| Ever since continuous lattice was put forward by Scott who achieves the Tur-ing Award, many researchers have been very interested in it. A continuous latticeis a complete lattice with special properties. In order to understand it clearly, re-searchers attempted to represent it by various modalities, such as, the descriptionby means of equations, topologies and information systems. Now, from the pointof view of set theory, an operator called quasi-closure operator is defined on thepower set of a nonempty set. By means of that, we build a bijective correspondencebetween continuous lattices and the sets of fixpoints with respect to the operators.If some additional conditions are given, then the sets of fixpoints are in one-to-one correspondence with completely distributive lattice or prime algebraic lattices.Moreover, we study continuous information systems via the operators, and showthat they are a representation of continuous lattices, and the strong continuous in-formation systems are a representation of completely distributive lattices. Anotherapproximable mapping is defined, and the equivalence of continuous informationsystems and continuous lattices is obtained from the logic point of view.Formal concept analysis is an order-theoretic method for dealing with scien-tific data, its basic concepts are formal contexts and formal concepts. When formalconcepts are extracted from a formal context, they form a lattice, that is, formalconcept lattice. There is a bijective correspondence between complete lattices andformal concept lattices. Combining rough set theory, Du¨ntsch and Yao proposedattribute oriented rough concept and object oriented rough concept in a formalcontext, and there is also a bijective correspondence between complete latticesand rough concept lattices. Later, Zhang introduced approximable concepts, andproved that the approximable concept lattices are a representation of algebraic lat-tices, and approximable concepts are in one-to-one correspondence with the pointsin the algebraic information system induced by a given formal context. Soon after,Lei and Luo developed approximable concepts by rough approximable concepts,and show that the latter has the same properties as the former. We continue thisresearch in formal concept analysis. Weakly approximable concepts are introducedas a generalization of approximable concepts, it is shown that the correspondinglattices and prime algebraic lattices (i.e., completely distributive algebraic lattices)can represent each other. Finally, we give a method of constructing a prime infor-mation system from a given formal context with its points are corresponding to weakly approximable concepts.Quantitative domain theory (sometimes called fuzzy domain theory) aims toprovide a mathematical model for concurrent systems. Fuzzy complete lattice, oneof classical structures in this area, is explored in this study. Inspired by the resultthat classical complete lattices are isomorphic to closure systems, we first inves-tigate the relationship between fuzzy complete lattices and fuzzy closure systems.Next, we combine the commons of fuzzy order and logic, propose the concept offuzzy information systems. It is shown that fuzzy information systems are a con-crete representation of fuzzy complete lattices. With the appropriate morphismsbeing defined, we obtain that fuzzy complete lattices and fuzzy information sys-tems are equivalent to each other.Fuzzy partially ordered set is another hot issue in quantitative domain theory.In this paper, we focus on its completions. Based on the previous works, we studythe characterizations of the Dedekind-MacNeille completion (abbreviated by DMcompletion) for a fuzzy partially ordered set in categorical terms. With appropriatemorphisms, we prove that the operator of DM completion is a covariant functorfrom the category of fuzzy posets to the category of fuzzy complete lattices. Underthe category of fuzzy complete lattices, DM completion is a free object over a fuzzyposet. We also show that the category of fuzzy complete lattices is a reflective fullsubcategory of the category of fuzzy posets. Besides, an equivalent description forDM completion is given. Relating to the formal concept analysis, we give otherdefinitions for the DM completion, the relationship between them and fuzzy roughconcepts is discussed.Fuzzy Scott topology and Scott convergence have been studied on fuzzy par-tially ordered sets in which every fuzzy directed subset has a fuzzy join, i.e., fuzzydcpos. Unfortunately, the results are not fit for general fuzzy partially orderedsets, even its continuity needs to be reconsidered. For this, we propose fuzzy way-below relation on fuzzy partially ordered sets, and investigate its continuity overagain. Next, we study fuzzy convergence structure and fuzzy Scott topology onfuzzy partially ordered sets. In the sequel, Scott convergence on fuzzy partiallyordered sets is built, and it can be applied to describe its continuity.This dissertation is supported by Hunan Provincial Innovation FoundationFor Postgraduate. This dissertation is typeset by software LATEX2ε. |
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