Uniquely Complemented Posets

We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.     

Relatively pseudocomplemented Hilbert algebras

We characterise those Hilbert algebras that are relatively pseudocomplemented posets.      

关于几乎唯一泛圈图

设G是阶为n的简单Hamilton图．若存在m(3(?)m＜n)使对每个l∈{3,4,…,n} -{m},G恰有一个长为l的圈且不含长为m的圈,则称G是几乎唯一泛圈图,用(?)k表示具有n k条边和恰有1/2(k 1)(k 2)个圈的简单H图的集合,用(?)_k~*表示具有n k条边恰有2~k k个圈的简单外可平面H图的集合,本文确定了(?)_k和(?)_k~*中所有几乎唯一泛圈图,并证明这些图都是简单MCD图,本文还构造了50个含有同胚于K_4的子图的几乎唯一泛圈图,并提出了若干问题和猜想。     

Orthocomplemented Posets with a Symmetric Difference

We endow orthocomplemented posets with a binary operation–an abstract symmetric difference of sets–and we study algebraic properties of this class, . Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative. The authors acknowledge the support of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic and the grant GAČR 201/07/1051 of the Czech Grant Agency.     

Z-Continuous Posets and Their Topological Manifestation

A subset selection Z assigns to each partially ordered set P a certain collection Z P of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general Z-level, by replacing finite or directed sets, respectively, with arbitrary Z-sets. This leads to a theory of Z-union completeness, Z-arity, Z-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general Z-setting as well. For example, we characterize Z-distributive posets and Z-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections Z, a poset P is strongly Z-continuous iff its Z-join ideal completion Z P is Z-ary and completely distributive. Using that characterization, we show that the category of strongly Z-continuous posets (with interpolation) is concretely isomorphic to the category of Z-ary Z-complete core spaces. For suitable subset selections Y and Z, these are precisely the Y-sober core spaces.     

Additive Macaulay Posets

An additive sequence of integers is a finite sequence in which the sum of any number of consecutive terms is less than or equal to the sum of the same number of initial terms in the sequence and greater than or equal to the sum of the same number of final terms in the sequence. If the final several terms in an additive sequence are greater than or equal to, in order, the initial several terms of a second additive sequence, then the juxtaposition of the two sequences is also additive. This simple fact has combinatorial corollaries.     

Characterizations of Standard Elements in Posets

The fundamental characterization theorem of standard elements in lattices is extended to posets. Several other characterizations of standard elements are obtained in a sectionally semi-complemented poset and also in an atomistic, dually sectionally semi-complemented poset.     

代数的局部完备集范畴和FS-局部dcpo范畴的笛卡儿闭性

本文引入了代数的局部完备集,FS-局部dcpo,局部稳定映射等概念．主要结果是：以局部Scott连续映射为态射的代数的局部完备集范畴,以局部稳定映射为态射的代数的局部完备集范畴以及以局部Scott连续映射为态射的FS-局部dcpo范畴都是笛卡儿闭范畴．     

A Note on the Homology of Signed Posets

Let S be a signed poset in the sense of Reiner [4]. Fischer [2] defines the homology of S, in terms of a partial ordering P (S) associated to S, to be the homology of a certain subcomplex of the chain complex of P (S).In this paper we show that if P (S) is Cohen-Macaulay and S has rank n, then the homology of S vanishes for degrees outside the interval [n/2, n].Research partially supported by the National Science Foundation and the John Simon Guggenheim Foundation.     

分层有限偏序集的直积

A new concept Graded Finite Poset is proposed in this paper.Through discussing some basic properties of it,we come to that the direct product of graded finite posets is connected if and only if every graded finite poset is connected.The graded function of a graded finite poset is unique if and only if the graded finite poset is connected.     

Mpi-空间与偏序集

通过建立mpi-空间和偏序集之间的对应关系,在同构意义下,得到无环mpi-空间的特征.利用这种持征,mpi-空间的一些结果可以转化到偏序集框架结构下进行研究.这些成果清楚地表明这里所提供的思想是研究mpi-空间的一个新方法.最后,概述了我们未来的一些工作.     

Three-Variable Equations of Posets

We find an independent base for three-variable equations of posets.     

A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for Cartesian powers P ⁿ of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L ⁿ . Namely, we show that under certain conditions the shadow of any initial segment of the order L ⁿ for n 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.     

Notes on sectionally complemented lattices. I

Summary In 1944, R.P. Dilworth proved (unpublished) that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In 1960, G. Grätzer and E. T. Schmidt improved this result by constructing a finite sectionally complemented lattice L whose congruence lattice represents D. In L, sectional complements do not have to be unique. The one sectional complement constructed by G. Grätzer and E. T. Schmidt in 1960, we shall call the 1960 sectional complement. This paper examines it in detail. The main result is an algebraic characterization of the 1960 sectional complement.     

On Complemented Subspaces of Some Koethe Spaces of Infinite Type

There is a basis for each complemented subspace of a Koethe space in the Dragilev class of type 1.     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Quasicontinuity of Posets via Scott Topology and Sobrification

In this paper, posets which may not be dcpos are considered. In terms of the Scott topology on posets, the new concept of quasicontinuous posets is introduced. Some properties and characterizations of quasicontinuous posets are examined. The main results are: (1) a poset is quasicontinuous iff the lattice of all Scott open sets is a hypercontinuous lattice; (2) the directed completions of quasicontinuous posets are quasicontinuous domains; (3) A poset is continuous iff it is quasicontinuous and meet continuous, generalizing the relevant result for dcpos. Supported by the NSF of China (10371106, 10410638) and by the Fund (S0667-082) from Nanjing University of Aeronautics and Astronautics.     

Notes on sectionally complemented lattices. II

Summary G. Grätzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Grätzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders.     

Order Varieties and Monotone Retractions of Finite Posets

In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence modular if and only if every idempotent subalgebra of P is connected. We give a polynomial time algorithm to decide whether or not a variety generated by an order-primal algebra admits a near unanimity function and so we answer a problem of Larose and Zádori.