Generalized Frame Areas

In view of topology, frames (or locales) arc generalization of open-set lattices of topological spaces, and in view of algebra, a frame (or locale) is a complete lattice L, satisfying the infinite distributive law: what the most important meaning for the generalization is that we can study the topological properties of frames (or locales) on the basis of intuitional logic by means of order algebra, and apply them to the category of topological spaces. In view of category, a frame (or locale) is a small complete (cocomplete) category L, such that the set of morphisms between two objects has at most one element, satisfying the condition that for any object a, the product functorα×(-) : L→L preserves coproducts. Hence it is a natural and interesting question whether we can generalize a frame (or locale) to a small category.In the thesis, we introduce the concepts of generalized frames and generalied frame homomorphisms which are strict generalizations of the corresponding concepts of frames in the sense of category. There exist a great number of non-trivial examples showing that generalized frames are not frames in general. So we can discuss the topological properties, algebraic properties and category properties of generalized frames by the tool of category theory. As small categories, generalized frames have many category properties such as completeness, cocompleteness and Cartesian closedness. We show that the category of frames is a reflective subcat-egory of the category of generalized frames (i.e., the category whose objects are generalized frames, and whose morphisms are generalized frame homomorphisms).We discuss the properties of points, prime elements and spectrums of generalized frames, and prove that the category of all points of a generalized frame and the category of all prime elements of the generalized frame are equivalent. The result differs from the classical result that the set of all points of a frame and the set of all prime elements of the frame are one-to-one in frame theory. Moreover, we discuss the functor relation between the category of generalized frames and the category of topology spaces, and show that the spectrums of generalized frames are sober.The definitions of nucleus functors, quotients, open quotients and closed quotients of generalized frames are introduced. We prove that the categpry N(A) of all nucleus functors of a generalized frame A (or the category Q( A) of all quotients of a generalized frame A) is generalized frame. This generalize the classical result that the lattice of all nuclei of a frame (or the lattice of all quotients of a frame) is frame.Moreover, we obtained the product and coproduct construction of the category of generalized frames. Finally, we discuss the separations and compactness of generalized frames.