The Relationship And Nature Of Pseudo-boolean Algebras And Related Distribution Lattices

As the algebraic structure of classical logic system, Boolean algebra is of great signification in the study of logic algebra. Pseudo-Boolean algebra is a generalization of Boolean algebra. In this paper, the properties of Pseudo-Boolean algebra and relevant distributive lattice were further discussed. By comparing the difference between a few of kinds of distributive lattices, such as soft-algebra, Heyting-algebra, R0algebra and DFI-algebra, the relations of Pseudo-Boolean algebra and them were discussed, And several sufficient and necessary conditions under which a Pseudo-Boolean algebra becomes a Boolean algebra were given. Secondly, The properties of the distributive lattices relevantly were discussed. The new structure was defined by expand the operation of the.Ro-unit interval, and the isomorphism of them was proved. The definition of the distance function is introduced into DFI-algebra, and some basic properties of the distance function of DFI-algebra were discussed. Finally, The corresponding metric space was established according to the distance function which introduced, And some properties were given.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:Preliminaries. We give the basic concepts and related theorems of lattice, distributive lattice, residuated lattice, Pseudo-Boolean algebra, distance function, which will be used in this paper.Chapter 2:The relation between Pseudo-Boolean algebras and a few kinds of distributive lattices. Firstly, the relation between Pseudo-Boolean algebras and soft algebra was given; Then, several sufficient and necessary conditions under which a Pseudo-Boolean algebra becomes a Boolean algebra was proved; finally, the relation between Pseudo-Boolean algebra and DFI-algebra was discussed.Chapter 3:The relation between Pseudo-Boolean algebras and Heyting algebra, R0-algebra and the relevant properties. Firstly, the relation between Pseudo-Boolean algebras and Heyting algebra, R0 algebra were discussed; Then, We expand the operation of Ro-unit interval, and the proof of the isomorphism relation was given.Chapter 4:The distance function of DFI-algebra. Firstly, The definition of a distance function in DFI-algebras was given. It's properties were discussed in this chapter, and some special properties of the distance function of some kinds of special DFI-algebras, for example, linear DFI-algebras were given; Then, A new operation was introduced in DFI-algebras, and the equivalent definition of the dis-tance function of DFI-algebras and some farther properties were given. Finally, the corresponding metric space was established according to the distance function introduced, some properties of the metric space of Pseudo-Boolean algebras were given.