Stratification structures on a kind of completely distributive lattices and their applications in theory of topological molecular lattices

A completely distributive lattice is also called a molecular lattice.The least and the greatestelements in a lattice are denoted by 0 and 1,respectively.For the empty subset L of alattice L,we assume that V=0 and ∧=1.The set of all the molecules（i.e.non-zero∨-irreducible elements）in a lattice L is denoted by M（L）. Let L1 and L2 be completely distributive lattices,f:L1→L2 and g:L2→L1 be order-preserving maps.If for any a∈L1 and any b∈L2,f（a）≤b if and only if a≤g（b）,then fis called the left adjoint of g,or g is called the right adjoint of f.The right adjoint of f

UNIFORMITIES ON COMPLETELY DISTRIBUTIVE LATTICES

In order to study fuzzy topology and general topology in view of topological lattice, many researches have been carried out. Because of the lack of reverse order involution, there are some fundamental works, such as the theory of uniformity and metrization to be considered. In this note, we introduce the notion of uniformity on completely distributive lattice in terms of cover systems, and show that it is a good extension, we obtain some properties and prove that the equivalence of complete regularity and uniformizability on completely distributive lattices.

CONTACT PROCESS ON HEXAGONAL LATTICE

In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.

Rough Class on a Completely Distributive Lattice

This paper generalizes the Pawlak rough set method to a completely distributive lattice. The concept of a rough set has many applications in data mining. The approximation operators on a completely distributive lattice are studied, the rough class on a completely distributive lattice is defined and the expressional theorems of the rough class are proven. These expressional theorems are used to prove that the collection of all rough classes is an atomic completely distributive lattice.

Notes on Preregular Operators in Banach Lattices

Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

PRODUCTS AND COPRODUCTS IN CATEGORY OF COMPLETELY DISTRIBUTIVE LATTICES AND THEIR STRUCTURES

Taking completely distributive lattices for objects and complete lattice homomorphisms for morphisms, we get a category Lat (see Refs. [1—3]). There is a primary question about Lat: Does every family of objects have product and coproduct in Lat? We prove that Lat has products and coproducts, and describe the intrinsic relation between

Cartesian closedness of categories of completely distributive lattices

The category of completely distributive lattices with Scott continuous functions is cartesian closed. Neither the category of completely distributive lattices with arbitrary union preserving mappings nor the category of completely distributive lattices with nonempty union preserving mappings is cartesian closed.

THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA

Let （L, 〈, V, A） be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by V and A respectively, is studied. We obtain： （i） the necessary and sufficient conditions for S（A,b）≠Ф; （ii） the necessary conditions for IS（A,b）｜ = 1. We also obtain the vector x ∈ Ln and prove that it is the largest element of S（A, b） if S（A, b）≠Ф.

The complete principal divisor lattices

In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.

RELATIVELY COMPLEMENTED DISTRIBUTIVE LATTICE

In this paper, we have proved that for a relatively complemented distributive lattice L,there exists one and only one completion ? of L such that for every ξ∈?, there exists afamily {S_α}_(α∈△) of non-vacuous subsets of L satisfying ξ= ∨[∧f(S_α)]. Such a completion? is called the entire completion of L. We have in this paper extended the homomorphic extension theorem of a generalizedBoolean lattice to that of a relatively complemented distributive lattice and proved that the lat-tice of the congruences on a relatively complemented distributive lattice is isomorphicwith the lattice of the convex sublattices containing a fixed element and that the entire comple-tion of a relatively complemented distributive lattice L, the lattice of the completable con-gruences on L and the lattice of the completable convex sublattices of L containing a fixed ele-ment are isomorphic.

ANALYTIC AND TOPOLOGICAL CHARACTERIZATIONS OF COMPLETELY DISTRIBUTIVE LAW

Using an algebraic property, the completely distributive law, we have ever given out a characterization of the semicontinuity of lattice-valued mappings. How about the inverse implication? That is to say, can we analytically characterize the completely distributive law? Moreover, can we characterize the completely distributive law in terms of fuzzy topology? The purpose of this note is to answer affirmatively these questions for the infinitely distributive lattices. This study connecting algebra with analysis and topology seems to be rather interesting.

基于一般二元关系粗糙近似算子的格结构研究

刻画了基于一般二元关系及悲观多粒度粗糙近似算子的完备格结构,证明了在选取蕴涵算子之后,基于一般二元关系的粗糙近似算子构成MV、R0与布尔代数。

CD格上的近似算子

Pawlak近似算子具有多种推广形式。讨论了完全分配格上的近似算子。通过近似空间中的不确定性映射,分别引入了三种形式的上近似算子及下近似算子,讨论了它们的基本性质及其与已有近似算子之间的关系。研究结果表明,目前文献中出现的多种近似算子可以作为完全分配格上近似算子的特例。

有界Heyting代数的扩张理想和稳定理想

运用泛代数的方法和原理深入研究有界Heyting代数的理想问题.在有界Heyting代数(H,≤,→,0,1)中引入了理想I关于H的子集的扩张理想和稳定理想概念,获得了它们的若干基本性质.系统讨论了由两类特殊扩张理想构成集合的格论特征,证明了:(1)有界Heyting代数(H,≤,→,0,1)的一个给定理想I关于H的所有子集的扩张理想全体之集EI(P(H))在一定条件下构成一个分配完备格,进一步构成一个Stone格和完备Heyting代数;(2)有界Heyting代数(H,≤,→,0,1)的关于一个给定子集A■H的稳定理想全体之集S_(Id(H))(A)构成一个完备Heyting代数.最后考察了商有界Heyting代数和乘积有界Heyting代数的扩张理想性质.

向量格中关于拉格朗日乘子存在性的证明

通过研究向量格中的Hahn-Banach定理,利用线性算子扩张的一般结果,进一步给出向量格中拉格朗日乘子存在性的证明.

L-模糊凸子域度及其诱导的L-模糊凸结构

在L-模糊凸子域的基础上,引入L-模糊凸子域度的概念,利用L-模糊集的四种截集给出L-模糊凸子域度的等价刻画,研究由L-模糊凸子域度诱导的L-模糊凸结构.

Pointwise uniformities and pointwise metrics on fuzzy lattices

UP till now there has been much spectacular and creative work about the theories of uniformi-ties and metrics in topological lattice.But this is mostly generalizations and developmentsof Hutton’s and Erceg’s pointless work.They cannot directly reflect the characteristics ofpointwise topology.In this note we shall set up a theory of pointwise uniformity and a theoryof pointwise metric on fuzzy lattices.

STRONGLY ALGEBRAIC LATTICES AND CONDITIONS OF MINIMAL MAPPING PRESERVING INFS

The tabor gives some characterizations of strongly algebraic lattices, and proves that thecategory of strongly algebraic lattices is complete and cocomplete. Finally, this paper gives thecomplete conditions under which the minimal mapping β: L→2L on a completely distributivelattice L preserves finite infs and arbitrary infs.

Limits in the category of topological molecular lattices

The researches of limits in categories are an important and essential problem in category theory. If the limit and colimit structures in a concrete category having limit and colimit are constructed, then a lot of properties in the category will become