A Number Of Topics. Cell And Super Cell Research

The concept of lattice was put forward by Dedekind firstly. The modern lattice theory came into being in the thirties of this century. The book 《lattice theory》 which was written by Birkoff enriched and developed the lattice theory further until the forties. In recent years, as order and partial ordered set theory were widely applied in the combinatorics, Fuzzy mathematics, computer science, and even in the social science, the lattice theory has became an important branch of mathematics. Since F.Marty advanced the hyperalgebraic system in 1934, the theory of hyperalgebraic has attracted a good deal of attention of many scholars. Therefore, many branches of hyperalgebras appeared, such as hypergroup, hyperring, hyper BCK-algebras, hyper BCI-algerbras and so on. The hyper algebraic system theory is used in many aspects of pure and applied mathematics. Proffessor Xin Xiaolong introduces the hyper algebraic system theory into the algebraic structure of lattice theory and put forward the concept of hyperlattice with a certain number of valuable results. In succession, professor ZhaoBin studies the distributive hyperlattice and the ideals of the hyperlattice. It deserves us much research on the hyperlattice which is the extension of lattice algebraic system.This thesis studies a certain number of lattice and hyperlattice algebraic characteristics, mainly discussing them from the following several aspects.In the second chapter, we define the lattice differential, study some properties of it and also obtain a certain number of equal proposition.In the third chapter, we give the concept of the filtered product and the ideal sum and set the chineses Remaider Theorem on the distributive lattice. meanwhile, the isomorphism theorem on the distributive lattice is made clear.In the forth chapter, first of all, we raise the definitions of the least element, the greatest element and the bounded hyperlattice and illustrates the reasonable elements by example. Then, we make a further study on the ideal of hyperlattice on the base of the definition of hyperlattice ideal made byprofessor ZhaoBin in the article.In the fifth chapter, we put forward the concepts of the power hyperlattice, distributive power hyperlattice and power hyperlattice direct product. Some worthy properties of power hyperlattice is also discussed in this chapter.