自反和超自反子空间格

本文讨论了子空间格的序和、序积的自反和超自反性以及它们对应的代数的超自反性。     

五角子空间格的性质

本文证明了若Ｌ为五角子空间格，则ａｌｇＬ和ａｌｇ（Ｌ≥Ｌ）的每一非０单元为秩１算子，另外也考虑了Ｌ和别的子空间格的序和所对应的代数中秩１算子和单元的关系     

J-子空间格代数上中心化子和广义导子的刻画

设L是Banach空间X上的J-子空间格,AlgL是相应的(J-子空间格代数.设φ:AlgL→AlgL是可加映射,对每个K∈(J)(L),dimK≥2.该文证明了下列表述等价:(1)φ是中心化子;(2)φ满足AB=0■φ(A)B=Aφ(B)=0;(3)φ满足AB+BA=0■φ(A)B+φ(B)A=Aφ(B)+Bφ(A)=0;(4)φ满足ABC+CBA=0■φ(A)BC+φ(C)BA=ABφ(C)+CBφ(A)=0.作为应用,得到AlgL上在零点广义可导的可加映射的完全刻画.     

超自反的子空间格

本文讨论了子空间格的自反性和超自反性，给出了子空间格是超自反的等价条件．     

原子Boolean格代数的导子

设L是Banach空间X上的原子Boolean子空间格,δ是algL的任一导子,则存在X中的一个稠定线性算子T,使得δ（A）＝AT—TA（A∈algL）在T的定义域D（T）上成立．另外,如果L还是一个有限格,并且对L的任一原子L,L＋L＇闭,则δ是连续的和内的．     

广义度量空间和偏序集

广义度量空间和偏序集都具有函数空间．而函数空间的存在为数学构造和计算提供了很大方便．本文还讨论了广义度量空间和偏序集之间的相互转化问题．     

线性空间的亚子空间

本文给出了线性空间中亚子空间的概念，讨论了亚子空间的性质和维数，并将其应用到有解的非齐次线性方程组中去．     

FI-格上的φ-导子

本文利用FI-格上的自同态对其上的导子进行了推广,探讨了FI-格上的φ-导子,并得到了FI-格上自映射成为φ-导子的充分必要条件。其次,定义了保序φ-导子、幂等φ-导子,讨论了保序φ-导子与φ-闭包算子之间的关系。最后,证明了可换FI-格上的保序φ-导子的不动点之集为其上的格滤子,并给出了FI-格上φ-导子不动点之集的等价刻画。这些结果进一步推广和丰富了格值模糊逻辑代数上导子的相关理论。     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

Intersection Numbers For Subspace Designs

Intersection numbers for subspace designs are introduced and q‐analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q‐analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2‐ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.     

Minimum Cutsets for an Element of a Subspace Lattice over a Finite Vector Space

Let L_n(q) denote the lattice of subspaces of ann-dimensional vector space over the finite field of q elements, ordered byinclusion. In this note, we prove that for all n and m the minimum cutsetfor an element A with is justL(A) if m < n/ 2, is U(A) if m > n/ 2, and both L(A) andU(A) if m = n/ 2, where L(A) is the collection of all such that and , and U(A) the collection of all such that and . Hence a finite vector space analog isgiven for the theorem of Griggs and Kleitman that determines all the minimumcutsets for an element of a Boolean algebra.     

分子格上伪补的构造

本文给出了分子格上伪补的构造定理，并证明了当一个非平凡分子格上有伪补，其既约因子上并不一定存在伪补，但其准既约因子上一定存在伪补，且任一分子格上的伪补都可表为其准既约因子上伪补的直积．     

Note on Reflexive Algebras with Non—distributive Invariant Lattices

§０．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＬｂｅａｃｏｌｌｅｃｔｉｏｎｏｆｃｌｏｓｅｄｓｕｂｓｐａｃｅｓｏｆａＨｉｌｂｅｒｔｓｐａｃｅＨｃｏｎｔａｉｎｉｎｇ｛０｝ａｎｄＨｗｈｉｃｈｆｏｒｍｓａｃｏｍｐｌｅｔｅｓｕｂｓｐａｃｅｌａｔｔｉｃｅｕｎｄｅｒｔｈｅｏｐｅｒａｔｉｏｎｓ∧（ｉｎｔｅｒｓｅｃｔｉｏｎ）ａｎｄ∨（ｃｌｏｓｅｄｌｉｎｅａｒｓｐａｎ）．ＷｅｄｅｎｏｔｅｂｙａｌｇＬｔｈｅａｌｇｅｂｒａｏｆａｌｌｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓｗｈｉｃｈｌｅａｖｅａｌｌｓｕｂｓｐａｃｅｓｉｎＬｉ…     

The determinants of certain matrices arising from the Boolean lattice

Let B be the Boolean lattice on an n-set with B=?B_i the rank decomposition. Let M(n,i) be the incidence matrix between B_i and B_n−i. We obtain a recursive formula for the determinant of the matrix M(n,i).     

On Subspace Lattices I. Closedness Type Properties and Tensor Products

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various “closedness” properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.     

Doubling Convex Sets in Lattices: Characterizations and Recognition Algorithms

We characterize lattices obtained from another lattice by a doubling of a convex set. This gives rise to a characterization of the class CN of lattices obtained by doublings of connected and convex sets when starting from a two-element lattice, and from this characterization result we derive an efficient recognition algorithm. This algorithm can be directly applied to the recognition of lattices in the subclasses of CN defined by giving some additionnal constraints on the convex sets used in the doublings.