偏序集的代数完备

引进一个偏序集的代数完备, 并且构造任意偏序集的一个代数完备.有最小元的并半格的代数完备正好是它的理想完备. 一个偏序集的代数完备同构于它的一个由下集作为元的完备格,并且这个完备格包含所有主理想. 基于代数完备的Galois联络的下扩张仍然是一个Galois联络.     

模糊DCPO范畴的一个笛卡尔闭的满子范畴

引入了有界完备模糊dcpo的概念,研究了有界完备模糊dcpo的基本性质。证明了当赋值格L是Frame时,以模糊Scott连续映射为态射的有界完备模糊dcpo范畴BC-FDCPO是以模糊Scott连续映射为态射的模糊dcpo范畴FDCPO的笛卡尔闭子范畴。同时还给出了模糊完备交半格、强模糊完备交半格的定义,并研究了它们与有界完备模糊dcpo之间的关系。     

φ-连续格的刻划与完全分配格的拓扑表示定理

本文在完备格中引入φ Ｓ集的概念，并在讨论φ Ｓ集族性质的基础上给出φ 连续格的一族拓扑及格论刻划，用局部超紧的Ｓｏｂｅｒ空间范畴给出完全分配格的拓扑表示定理     

分子格的既约度,分解度与分子格乘积的既约性

为探讨分子格的乘积的既约性和乘积的既约分解,文献[1]提出了分子格的既约度和分解度的概念,本文是[1]的继续,进一步给出了分子格的既约度与分解度的一些性质,证明了关于分子格乘积的全体主子格之集的基数的一个定理,以及得出了一族分子格的乘积是既约分子格的一个充要条件,此外,本文还证明了分子格范畴中乘积对上积的完全分配性是自然同构,改正了[1]中对这一结论的证明。     

模糊集范畴及集合范畴上的模

本文构造了在完备格上模糊集范畴.利用极小扩展原则和范畴的性质,获得了函子Uα构成集合范畴上的模结构,推广了P.Eklund的结论.     

完全分配格的谱论与拓扑分子格

本文借助于完全分配格的谱理论是首先证明分子格范畴同构于某连续偏序集范畴子范畴,然后利用上述同构,证明分子格上余拓扑同构于其中分子之集上。与分子序密切相关的某分明拓扑,从而就给“重域”“远域”这两个基本概念以合理解释,并证明许多拓扑分子格性质的研究可以化为相应的拓扑空间性质的研究.“重域”概念的引入,使 fuzzy 拓扑学的研究发生了根本变化,导致了有点派的兴起。而“远域”的引入,则导致因更广的拓扑分子格理论的产生,从而把 Fuzzy 拓扑为学纳入了拓扑格的范畴.本文中我们首先建立分子格范畴与连续偏序集范畴某子范畴的同构,从而把二者的研究紧密结合起来,然后借助上述同构把拓扑分子格中的问题的研究化为连续偏序集中问题去考虑,通过这种转化,我们将会看到,“重域”,“远域”等基本概念确为 fuzzy 拓扑学,拓扑分子格中唯一合理的点与集合的邻属关系,而择一原理这条fuzzy 拓扑学中的基本原理成立的原因也就变得很清楚。本文中凡未定义的概念请参看[4].     

ψ—连续格的刻划与完全分配格的拓扑表示定理

本文在完备格中引入ψ－Ｓ集的概念，并在讨论ψ－Ｓ集族性质的基础上给出－ψ－连续格的一族拓扑及格论刻划，用局部超紧的Ｓｏｂｅｒ空间范畴给出完全分配格的拓扑表示定理。     

内射拓扑分子格

本文证明了任给T_O拓扑分子格(L,η),以下三条等价:(1)(L,η)为正则内射拓扑分子格;(2)L为完备集环且其完备余素元集ht(L)形成一连续格,余拓扑η为该连续格ht(L)上的Scott闭集格;(3)存在T_O内射拓扑空间(X,Τ),(L,η)同胚于(P(X),Τ~c)在拓扑分子格范畴中的Sober化。此外,还给出了正则内射拓扑分子格、(一般)内射拓扑分子格以及正则内射分子格的一般结构。作为应用,重新证明了有指数元的拓扑分子格的结构。     

半极小集及其应用

在完备格中,引入半极小集概念,得到完备格L是半连续格当且仅当每一个元都有半极小集这一结论,同时给出半极小集的一些刻画性质.在半Scott连续映射的帮助下,给出了保半极小集的一些映射性质,并用这些性质证明了半连续格的子半格、商半格还是半连续格.     

Quantic格范畴

系统研究了Quantic格范畴。证明了Quantic格范畴有等子、余等子。给出了Quantic格范畴中的极限和逆极限结构,从而说明了Quantic格范畴是完备范畴。     

Limit分子格

通过在完全分配格上引入理想收敛,给出ｌｉｍｉｔ分子格及其范畴,证明了其是包含拓扑分子格范畴为全反射范畴的笛卡儿闭范畴．     

Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Sober拓扑分子格

A topological molecular lattice (TML) is a pair (L, T), where L is a completely distributive lattice and r is a subframe of L. There is an obvious forgetful functor from the category TML of TML‘s to the category Loc of locales. In this note,it is showed that this forgetful functor has a right adjoint. Then, by this adjunction,a special kind of topological molecular lattices called sober topological molecular lattices is introduced and investigated.     

L-filters and LF-topologies

For a complete quasi-monoidal lattice L, with some additional properties, we consider the category of L-filtered sets with fixed basis L, and we find out a functorial relation from this category to that of the L-fuzzy topological spaces. On the other hand, a morphism in the category of complete quasi-monoidal lattices allows us to establish a pair of adjoint functors (change of basis) between the corresponding categories of L-filterd sets.     

Z-半连续格

作为连续格和半连续格的公共推广,引入了广义理想子系统Z、Z-半连续格及强Z-连续格的概念,讨论了它们的基本性质和Z-半连续格的函数空间的结构,给出了强Z-连续格到方体的嵌入,证明了当子系统Z满足一定条件时,Z-半连续格范畴SCLZ是笛卡儿闭的。     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Graphical algebras — a new approach to congruence lattices

In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.     

Z-连通集系统及其范畴特征

本文引入了Z-连通集系统的概念,讨论了Z-连通连续偏序集的一系列性 质,证明了Z-连通连续偏序集范畴对偶等价于完全分配格范畴的一个满子范畴.