格中两类分配等式的关系研究

针对离散数学经典教材中提出的"交运算对并运算的分配等式和并运算对交运算的分配等式是等价的"这一结论,分析了一种常见的错误证明,通过一个反例说明该结论在一般的格中不一定成立,进一步证明这两个分配等式在且仅在模格中是等价的,并提出利用定义判断一个模格是否是分配格的简便算法.作为一个应用,重新证明了该教材中的一条定理.     

布尔矩阵的指标格的性质

介绍了布尔矩阵的行零元、列零元和相容子矩阵的定义并讨论了它们的性质,给出了布尔矩阵的指标格分别为分配格、半分配格和半模格的等价条件.     

具有析取性质的分配格的拓扑表示及性质

本文首先给出了一般格成为析取分配格的几个条件，并证明了在满足并无限分配律的条件下，具有析取性质的分配格与Ｂｏｏｌｅ格是等价的。文章还给出了析取分配格的拓扑表示。     

有1模格的另外两种等价定义

对具有单位元的模格,本文给出了两种更为简洁的等价条件,并根据模格所满足的幂等律、交换律、结合律、吸收律和模律,证明了有1模格的各种定义之间的等价性。     

自由格的若干定理

格的一个等式类Ｋ上由一个弱偏格生成的自由格的定义可以仿照Ｋ上由偏格生成的自由格的定义来给出．本文给出了Ｋ上由弱偏格生成的自由格的存在定理，并对任一非序集生成的自由分配格的结构作了两点刻画．     

关于有1模格的等价定义

进一步讨论有1模格的等价定义问题,得到并证明了一个(2,2,0)型代数成为有1模格的一个充分必要条件.这样大大简化了有1模格的等价定义.     

每个元有上覆盖的紧生成格的结构

Dilwrorth与Crawleyl973年提出能否去掉上半模格条件来刻画元素的不可约完全交既分解问题以及能否去掉强原子格的条件刻画紧生成格结构的问题,本文首先证明了每个元有上覆盖的紧生成格L中任意元有不可约完全交既分解,从而肯定地回答了Dilworth与Crawley上述第一个问题.之后,在每个元有上覆盖的紧生成格中引入局部强模格与局部强分配格的概念,研究了局部强模格中独立集的特性以及局部强模格与局部分配格的结构,从而部分解决了Dilworth与Crawley上述第二个问题.     

有1分配格的若干种等价定义

对有单位元1的分配格,给出若干种等价定义,并根据分配格所满足的幂等律、交换律、结合律、吸收律和分配律证明了最新定义与原始定义的等价性。     

完备分配格L上的S-幂零矩阵

利用完备分配格L上的s-模定义格L上的矩阵运算,给出这些运算的一些基本性质,并且定义格L上的s-幂零矩阵,给出一些全新的结果.     

对称扩展的有界分配格的同余关系及其应用

给出了对称扩展的有界分配格的定义,即带有满足一定条件的一元运算的有界分配格.然后给出了这种分配格上的主同余的等式刻划及其可补性.最后,讨论了对称扩展的有界分配格的次直不可约性。     

Finiteness and Infiniteness of 3-Generated Lattices with Distributive Elements Among Generators

Siberian Mathematical Journal - We consider 3-generated lattices whose generators are distributive, dually distributive, right modular, dually right modular elements, or elements possessing a...     

An Analogue of Distributivity for Ungraded Lattices

In this paper, we study lattices that posess both the properties of being extremal (in the sense of Markowsky) and of being left modular (in the sense of Blass and Sagan). We call such lattices trim and show that they posess some additional appealing properties, analogous to those of a distributive lattice. For example, trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Any distributive lattice is trim, but trim lattices need not be graded. The main example of ungraded trim lattices are the Tamari lattices and generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim; we conjecture that all Cambrian lattices are trim.     

Semimodular Lattices and Semibuildings

In a ranked lattice, we consider two maximal chains, or flags to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a Jordan-Hölder permutation between any two flags. This permutation has the properties of an S_n-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an S_n-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits' result that S_n-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).     

On Lattices of Uniformities

We study lattice theoretical properties of lattices of uniformities such as modularity, distributive laws and the existence of (relative) complements. For this the concepts of permutable uniformities (see Definition 3.1) and independent uniformities (see Definition 4.1) are important. Moreover, we show that e.g. the lattice of all lattice uniformities on a lattice L is a closed sublattice of the lattice of all uniformities on L.     

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki? have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.     

Collection of Finite Lattices Generated by a Poset

It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.     

Z-Join Spectra of Z-Supercompactly Generated Lattices

The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces.     

格的内蕴拓扑与完全分配律

本文以Ｓｃｏｔｔ拓扑，Ｚａｒｉｓｋｉ拓扑，区间拓朴等刻划了完全分配律；获得了格的特殊元的存在性与区间拓扑连通性等之间的众多制约关系，给出了若干应用实例和重要反例．     

Distributive Envelopes and Topological Duality for Lattices via Canonical Extensions

We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of ‘admissibility’ to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.