FS-相容Domain的定向完备化及相关范畴性质

引入FS-相容Domain概念，研究FS-相容Domain的性质，主要结果有：（1）FS-相容Domain的收缩核与连续函数空间还是FS-相容Domain；（2）FS-相容Domain是有限生成上集，从而是Scott紧的；（3）FS-相容Domain的定向完备化是FS-Domaln；（4）有最大元的FS-Domain去掉最大元后是FS-相容Domain；（5）证明了以Scott连续映射为态射，FS-相容Domain为对象的范畴FS-CDOM是笛卡儿闭范畴并以FS-Domain范畴FS-DOM作为满的反射子范畴。     

FS-偏序集和连续L-偏序集

引入了FS-偏序集和连续L-偏序集概念,探讨了FS-偏序集和连续L-偏序集的性质.主要结果有(1)每一FS-偏序集都是有限上集生成的,因而是Scott紧的;(2)证明了FS-偏序集(连续L-偏序集)的定向完备化是FS-偏序集(连续L-偏序集);(3)一个偏序集是一个FS-Domain当且仅当它为Lawson紧的FS-偏序集;(4)FS-偏序集(连续L-偏序集)去掉部分极大元后还是FS-偏序集(连续L-偏序集).     

一类Domain范畴及其笛卡儿闭子范畴

本文给出一类新的 Ｄｏｍａｉｎ范畴 ＡｌｇＤ■（即以代数 Ｄｏｍａｉｎ为对象,保逼近序的Ｓｃｏｔｔ连续函数为态射的范畴）及其满子范畴ＡｌｇＤ■（即以有底（最小元）的代数Ｄｏｍａｉｎ为对象,保逼近序的Ｓｃｏｔｔ连续函数为态射的范畴）,并且讨论它们的极大的笛卡儿闭的满子范畴．     

Representation theorems for directed completions of consistent algebraic L-domains

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory. Received January 8, 2005; accepted in final form June 15, 2005.     

QFS-Domains and their Lawson Compactness

In this paper, concepts of quasi-finitely separating maps and quasi-approximate identities are introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are: (1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains, are all QFS-domains; (2) QFS-domains are compact in the Lawson topology; (3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is compact in the Lawson topology; (4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains.     

笛卡儿闭Domain范畴的两个重要性质

对于CONT的任意一个笛卡儿闭的满子范畴C,构造CONT的两个新的满子范畴R-C(以C对象的收缩为对象的范畴)和B-C(以C扩张序列的双极限为对象的范畴),并证明了它们都是笛卡儿闭范畴。由于R-C和B-C都包含C为其满子范畴,利用上述结果可得CONT的极大笛卡儿闭子范畴必对收缩和双极限封闭。本文还从范畴笛卡儿闭性的角度给出了Domain理论中一个公开问题的等价描述。     

相容连续Domain的不变性

引入了Scott相容连续映射与商相容Domain等概念,研究了Scott相容连续映射保局部基与保waybelow序及保局部基与保紧元之间的关系,证明了相容连续Domain或相容代数Domain在保局部基的Scott相容连续满映射下保持不变.     

连续Domain的遗传性及其不变性

引入Domain子空间的概念,得到Scott开集和闭集都是Domain的子空间。证明连续Domain或代数Domain对开子空间和闭子空间都是可遗传的。证明连续Domain或代数Domain在保Waybelow序的Scott连续映射下保持不变。     

连续信息基

D.Scott在20世纪60年代引入了domain理论,并给出了两种等价形式——信息系统和邻域系。在型论(typetheory)基础上,G.Sambin提出了形式拓扑(Formal topology)理论,并证明了一元形式拓扑与代数Scott domain等价,从而推出代数信息基的概念,同时作为信息系统与邻域系的推广。本文从信息基的观点出发,提出了连续信息基的概念,证明了它与连续Scott domain的等价性。     

拟连续Domain与广义完全分配格

本文证明了(1)在合适的态射下,拟连续domain范畴与广义完全分配范畴等价;(2)对有界完备的拟连续domain P,(P,σ(P))为极大极限空间.