Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σZ(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σZ(P)是Sober的当且仅当σZ(P)具有Rudin性质,P贼予Z-Lawson拓扑λZ(P)是pospace,且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λZ(P))为严格完全正则序空间.     

Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σ_z(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σ_z(P)是Sober的当且仅当σ_z(P)具有Rudin性质,P赋予Z-Lawson拓扑λ_z(P)是pospace;且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λ_z(P))为严格完全正则序空间。     

拟Z-代数domain

基于一般子集系统Z,引入拟Z-代数domain的概念,研究了拟Z-代数domain的一些映射性质,并讨论了拟Z-代数domain与拟Z-连续domain之间的关系及拟Z-代数domain的乘积。     

强Z-连续domain和Z-代数domain的刻画定理

给出强Z-连续domain和Z-代数domain的一个刻画及一个范畴性质--余反射性质.     

Rudin性质与拟Z-连续Domain

对一般子集系统Z,引入了Rudin性质,给出了它的映射式刻划.作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了Rudin性质及其映射式刻划在拟Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟Z-连续Domain情形.     

Rudin性质与拟Z-连续Domain

对一般子集系统 Z,引入了 Rudin性质,给出了它的映射式刻划,作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了 Rudin性质及其映射式刻划在拟 Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟 Z-连续 Domain情形。     

局部Z-空间和Z-连续空间

对于一般广义子集系统Z,引入了局部Z-空间和Z-连续空间的概念,讨论了局部Z-空间的基本性质;基于收敛网,给出了局部Z-空间的等价刻画,证明了X为Z-连续空间当且仅当X为局部Z-空间。     

Z-连通连续偏序集的特征和浓度

引入了Z-连通连续偏序集的局部基和稠密子集的概念,给出了局部基的一些刻画,在此基础上定义了Z-连通连续偏序集的特征和浓度。证明了Z-连通连续偏序集的特征和浓度与Z-连通连续偏序集带上Scott拓扑时的拓扑空间的特征和浓度相等,它们分别小于Z-连通连续偏序集带上Lawson拓扑时拓扑空间的特征、浓度。     

Z-连续格的函数空间

若 Z为并完备的子集系统 ,且 IZ( L)关于集合的包含关系构成完备格 ,则 :( 1 ) Z-连续格的函数空间仍为 Z-连续的 ;( 2 )对于 Z-连续格范畴 ZL ,定义了一函子 F:ZL× ZL→ ZL.     

关于交半格同态构成的函数空间与FS-交连续Domain

定义了一类序结构-FS-交连续domain,讨论其相关性质并证明:(1)FS-交连续domain关于由Scott连续且保持非空有限交运算的函数构成的函数空间封闭,以(代数)FS-交连续domain为对象、以Scott连续函数为态射的范畴是Cartesian闭范畴;(2)任意分配可乘的有界完备domain是FS-交连续domain,从而紧连续dcpo的Smyth幂domain是FS-交连续domain.这些结果表明,FS-交连续domain是关于保非空有限交的连续映射构成的函数空间封闭的最恰当序结构.     

连续偏序集及其Smyth幂的几个等权定理

推广连续D om a in的权的概念到连续偏序集上,探讨连续偏序集的权、相应内蕴拓扑的权、定向完备化的权以及Sm yth幂D om a in的权间的关系。得到了几个等权定理:(1)连续偏序集的权与其上Scott拓扑、L aw son拓扑的权相等;(2)连续偏序集的权与其定向完备化的权相等;(3)无穷连续D om a in的权与其Sm yth幂D om a in的权相等;(4)有限D om a in的权小于或等于它的Sm yth幂D om a in的权。     

Almost Integrally Closed Domains

Abstract

A domain R is called almost integrally closed if R is integrally closed in R _P for each nonzero P ∈ Spec(R). Arbitrary quasilocal domains of (Krull) dimension 1 and arbitrary integrally closed domains are examples of almost integrally closed domains. There are no other examples in the contexts of Noetherian, one-dimensional or pseudo-valuation domains, as a consequence of the fact that any almost integrally closed domain that is not integrally closed has at most one height 1 prime ideal. However, a pullback example shows that a non-integrally closed domain that is almost integrally closed need not be semiquasilocal or of dimension at most 1. By analyzing the behavior of the almost integrally closed property for CPI-extensions, we obtain a characterization of the almost integrally closed locally divided domains. Applications are given to the case of G-domains. It also follows that if a divided domain R is not a field, then R is almost integrally closed if and only if some (resp., each) nonzero P ∈ Spec(R) is such that R _P is almost integrally closed and R is integrally closed in R _P .     

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。     

Some Categorical Aspects of Information Systems and Domains

Categories based on the Vickers's continuous information systems and the related categories of continuous domains (algebraic domains, Scott domains, continuous lattices etc.) are shown to be both Kleisli and Eilenberg–Moore categories of a monad of ideals. Further, the functor of ideals is shown to be a completion in the sense of Brümmer, Giuli and Herrlich.The first author would like to express his thanks for the support by the Italian C.N.R. and by the project IM 0021620808 of the Ministry of Education of the Czech Republic.     

t-Schreier Domains

We study, under the name t-Schreier, the class of those integral domains whose group of t-invertible t-ideals satisfies the Riesz interpolation property. The so-called Prüfer v-multiplication domains (PVMDs) and Prüfer domains are special cases of t-Schreier domains. We show that, while a number of results known for Prüfer domains and PVMDs hold for these domains, the t-Schreier domains have a remarkable capability of unifying various results in that the results proved for t-Schreier domains can also be translated to results on pre-Schreier domains and hence on GCD and Bezout domains.