On The Order Structure Properties Of Ω-categories

Let (Ω,Ω,I) be a commutative, unital quantale. Then (Ω,Ω,I) is a small,complete, symmetric, monoidal closed category from the viewpoint of categorytheory. A category enriched in (Ω,Ω,I) is called anΩ-category for short.Ω-categories provide quantitative models for semantic of program languages,and have received wide attention in quantitative domain theory. Because domaintheory focuses on convergence and approximation of information, directed com-pleteness ofΩ-categories has received more attention in the literature. However,since the complete latticeΩis much more complicated than the Boolean algebra2, several quite diΩerent notions of directed completeness forΩ-categories havebeen developed by diΩerent authors. One task of this thesis is to characterizethe directed completeness in the framework ofΦ-cocompleteness developed forenriched categories.Since (Ω,Ω,I) is meant to play the role of the table of truth-values of amany valued logic, a category enriched inΩcan be treated as a many valuedpreordered set. Thus, the study ofΩ-categories is also a study of many valuedorder structures. The second task of this thesis is to apply categorical methodsto systematically investigate the many valued order structures ofΩ-categories,including: completeness, continuity and complete distributivity.It should be emphasized that the theory of many valued ordered sets is not astraightforward generalization of its classical counterpart. In fact, since the com-plete latticeΩis much more complicated than that the Boolean algebra 2, many easy problems in the classical case turn out to be much complicated in the manyvalued setting. For example, a preordered sets is tensored if and only if it has aleast element, but it is diΩcult to find such simple description forΩ-categories.Another example is that the opposite of every classical completely distributivelattice is also completely distributive, however, the complete distributivity of theopposite of a completely distributiveΩ-lattice depends on the logic property ofΩ, the table of truth-values! Precisely, it is shown that whenΩis a commuta-tive Girard quantale, the opposite of every completely distributiveΩ-category isalso completely distributive. Moreover, ifΩis a complete residuated lattice, thiscondition is also necessary.