定向极小集及其应用

文［５］在完备格上给出了ψ－极小集与ψ－极大集理论。本文研究了当ψ是定向集族（过滤集族）时的ψ－极小集（ψ－极大集），并借助于它们给出了完全分配格的若干刻划。     

关于定向极小集

本文对定向极小集作了进一步的研究，得到一系列重要性质，文章最后给出连续格为完全分配格的一个充分条件．     

关于定向极小集

本文对定向极小集作了进一步的研究，得到一系列重要性质，文章最后给出连续格为完全分配格的一个充分条件。     

完全分配格中极小族、极大族的刻划

王国俊在[3]中对B.Hutton在[1]中提出的极小族的概念给出了一个较强的定义并以此为工具讨论了Fuzzy格的构造。本文采用[3]的定义对完全分配格上的极小族,极大族给出了进一步的刻划,使其直观化,具体化。我们还给出了完全分配格的一个构造定理。     

关于格的极小生成元集

本文证明了格的极小生成元集一定是最小生成元集且只能是非零完全并既约元全体，证明了分配格具有最小生成元集的必要条件是它满足并无限分配律．本文还证明了完全Ｈｅｙｔｉｎｇ代数具有最小生成元集当且仅当它是强代数格，证明了完备格是强代数格当且仅当它和它的对偶格均是具有最小生成元集的分配格．     

完全分配格上的弱辅助序与广义序同态

为研究格上的拓扑学,王国俊在[1]中定义了完全分配格上广义序同态概念,并得到一系列重要的结果。刘应明在这方面也进行了深入的研究。本文利用完全分配格上一个逼近的弱辅助序给出广义序同态的一个内在的特征性质与极小集的刻划,并得到保极小集映射的两个等价条件。在此基础上,我们建立了广义序同态的新的扩张定理,然后讨论了以广义序同态为态射的完全分配格范畴的对偶定理,并在乘积范畴上引进一个重要而有趣的函子。     

广义半Smooth格

引入了广义半Smooth格和广义半Smooth代数格的概念,讨论了它们的一些基本性质,证明了完备格L是广义完全分配格当且仅当L是拟连续的广义半Smooth格。     

广义Smooth格

作为由Weng所引入的smooth格和scott紧生成格的推广,引入了广义smooth 格和广义smooth代数格,讨论了它们的一些基本性质,证明了完备格L是广义完全分配格当且仅当L是广义smooth格和广义连续格.     

商拓扑分子格的分解刻划

本文是王国俊的文[3]和[4]的继续和深入。文中给出了一般集上等价关系在完全分配格上推广——正则等价关系,并利用完全分配格的极小族刻划(见[3])证明了在正则等价关系诱导的商上可定义序,商按所定义的序成为完全分配格,称它为商分子格。由此可得自然商 TML,且文[4]中列入的商 TML 恰是自然商 TML(同胚视为一致),从而给出了商 TML 的分解刻划。     

Lawson-Hoffmann对偶定理的推广

本文讨论了Z－连续偏序集的一系列性质，主要证明了Z－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．     

Common fixed point for generalized -weak contractions

In this paper we introduce the class of generalized (ψ,φ)-weak contractive mappings. We establish that these mappings necessarily have a unique common fixed point in complete metric spaces. This result generalizes an existing result in metric spaces.     

L-拟序集上的广义Alexandroff拓扑与广义超度量空间上的广义Alexandroff拓扑

证明[10]中定义的L-拟序集上的广义Alexandroff拓扑是[2]中定义的广义超度量空间上的广义Alexandroff拓扑的推广，并且广义超度量空间中有关广义Alexandroff拓扑的许多性质都可以推广到L-拟序集中。     

The existence of generalized mix functions

To enrich the message space of a cipher and guarantee security, Ristenpart and Rogaway defined mix functions on two sets of equal size. To mix inputs from two sets of different sizes, Stinson generalized the definition of mix functions (called generalized mix functions), and established an existence result for generalized mix functions with 10 undetermined pairs of input sizes. In this paper, we complete the solution to the existence problem for generalized mix functions.      

Common Fixed Point Theory for Multivalued Contractive Maps of Reich Type in Uniform Spaces

This article presents common fixed point theory for multivalued maps defined on complete gauge spaces. Our analysis is elementary and relies only on properties of the generalized Hausdorff pseudometric.     

Similarity of L-fuzzy Relations Based on L-topologies Induced by L-fuzzy Rough Approximation Operators

This paper proposes similarity of L-fuzzy relations based on L-topologies induced by L-fuzzy rough approximation operators. First, the notion L-fuzzy rough set is generalized and the relationship between generalized L-fuzzy rough sets and L-topologies on an arbitrary universe is investigated. It shows that Alexandrov L-topologies can be induced by L-fuzzy relations without any preconditions. Second, the concept of similarity of L-fuzzy relations is introduced and variations of an L-fuzzy relation are investigated. Third, algebraic structures on similarity of L-fuzzy relations are obtained. Finally, we prove that the subset of the transitive L-fuzzy relations similar to a fixed L-fuzzy relation is a complete distributive lattice.     

广义凸模糊映射

给出广义凸模糊映射、广义弱凸模糊映射等概念和若干特例。其次,构造集合Axf,y、Af,证明当f为下半连续广义弱凸模糊映射时Afx,y为闭弱凸集,进而得到广义凸模糊映射的充分条件。最后,给出广义凸模糊映射的性质,并指出半严格广义凸模糊映射成为严格广义凸模糊映射的条件。     

Existence of Solutions of Generalized Vector Variational Inequalities in Reflexive Banach Spaces

The purpose of this paper is to study the solvability for a class of generalized vector variational inequalities in reflexive Banach spaces. Utilizing the KKM-Fan lemma and the Nadler’s result, we prove the solvability results for this class of generalized vector variational inequalities for monotone vector multifuctions. On the other hand, we first introduce the concepts of complete semicontinuity and strong semicontinuity for vector multifunctions. Then we prove the solvability for this class of generalized vector variational inequalities without monotonicity assumption by using these concepts and by applying the Brouwer fixed point theorem. The results in this paper are extension and improvement of the corresponding results in Huang and Fang (2006).     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.