The Structures Of Two Kinds Of Lattices And Their Applications

In this paper, the structures of two kind of lattices and their applications are investigated. First, a necessary and sufficient condition that a strongly atomic algebraic lattice is semimodular is shown by using the term of cell-lattices. Next, the notion of a weak coordinatization of a finite atomic lattice which has weaker hypotheses than a coordinatization is introduced, and the characterizations of all such weak coordinatizations are showed. Then the concept of a finite super-atomic lattice is given, and the structures of L(n) are discussed by their super-atomic lattices, and an algorithm for calculating all the super-atomic lattices in L(n) are proposed. This paper finally presents a specific labeling of finite atomic lattice and obtains the conditions that the specific labelings of finite atomic lattices are the weak coordinatizations or the coordinatizations by using the terminology of super-atomic lattices.