Quantized Domain Theory Based On Continuous Triangular Nor

From the point of view of mathematics,domain theory investigates the connection between order and topology,in which Scott topology plays a primary role.Similarly,in the theory of quantitative domains,the investigation of the connection between quantitative order and quantitative topology,the quantitative Scott topology in particular,is a central topic.In 1973,Lawvere interpreted metric spaces as categories enriched over the quantale([0,∞]op,+,0),applied a categorical approach to research of metric spaces,and extended some notions of metric spaces,such as the Cauchy completeness,to general categories.The implications of Lawvere’s insight are profound.Replacing([0,∞]op,+,0)with a commutative unital quantale,categories enriched over the quantale can be recognized as quantitative partial orders valued in the quantale,which is the main study object of quantitative domain theory.In this thesis,we take fuzzy ordered sets valued in the unit interval endowed with a continuous triangular norm as a quantitative version of partially ordered sets,and study fuzzy domains and fuzzy Scott topologies.Fuzzy ordered sets are a special kind of enriched categories.Making use of the general theory of colimits with respect to a class of weights in enriched category theory,we postulate fuzzy domains with respect to a class of weights by requiring the existence of certain adjunctions,in which the class of weights plays a role as fuzzy directed lower sets.We take flat ideals and irreducible ideals as fuzzy directed lower sets.With help of the ordinal sum theorem of continuous triangular norms,we prove that the unit interval with the canonical fuzzy order is a domain with respect to flat ideals if and only if the implication operator of the continuous triangular norm is continuous at each point off the diagonal.Two functors from the category of fuzzy orders to fuzzy topological spaces are introduced;one of them is a fuzzy version of the Scott topology functor based on flat ideals,and the other is based on irreducible ideals.The codomain of these functors can be restricted to simultaneously reflective and coreflective full subcategories of fuzzy topological spaces.It is proved that(i)fuzzy the Scott topology functor based on flat ideals is full if and only if the continuous triangular is Archimedean;(ii)the fuzzy Scott topology based on flat ideals of every fuzzy domain based on flat ideals is sober.When we study fuzzy Scott topology based on irreducible ideals,we introduce the theory of Kleisli monoids of an order-enriched monad in monoidal topology,and prove saturated prefilters give rise to a monad if and only if the implication operator of the continuous triangular norm is continuous at each point off the diagonal.The resulting monad is order-enriched,and its Kleisli monoids are precisely fuzzy topological spaces with a conical neighborhood system.Hájek’s BL-logic is essentially a fuzzy logic based on continuous triangular norm,thus the continuous triangular norm based the theory of quantitative domains is a quantitative theory that corresponds to BL-logic.Our investigation will enhance the understanding of the connection between the theory of quantitative domains and fuzzy logic.The fuzzy directed lower sets taken in this thesis,i.e.,flat ideals and irreducible ideals,have a natural connection to fuzzy topology.The fuzzy Scott topology functor based on flat ideals and the fuzzy Scott topology functor based on irreducible ideals play an important role in the research of the connection between quantitative order and quantitative topology.The filter monad is helpful to study order and topology.The saturated prefilter monad,as a quantitative version of filter monad,will also be helpful to study quantitative order and quantitative topology.