伪Scott拓扑与伪Scott开滤子

讨论抽象基(特别是偏序集带辅助序)上的伪Scott拓扑与伪Scott开滤子集的一些基本性质, 推广了Domain理论中一些熟知的结论,证明了抽象基上的伪Scott拓扑是完全分配格;若在偏序集P上赋予辅助关系＜, 则其上伪Scott开滤子之集是连续domain.     

拟信息基,ω-代数Cpo,和SFP domain

G.Sam b in引入了(代数)信息基的概念,并证明了代数Scott D om a in范畴和信息基范畴是等价的.B.R.C.Bedrega l给出了ω-代数cpo和SFP dom a in的刻划.而G.Q.Zhang通过序结构给出了SFP dom a in的刻划.本文将引入了拟信息基的概念并给出了ω-代数cpo和SFP dom a in的刻划.     

完全分配格与点格

本文第一部分利用完备格上的上拓扑子基,给出完全分配格与点格的若干新刻划,并讨论其上的 Scott 拓扑与 Lawson 拓扑的基与子基的构造.第二部分讨论点格与代数格的关系,证明了 L 是点格当且仅当 L 为代数格且 L~(op)为完全 Heyting代数,并证明了代数偏序集范畴与点格范畴是等价的.     

性质M_F与拟连续Domain中的Scott上紧集

对拟连续domain引入了关于基的性质MF,证明了对拟连续domain P,任两个Scott紧上集的交是Scott紧当且仅当P关于某一(任一)基具有性质MF.     

半连续格上的半基和局部半基

在完备格上引入了半基和局部半基的概念,给出了半基和局部半基的性质及若干等价刻画,证明了一个完备格是半连续格当且仅当它具有半基也当且仅当它每点有局部半基。在此基础上定义了半连续格的权和特征,探讨了半连续格的权和特征与其上赋予半Scott拓扑和半Lawson拓扑时的拓扑空间的权和特征的关系。解决了文献[8](赵彬,刘妮.连续Domain的特征和浓度,陕西师范大学学报,2002,30(2):1~6)中提出的一个问题。     

超连续domain上的投射北大核心

本文给出了一个反例说明超连续domain L在Scott连续闭包算子c下的像c(L)不一定是超连续domain,证明了若超连续domain L上的Scott连续投射p有上伴随或有下伴随,则p(L)是超连续domain;若超代数domain L上的Scott连续闭包算子c有上伴随或有下伴随,则c(L)是超代数domain.     

超连续domain上的投射

本文给出了一个反例说明超连续domain L在Scott连续闭包算子c下的像c(L)不一定是超连续domain,证明了若超连续domain L上的Scott连续投射p有上伴随或有下伴随,则p(L)是超连续domain;若超代数domain L上的Scott连续闭包算子c有上伴随或有下伴随,则c(L)是超代数domain.     

L-Fuzzy Domain上的广义Scott拓扑

本文研究了L-fuzzy domain上的广义Scott拓扑,利用[1]中引入的L-fuzzy domain.获得了其上的广义Scott拓扑,它是Domain上Scott拓扑的推广,证明了一个L-fuzzy单调映射是L-fuzzy Scott连续映射当且仅当它关于L-fuzzy domain上的广义Scott拓扑连续.     

软泛代数及其整体结构性质

提出了软泛代数概念,将已有的软群、软环等概念统一纳入这一框架中,从整体上研究了软泛代数的序结构性质,证明了固定指标集和T-代数后,相应的软T-代数全体以点式序形成代数格.引入了Scott连续软泛代数概念,证明了从代数紧拓扑空间到给定T-代数的Scott连续软T-代数的全体以点式序形成代数格.     

S_1-代数偏序集

本文引入了S_1-代数偏序集的概念,证明了偏序集为S_1-代数的当且仅当其上S_1-拓扑为强代数格,并证明了S_1-连续domain的S_1-连续收缩仍为S_1-连续domain.     

Choice-Free Dualities for Domains

A basic tool in domain theory and point-free topology are (Scott) open filters in a partially ordered set. A systematic investigation of that concept shows that central notions and facts like Lawson’s famous self-duality of the category of continuous domains may be established without invoking any choice principles, if only continuous domains are replaced by so-called δ-domains, which coincide with the former in the presence of dependent choices. Many of the conclusions remain valid for the more flexible notion of ζ-domains, comprising important variants such as algebraic or hypercontinuous domains.     

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。     

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．     

相容连续Domain的遗传性

定义相容Domain的子Domain与子空间等概念,得到Scott开集和Scott闭集都是相容Domain的子Domain与子空间的结论,证明了相容连续Domain或相容代数Domain对开子空间和闭子空间都是可遗传的。     

Injectivity of directed complete poset acts

Action of a monoid on a set have always been of interest for mathematicians and computer scientists. On the other hand, Domain Theory, which studies directed complete partially ordered sets, was introduced by Scott in 1970 when he explored the possibility of using ordered topological spaces to give meaning first to typed and then untyped λ-calculi.

In this paper, combining the above two notions, we consider actions of a monoid on domains and study the algebraic notion of injectivity with respect to monomorphisms and embeddings in the category so obtained.     

Uniform domain representations of ℓp ‐spaces

The category of Scott‐domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach‐space is representable over a separable domain. A large class of topological spaces, including all Banach‐spaces, is representable by domains, and in domain theory, there is a well‐understood notion of parametrizations over a domain. We explore the link with parameter‐dependent collections of spaces in e. g. functional analysis through a case study of ?^p ‐spaces. We show that a well‐known domain representation of ?^p as a metric space can be made uniform in the sense of parametrizations of domains. The uniform representations admit lifting of continuous functions and are effective in p. Dependent type constructions apply, and through the study of the sum and product spaces, we clarify the notions of uniformity and uniform computability. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.