Weakly Orthomodular and Dually Weakly Orthomodular Lattices

We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product.     

Stone''''s representation theorem in fuzzy topology

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0 ∈ L is a prime or 1 ∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L-locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L-topological spaces.     

A characterization of neutral elements by the exclusion of sublattices

The main result of this paper gives a characterization of neutral elements in lattices by the exclusion of seventeen types of sublattices containing it. The proof involves several generalizations of distributive, standard and neutral elements in lattices which are introduced using certain modular identities.     

分配格与模格的另一二条件等价定义

针对分配格与模格的格等式定义问题,得知了二条件是定义分配格与模格的最少条件,并进一步证明了Sholander's basis是定义分配格的最短最少变量格等式,最后又从分配格和模格的基本定义出发给出了新的分配格的二条件和三条件等价定义等式及模格的二条件与三条件等价定义等式.     

Stone style duality for distributive nearlattices

The aim of this paper is to study the variety of distributive nearlattices with greatest element. We will define the class of N-spaces as sober-like topological spaces with a basis of open, compact, and dually compact subsets satisfying an additional condition. We will show that the category of distributive nearlattices with greatest element whose morphisms are semi-homomorphisms is dually equivalent to the category of N-spaces with certain relations, called N-relations. In particular, we give a duality for the category of distributive nearlattices with homomorphisms. Finally, we apply these results to characterize topologically the one-to-one and onto homomorphisms, the subalgebras, and the lattice of the congruences of a distributive nearlattice.     

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.     

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.     

On the level decompositions by the range of an <Emphasis Type="Italic">L</Emphasis>-fuzzy topological space

This paper is on the level decompositions corresponding to the elements in the range of an L-fuzzy topology on a given set and an investigation into the lattice structure of the same. In general, this lattice is not complete and distributive. However, certain necessary and sufficient conditions for it to be modular, distributive, complete and complemented are derived. Atoms and dual atoms of it along with the conditions for their existence are obtained. Certain related properties of it are also discussed here.     

Subdiagrams equal in number to their duals

In the middle 1930's, the early days of combinatorial lattice theory, it had been conjectured thatin any finite modular lattice the number of join-irreducible elements equals the number of meetirreducible elements. The conjecture was settled in 1954 by R. P. Dilworth in a remarkable combinatorial generalization. Quite recently we have been led to this question. Which ordered setsS satisfy the property that,for any modular lattice M, the number of subdiagrams of M isomorphic to S equals the number of subdiagrams of M dually isomorphic to S?     

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．     

Z_S-相容连续偏序集及其相关范畴性质

引入了Zs-相客集系统的概念,讨论了Zs-相客连续偏序集的一系列性质.证明了Zs-相容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.     

The lattice of closure relations on a poset

In this paper we show that the set of closure relations on a finite posetP forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison [EJ]). We also characterize the modular elements of this lattice (whenP has a greatest element) and compute its characteristic polynomial.Presented by R. W. Quackenbush.     

Group-valued modular functions

Modular functions on a lattice (m(x)+m(y)=m(x∪y)+m(x∩y)) live on modular lattices in that they are induced by modular functions on a quotient modular lattice. Those which identify pairs of the distributive inequality live on distributive lattices in the same sense. The structure of all modular functions on a lattice of finite height is determined. The “distance function” derived by Kranz from a modular function is shown to satisfy the triangle inequality. Presented by E. Nelson.     

Z-连通集系统及其范畴特征

本文引入了Z-连通集系统的概念,讨论了Z-连通连续偏序集的一系列性 质,证明了Z-连通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.     

Comparability graphs of lattices

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is (d+1)-connected.In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d∈N₀). Then the comparability graph of L is (d+1)-connected if and only if L has no simplicial elements, where z∈L is simplicial if the elements comparable to z form a chain.     

Generalized Priestley Quasi-Orders

We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991; Schmid, Order 19(1):11–34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the well-known characterization (Priestley, Proc Lond Math Soc 24(3):507–530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147–151, 1974).     

On the Distributivity and Modularity Properties of the Lattice of Fitting Classes

Mathematical Notes - It is proved that the lattice of all Fitting classes of finite groups is not distributive, and conditions under which Fitting classes satisfy the distributive and modular laws...     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

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.     

Graded left modular lattices are supersolvable

We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [MT] by combining results of McNamara [Mc] and Liu [Li]. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.Received May 24, 2004; accepted in final form October 12, 2004.