一致Locale的乘积

本文利用Ｌｏｃａｌｅ上的收敛结构研究一致Ｌｏｃａｌｅ的乘积结构，证明了一致Ｌｏ ｃａｌｅ的完备性关于Ｌｏｃａｌｅ积和弱积封闭．     

Locale 的内部与边界

本文定义了Ｌｏｃａｌｅ的内部算子与边界算子，详细讨论了这两个算子的性质，进一步得到了Ｌｏｃａｌｅ形式的Ｋｕｒａｔｏｗｓｋｉ定理。     

Locale范畴中的零维性

本文讨论ｌｏｃａｌｅ的零维性质，主要结果有：（１）给出ｌｏｃａｌｅＡ的核映射（ｎｕｃｌｅｕｓ）构成的ｌｏｃａｌｅＮ（Ａ）中上确界的点式刻划，并得到了Ｎ（Ａ）的紧性与Ａ的紧性之间的关系；（２）给出零维ｌｏｃａｌｅ与ｃｏｈｅｒｅｎｔｌｏｃａｌｅ之间的关系，以及零维ｌｏｃａｌｅ的紧零维反射；（３）给出零维ｌｏｃａｌｅ范畴在ｌｏｃａｌｅ范畴中的刻划．     

无点滤子收敛

我们在ｌｏｃａｌｅ上定义了一种新的滤子收敛概念．对空间式ｌｏｃａｌｅ,这种收敛与拓扑收敛等价,并且适应于任意ｌｏｃａｌｅ．作为应用,我们给出了ｌｏｃａｌｅ紧的程度的刻划和 Ｃａｕｃｈｙ完备性的描述．     

Locale范畴的反射子范畴

本文研究ｌｏｃａｌｅ范畴的反射子范畴，给出反射子范畴的刻划定理，从一般的ｌｏｃａｌｅ出发，完全构造性地给出了ｌｏｃａｌｅ的正则反射、完全正则反射和零维反射的构造．     

L—fuzzyLocale理论与分配格的L—fuzzy拓扑表示

首先文中引入了Ｌ—ｆｕｚｚｙｌｏｃａｌｅ范畴，并证明了该范畴与满层Ｌ—ｆｕｚｚｙ拓扑空间范畴的关系类似于ｌｏｃａｌｅ与拓扑空间的联系．其次，文中建立了分配格的ｌｏｃａｌｅ式ｆｕｚｚｙＳｔｏｎｅ表示，并且与经典结果一致，任一分配格的Ｌ—ｆｕｚｚｙｌｏｃａｌｅ表示的点空间就是它的Ｌ—ｆｕｚｚｙ谱空间．     

L-fuzzy Locale理论与分配格的L-fuzzy拓扑表示

首先文中引入了Ｌ-ｆｕｚｚｙｌｏｃａｌｅ范畴，并证明了该范畴与满层Ｌ-ｆｕｚｚｙ拓扑空间范畴的关系类似于ｌｏｃａｌｅ与拓扑空间的联系．其次，文中建立了分配格的ｌｏｃａｌｅ式ｆｕｚｚｙＳｔｏｎｅ表示，并且与经典结果一致，任一分配格的Ｌ-ｆｕｚｚｙｌｏｃａｌｅ表示的点空间就是它的Ｌ-ｆｕｚｚｙ谱空间．     

Locale的正则紧反射

ｌｏｃａｌｅ的正则紧反射函子的构造的明确描述问题是由ＢａｎａｓｃｈｅｗｓｋｉＢ．和ＭｕｌｖｅｙＣ．Ｊ．于１９８０年提出的,十多年来一直没有进展。本文通过在ｌｏｃａｌｅ上引入一种二元关系,给出了ｌｏｃａｌｅ的正则紧反射函子的构造性描述。     

Locale 的内部与边界

本文定义了ｌｏｃａｌｅ的内部算子与边界算子,详细讨论了这两个算子的性质,进一步得到了ｌｏｃａｌｅ形式的Ｋｕｒａｔｏｗｓｋｉ定理。     

Weakly Unconditional Cauchy Series on Locally Convex Spaces

Ｗｅａｋｌｙ　Ｕｎｃｏｎｄｉｔｉｏｎａｌ　Ｃａｕｃｈｙ　Ｓｅｒｉｅｓ　ｏｎ　Ｌｏｃａｌｌｙ　Ｃｏｎｖｅｘ　ＳｐａｃｅｓＬｉＲｏｎｇｌｕ（李容录）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｒｂｉｎＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，...     

弱空间式Locale

空间式locale范畴SLoc是locale范畴Loc的余反射满子范畴,但对locale乘积不封闭．本文引入弱空间式locale,证明弱空间式locale范畴WSloc为范畴Loc的余反射满子范畴,且对locale秉积封闭．还证明了一个locale A是空间式的当且仅当它的枝映射localeN(A)是弱空间式的;一个空问式locale的每一个子locale都是空间式的当且仅当它的每一个子locale是弱空间式的．最后,证明了弱空间式性在定向函子下保持不变．     

Nearness, Subfitness and Sequential Regularity

In the point-free context, the structure of nearness has been so far studied in the regular case only. Here we answer the question as to how far beyond that one can go. It turns out that a frame (locale) (quasi-)admits a nearness iff it is subfit. Unlike in the case of spaces, where admitting nearness is a hereditary property, subfitness is not; therefore, also the hereditary subfitness (here called sequential regularity for reasons obvious from the properties presented) is studied. It is weaker than regularity and seems to be of some interest also in the spatial case.     

关于Locale Wallman紧化的一点注记

本文证明了在正规locale范畴中,Banaschewski-Mulvey形式的紧正则反射与Johnstone形式的Wallman紧化一致,从而推广了Johnstone在文献[4]中的主要结果.     

Completely regular paracompact reflection of locales

We give an explicit construction of the completely regular paracompact reflection pL of a completely regular locale L described as a sublocale of the Stone-tech compactification βL of L.     

A presentation of the assembly of a frame by generators and relations exhibits its bitopological structure

We present the assembly of a frame (the frame of its nuclei) by generators and relations. The particular type of presentation is due to Jung and Moshier and can be understood as a point-free analogue of taking the common refinement of two topologies. We prove that our construction indeed presents the assembly by showing that both frames have the same universal property. In locale-theoretic terms, our result can be understood as follows. While the assembly of a locale is by construction zero-dimensional, our presentation contains as additional information the order of specialisation of the original locale. We find that the elements of the original frame play the role of upper opens, while filters of the frame take the role of lower opens.     

Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid , which we call ‘ -order’. This requires a theory of semicategories enriched in the quantaloid , that admit a suitable Cauchy completion. There is a quantaloid of -orders and ideal relations, and a locally ordered category of -orders and monotone maps; actually, . In particular is , with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general -orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of . Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.Mathematics Subject Classifications (2000) 06F07, 18B35, 18D05, 18D20.