有限格，闭包系统和闭包算子

本文引进了新的闭包系统,新的闭包算子等概念,研究了它们之间的相互关系,给出了由闭包系统来表示有限原子格的表示定理,证明了分别以这些数学结构为对象,以它们之间的同态映射作为态射,所对应的格范畴和对应的闭包系统范畴是范畴等价的.     

Quantaloid-enriched 范畴上的闭包算子与闭包系统

In this paper, we introduce the fundamental notions of closure operator and closure system in the framework of quantaloid-enriched category. We mainly discuss the relationship between closure operators and adjunctions and establish the one-to-one correspondence between closure operators and closure systems on quantaloid-enriched categories.     

Lattice tensor products. II

G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A?B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A?B, obtaining the isomorphism Id(A?B)≌A?Id B. This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A?B introduced in Part I of this paper.     

Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

Lattice Tensor Products i. Coordinatization

G. Grätzer and F. Wehrung introduced the lattice tensor product, A B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A B is a congruence-preserving extension of B. The lattice A B is defined as the set of certain subsets of A B; there is no easy test when a subset belongs to A B. A special case, M ₃B, was earlier defined by G. Gräatzer and F. Wehrung as M ₃, the it Boolean triple construct, defined as a subset of B ³, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A, we can ``coordinatize" A B, that is, represent A B as a subset of B ⁿ (where n is the number of join-irreducible elements of A), and provide an effective criteria to recognize the n-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we reprove a special case of the above result: for a finite simple lattice A, the lattice A B is a congruence-preserving extension of B.     

Closure Operators with Respect to a Functor

A notion of closure operator with respect to a functor U is introduced. This allows us to describe a number of mathematical constructions that could not be described by means of the already existing notion of closure operator. Some basic results and examples are provided.     

奇型Sturm-Liouville微分算子的限界自伴扩张

本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件.     

对称算子自共轭扩张的酉参数化描述(英文)

利用构造性的方法,给出了边值空间理论中几个结果新的证明,其中,边值空间理论是有关对称算子自共轭扩张的一种方法.同时,得到了几个新的结果.如发现了一般的边界三元组所具有的结构.进一步地,利用这个结果证明了辅助Hilbert空间H上的酉变换与亏空间K-和K+之间的等距同构映射间存在一个双解析的映射.发现并证明了一般边界条件:B(ψ):=MΓ1ψ+NΓ2ψ=0(其中M,N是阶数为亏指数的方阵)是自共轭的充要条件以及相应的酉变换和边界映射.     

等距算子的延拓、逼近及相关问题

本文介绍了等距算子从单位球面、从区域及从保距离1等条件下的各种等距廷拓问题；并介绍了(线性与非线性算子)等距算子的弱扰动、强扰动及渐近等距算子等问题．文中从历史到现状，概括了重要的研究结果，并指出了一些尚未解决的遗留问题．     

格值命题逻辑系统LP(X)的闭包算子

In this paper, closure operators of lattice-valued propositional logic LP(X) are studied. A family of classical closure operators are defined and the relation between them and closure operators of LP(X) is investigated. At the same time, a tool for checking compactness of LP(X) is given.     

J-对称微分算子的J-对称扩张的J-辛几何刻画

本文利用J-辛几何，刻画了J-对称微分算子的J-对称扩张。     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

Compactness, Perfectness, Separation, Minimality and Closedness with Respect to Closure Operators

In this paper, the characterization of closed and strongly closed subobjects of an object in categories of various types of filter convergence spaces is given and it is shown that they induce a notion of closure. Furthermore, each of the notions of compactness, perfectness, separation, minimality and absolute closedness with respect to these two new closure operators are characterized in these categories and some known results are re-obtained.     

Multiplication and Compact-friendly Operators

During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set.     

由子基生成的内部算子和闭包算子

本文研究粗糙集与拓扑空间的关系,统一地使用拓扑空间中的集合关于子基的内部和闭包来研究粗糙集理论和覆盖广义粗糙集理论中的下近似集和上近似集,以及由它们导出的关于子基的开集,导集,闭集,边界．研究这两个概念及由它们导出的相关概念的性质不仅对于粗糙集理论,而且对于拓扑学本身都有重要的理论和实际应用意义．     

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

In this paper we show that each factorization structure on a small category , satisfying certain conditions, yields a presheaf on and a morphism of presheaves . We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on (b) subobjects of (c) morphisms from to (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on .     

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.     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．