Research On C - Filter In Strong And Half Grid And Its Application

Lattice theory as an algebra system is introduced with the development of clas-sical logic algebra and universal algebra. French mathematician Charles Ehressman thought that the lattice with some kind of distribution (such as complete Heyting algebra) itself can be studied as a kind of generalized topological structure, not just the viewpoints and methods of lattice theory. The structure of C-ideal in a meet semi-lattice suggested by P.T.Johnstone is the important research tool for Frame theory (or Locale). Given the co-Frame structure and Frame structure connect with each other and promote each other. Therefore, this article introduced the concept of C-filter in enriched join-semi-lattice, and the properties of the co-Frame structure is studied by the tool of C-filter in join-semi-lattice, and then the co-product object and other category Properties in co-Frame category are discussed.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:Preliminaries. we will list some concepts and properties which are closely related to this thesis, which give a necessary perparation for the following chapters, such as join-semi-lattice, co-Frame, join-semi-lattice homomorphism, co-Frame homomorphism, category and co-product, etc.Chapter 2:C-filters in enriched join-semi-lattice and co-Frame induced by C-filters. In this chapter, Firstly, the concept of upper covering relation is introduced in a join semi-lattice. Secondly, the enriched join-semi-lattice is imported by the upper covering relation, and the upper coverage is defined in an enriched join semi-lattice. and then, the concept of C-filters is introduced in enriched join semi-lattices by upper coverage. Finally, it is proved that the family CFil(S) consisting of all C-filters of enriched join-semi-lattice S is a co-Frame.Chapter 3:The properites of the join semi-lattice homomorphisms of simple-up-set-valued mapping. In this chapter, the first, this conclusion that the up set ↑x={y∈S|x≤y} generated by the single point set x∈S in enriched join semi-lattices is a C-filter for any up-coverage C is given, and the properties of simple-up-set-valued mapping u:Sâ†'CFil(S), (?)x∈S, u(x)=↑x={y ∈S|x≤y} are investigated. It is proved that the simple-up-set-valued mapping is a join semi-lattice homomorphism that is keeping upper covering. Then, any join-semi-lattice homomorphisms keeping upper covering are obtained by compositing simple-up-set-valued mapping u:Sâ†' CFil(S) and co-Frame homomorphism h:CFil(S)â†'A. At last, the equivalent form of co-Frame is given by C-filter.Chapter 4:The co-product of co-Frame category. Firstly, it is proved that the subset A of direct product Πλ∈ΓAλof a family of co-Frames{Aλ|λ∈Γ} whose elements have finite non-zero coordinates is an enriched join-semi-lattice, and further it is proved that A is the co-product object of family of co-Frames {Aλ|λ∈Γ} in the join-semi-lattice category. Secondly, an up-coverage C* of A is defined by up-covering relationship of Aλ(λ∈Γ). Moreover, it is proved that the co-Frame C*Fil(A) derived by up-coverage C* of the enriched join-semi-lattice A is the co-product object of family{Aλ|λ∈Γ} of co-Frames in co-Frame category; and the compositing mapping between a simple-up-set-valued mapping u:Aâ†'C*Fil(A) and the standard embedding mapping qλ:Aλâ†'Πλ∈ΓAλa family of homomorphism in co-Frame category.Chapter 5:Some properties of co-Frame category. Firstly,it is proved that co-Frame category has equalizer and co-equalizer. At Last, intersections and pullback square are given in co-Frame category.