Some Properties Of Closed Lattice And The Lattice’s Representation By The Family Of Subsets

In this paper, some equivalent characterizations of the closed lattices aregiven and the relationship between the Locale and the closed lattice areinvestigated. Moreover, we also propose a special type of closure system on aset X, named super-meet-structure, and discuss how to represent lattices by thefamily of subsets.In chapter one, we describe the development of lattices as well as posets.In recent years, posets and lattice theory use more widely in combinatorialmathematics, fuzzy mathematics and theoretical computer science, and even thesocial sciences, but also greatly promoted the development of the concept ofself-discipline, making mathematics and theoretical computer science majorstudy. At First, we cite many domestic example which the lattice combine withother relevant discipline. Then, some results on the structures of posets aregiven. At last, we provide some preliminary knowledge about posets, lattices,category and so on.In chapter two, we investigate some properties of the closed lattices andgive their characterizations. The relationship between the Locale and the closedlattice are discussed. We define the complete sub-lattice of closed lattice and itis still closed when satisfy certain conditions. We prove that the cartesianproduct of the close lattice is still closed lattice. Moreover, We obtain the resultthat the image of the closed lattice under a surjective map preserving arbitrarysups is still the closed lattice. Finally, we study the property of the image of theclosed lattice under the closure operator.In chapter three, we prove that the category which as a lattice for an object,a morphism for preserving finitely sup and meet is equivalent to the F-LATcategory from the perspective of the category.we introduce the concept of A-lattice, and establish a one to one correspondence between lattice and A-lattice.Super-meet-structure is also introduced and can be used to represent the lattice.That means the lattice can be represented in form of the family of subsets.