Li pointed out the relation between the relative convex sublattic and the distributive power lattice,and discussed ideal and filter in a relative convex sublattice.On the basis of previous studies,the research was further carried out for the properties of relative convex sublattice,the relation between relative ideal(M]_T and relative filter[M)_T of a class of relative convex sublattice and the relation between relative convex sublattice and power lattice.The thesis is mainly divided into four parts:Part I:Prior knowledge.It introduces the historical background,research significance,wide applications and innovations of lattices and relative convex sublattices.Some basic definitions,lemmas,properties and theorems involved in the thesis are listed,including lattices,sublattices,distributive lattices,the direct product lattice,(prime)ideal and(prime)filter of lattices,the relative(prime)ideal and(prime)filter of lattices,and other definitions and relevant conclusions.Part II:Properties of relative convex sublattices.The union(intersection)of the relative convex sublattice of distributive lattices is a relative convex sublattice.It is proved that the direct product of the relative convex sublattice of two lattices can become a relative convex sublattice of the direct product of lattice from both positive and converse aspects.Then ecessary and sufficient conditions for a sublattice to be a relative convex sublattice are proved.Part III:Relative ideal or relative filter of relative convex sublattices.The expressions of the relative ideal(M]_T,relative filter[M)_Tand relative minimum ideal in relative convex sublattices are given.It is shown that the relative convex sublattice can be expressed as the intersection of the relative ideal(M]_T and the relative filter[M)_T.The necessary and sufficient conditions for a relative ideal to become a relative prime ideal are given.By means of relative convex sublattices,the relationship between the lattice prime ideal(prime filter)and the relative prime ideal(filter)is established.The necessary and sufficient conditions for homomorphic nuclei of relativistic prime ideals are formulated.Part IV:Relationship between relative convex sublattices and power lattices.The sufficient and necessary conditions for a relative convex sublattice to become a power lattice are proved.It is shown that lattice homomorphism mapping exists between a relative convex sublattice and a power lattice.The conditions for the relative ideal and the relative filter of a relative convex sublattice to become a power lattice are established.