On Completions And Cartesian Closedness Of Posets

Domain theory,as a branch of order theory,has been widely applied to various areas of mathematics,logic and computer science.In domain theory,one of the most fundamental concepts is the way below relation,which is defined by directed sets and their joins.In this paper we introduce a new relation on posets,called?-approximation.A new concept of continuity of posets arises naturally,called?-continuity.Topological characterizations of ?-continuous posets are put forward,that is,a poset P is ?-continuous iff its ?-topology lattice is completely distribu-tive.We also present a new type of dcpo-completion of posets which is called Do-completion.Connections between ?-continuity and D?-completion of posets are investigated.We prove that a poset P is 0-continuous iff its D?-completion is a domain.The concepts of quasi ?-continuity and meet ?-continuity are introduced,we show that a poset P is ?-continuous iff it is a quasi ?-continuous and meet?-continuous poset.Replacing directed sets by other classes of subsets one has several types of"way below" relation and "continuity".The notion of a chain-continuous poset is obtained by taking all nonempty chains instead of directed sets.Defining the"way below" relation by arbitrary subsets leads to completely distributive lattices.The concepts of subset system Z and Z-continuity were introduced to cover all these cases by one notion.In this paper we introduce the notion of Z?-continuity as a generalisation of precontinuity,complete continuity and S2-continuity.For any subset system Z,we define a new type of completion,called Z?-completion,extending each poset P to a Z-complete poset.We prove that if Z is an HUL-system and P a Z?-continuous poset,then the Z?-completion of P is also Z??continuous,and a Z-complete poset L is a Z?-completion of P iff P is an embedded Zs-basis of L.The Dedekind-MacNeille completion is a special case of the Z?-completion.Since the D?-completion and Z?-completion are defined by universal properties,one natural question is how to characterize these completions.We define one type of subsets of a poset,by a subset,system Z and a subset selection r,the family of this type of subsets is called Zr-completion.We show that the Zr-completion varies depending on the choosing of Z and T,including the D?-completion and Z?-completion.Domain theory was introduced as models for the denotational semantics of programming languages.It is well known that the category CONT? of pointed domains and Scott continuous functions is not cartesian closed.Finding the max-imal cartesian closed full subcategories of CONT? used to be a long-standing problem.It turns out there are exactly two maximal cartesian closed full sub-categories of CONT?:L,the category of L-domains,and FS,the category of FS-domains.The cartesian closed category RB of retracts of bifinite domains and Scott continuous functions is a full subcategory of FS.However,it is still an open question whether or not FS and RB coincide.Passing from pointed domains to general domains,there are four maximal cartesian closed subcategories of the category CONT of domains and Scott continuous functions:F-L,F-FS,U-L and U-FS.And,it is still unknown whether cartesian closed categories:F-RB and U-RB(also denoted by cUB and cFB)are maximal ones of CONT.To gen-eralize the properties of domains to that of continuous posets,it is natural to ask what cartesian closed categories there are that consist of continuous posets rather than continuous dcpos.The category DCPO of dcpos and Scott continuous func-tions is cartesian closed,but the category POSET of posets and Scott continuous functions is not cartesian closed.Let P be a cartesian closed full subcategory of POSET,and let C be a subcategory of the category CONT.We define C-P to be the category of objects from P whose D-completion is isomorphic to an object from C and Scott continuous functions.The category C-P is always a subcatego-ry of the category CONTP of continuous posets and Scott continuous functions.We prove that if C is a cartesian closed full subcategory of F-L,U-L,F-RB or U-RB,then the category C-P is also cartesian closed.This allows the rich theory of cartesian closedness that has been developed in the setting of domains to be transposed to the setting of continuous posets.It is known that the category of consistcent directed complete posets and Scott continuous functions,denoted by CDCPO,is cartesian closed.In particular,we have the following cartesian closed categories:F-L-CDCPO,U-L-CDCPO,F-RB-CDCPO,U-RB-CDCPO,F-aL-CDCPO,U-aL-CDCPO,F-B-CDCPO,U-B-CDCPO,etc.If the cat-egories FS and RB coincide,then it leads directly to the most potential of this uniform way of finding new cartesian closed categories of continuous posets:for every cartesian closed full subcategory C of CONT,the category C-P is also cartesian closed.