The Nature Of Power Lattice And Fuzzy Power Lattice

Lattice that having the order relation is different from the algebraic systems of group,ring and field, that is to say, Lattice is a special partially ordered set. Because it has important applications in computer theory, security, switching theory, logic design and many science engineering fields, becoming an important branch of the modern mathematics.In addition to distributive lattice, modular lattice and complete lattice, the power lattice is another important lattice. In the non empty power set of lattice defining new operation, making it to be a lattice. It suggests power lattice.The mathematics structure of the universe by promoting to the power set, resulting in a new structure, is the improvement of the generation number structure in mathematics, that is to say, the power lattice is the product of the improvement of algebraic structure, a successful example of combing lattice with power set. The fuzzy power lattice, as a subclass of power lattice, can not only maintain certain characteristics of it, but also is different. It is important to learn power lattice and fuzzy power lattice. On the one hand, power lattice and fuzzy power lattice provide the algebraic semantics for logic structure; on the other hand, they generalize the algebraic structure which is widely used, having been widely used in computer theory,information processing artificial intelligence.The paper is on the basis of previous studies, by utilizing the method of combination of partially ordered set and power set, conducts in-depth exploration and promotion algebraic properties of power lattice and fuzzy power lattice, as the ideal and homomorphism. Some important conclusions are obtained.The article is divided into three parts:The first part: preliminary knowledge.The historical background, researches status and innovation of power lattice and fuzzy power lattice are introduced of the paper; the power lattice, fuzzy power lattice of the basic concepts, lemmas and results are given, The definition and related conclusions including:partially ordered set, lattice, ideal of lattice, homomorphism of lattice; power lattice,homomorphism of power lattice, fuzzy power lattice, ? cut set of fuzzy power set and so on.The second part: the nature of power lattice.The equivalence of algebraic lattice and partial lattice is extended to power set. It is proved that the algebraic power lattice and partial power lattice are equivalent; The necessary and sufficiency conditions are introduced for power set to become power lattice,and an example is given to illustrate the difference between the two; According to the meet of power lattices is power lattice, the join of power lattices whether it is a power lattice is discussed; Based on distributive lattice, power lattice is thought of a distributive lattice,whether the power lattice of a chain is a chain or not is discussed; The ideal of power lattice and prime ideal form and as the sufficiency and necessary conditions of the ideal of power lattice are given, the introduction of power lattice induced by the ideals of distributive lattices; the nature of power lattice of meet and join homomorphism and congruence relation are also described.The third part: the nature of fuzzy power lattice.The nature of power lattice is extended to fuzzy power lattice, the condition of the construction form of the fuzzy power set becoming to fuzzy power lattice is introduced; the relation of the fuzzy power lattice and the type of construction type of the fuzzy sub-lattice is given; the ideal of the fuzzy power lattice and form of prime ideal and the relationship of a prime ideal of the lattice and prime ideal of the fuzzy power lattice is also given.It is proved that the congruence relation of fuzzy power lattice can be determined by the homomorphism of the lattice.