Domains And Information Systems

It is an important method to apply topologies and domains to study the logical structure of information.After recalling the main results of information systems,we introduce a new concept 'theory' on a logical information structure.As long as the compatible sets are well defined,a theory,in which the finite non-contradiction is identified, will be encircled.We show that the notion of 'theory' which refines the concept of ideals of a semilattice,is just an invariable structure throughout transformations among formal contexts,lattices and domains.The dissertation introduces the notions of quasi-information systems and information domains,and exhibits the equivalence between the quasi-information system category and the information domain category.In this sense,the information domain is an infinite mirror of finite knowledge and the general information system is the finite cognition of infinite knowledge.Characteristics of information domain images of some information systems are investigated by our powerful function tools.We obtain some sufficient and necessary function conditions for entailment with special properties such as transition and interpolation. A set S is a point of a quasi-information system I if and only if S is a fixed point of the information domain induced by I.All points of a quasi-information system which may admit non-transition or non-interpolation form a Scott domain if the set of all post-fixpoints is a meet-complete semilattice.After the concept of algebraic completion of a poset is introduced,a concrete algebraic completion of a poset is constructed by cut,which extends the MacNeille completion. If a poset is explained into a set of propositions in the view of logic,then the algebraic completionis closed for the finite deduction system in logic,therefore,the algebraic completion reflects the logically closed extension of preliminary propositions.We investigate the intrinsic relations among algebraic completion,MacNeille' completion and ideal completion.Unlike the canonical extension,the number of algebraic completion of a poset may be more.But if a poset is a join semilattice,the algebraic completion is unique under order-isomorphism.Furthermore,an algebraic completion is equivalent to a down-set algebraic lattice which contains all principal ideals under order isomorphism.In the sense of equivalence,an algebraic completion could be regarded as the composition of an ideal completion and a join completion.The dissertataion reveals that the set of△-ideals of a poset generalizes the usual Scott topology.Though the former isn't frequently a topology,some skills of the classi- cal domain theory are able to be applied to strongly Z-precontinuous posets.If P is a strongly Z-precontniuous poset,then {(?)~Z u|u∈P} is join dense in the set consisting of all complements of△-ideals,which is analogous to the Scott topology of a domain. The category FSBP in which objects are finitely separating and upper bounded posets and arrows are D~△-continuous functions between them,is a cartesian closed category. Clearly this category is not a subcategory of category DCPO in which all dcpos act as objects and Scott continuous functions as arrows.The dissertation establishes a functor between the group category and the completely distributive complete lattice category,and proposes that the duality of cyclic poset of a group G is an algebraic domain if and only if the identity of group G is an algebraic element.