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.     

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．     

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.     

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.     

Three-Variable Equations of Posets

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

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.     

A Characterisation of Posets that Are nearly Antichains

We present a characterisation of posets of size n with linear discrepancy n − 2. These are the posets that are “furthest” from a linear order without being an antichain. This revised version was published online in September 2006 with corrections to the Cover Date.     

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.     

From Leibniz Homology to Cyclic Homology

For an algebra R over a commutative ring k, a natural homomorphism _*: HL_*+1(R) HH_* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map _* is surjective when R = gl(A), A an algebra over a characteristic zero field. If f: A B is an algebra homomorophism, the relative groups HL_* (gl(f)) are studied, where gl(f): gl(A) gl(B) is the induced map on matrices. Letting HC_* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL_*+1(gl(f)) HC_* (f) in the characteristic zero setting.     

Large-Scale Regularities of Lattice Embeddings of Posets

Let N denote the set of natural numbers and let P =(N ^k , ) be a countably infinite poset on the k-dimensional lattice N ^k . Given x N ^k , we write max(x) (min(x)) for the maximum (minimum) coordinate of x. Let be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to N ^k and edge set equal to the set of all (x, y) such that max(x) max(y) and x and y not comparable in P. For any subset D N ^k , we let P _D and _D denote the restrictions of P and to D. Points x N ^k with min(x) = 0 will be called boundary points. We define a geometrically natural notion of when a point is interior to P or relative to the lattice N ^k , and an analogous notion of monotone interior with respect to or _D . We wish to identify situations where most of these interior points are exposed to the boundary of the lattice or, in the case of monotone interior points, not concealed very much from the boundary. All of these ideas restrict to finite sublattices F ^k and/or infinite sublattices E ^k of N ^k . Our main result shows that for any poset P and any arbitarily large integer M > 0, there is an F E with F = M where, relative to the sublattices F ^k E ^k , the ideal situation of total exposure of interior points and very little concealment of monotone interior points must occur. Precisely, we prove that for any P =(N ^k , ) and any integer M > 0, there is an infinite E N and a finite D F ^k with F E and F = M such that (1) every interior vertex of P _{E k} or _{E k} is exposed and (2) there is a fixed set C E, C k ^k , such that every monotone-interior point of _D belonging to F ^k has its monotone concealment in the set C. In addition, we show that if P ¹ =(N ^k , ₁),..., P ^r =(N ^k , _r ) is any sequence of posets, then we can find E,D, and F so that the properties (1) and (2) described above hold simultaneously for each P ⁱ . We note that the main point of (2) is that the bound k ^k depends only on the dimension of the lattice and not on the poset P. Statement (1) is derived from classical Ramsey theory while (2) is derived from a recent powerful extension of Ramsey theory due to H. Friedman and shown by Friedman to be independent of ZFC, the usual axioms of set theory. The fact that our result is proved as a corollary to a combinatorial theorem that is known to be independent of the usual axioms of mathematics does not, of course, mean that it cannot be proved using ZFC (we just couldn"t find such a proof). This puts our geometrically natural combinatorial result in a somewhat unusual position with regard to the axioms of mathematics.     

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.     

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.     

The Semimodularity of Cardinal Power of Posets

LetX,Ybeanyposets,Y'denotethecardinalpowerwithbaseYandexponentX.Fortherelatedconceptslsymbolsorterminologies,seeF1],[2jor[3].lnL1],someexamplesaregaven,whichshowthatthesemimodularityofX,YandYxareindepentent.AndaquestionisaskedthatwhenthecardinalpowerYxwithbaseYandexpo-nentXissemimodular.Wenowanswerthequestioninthispaper.LemmaLetX,Ybeanyposets,f,geyx,thenthefollowingconditionsareequivalent:(l)fiscoveredbyg(denotedbyf,g);(2)Thereexists'anelementxoeX,suchthatf(x,),g(x,),andf(x)=g(x)(foran…     

有限偏序集的Cartesian积的收缩

通过对有限偏序集的Cartesian积的收缩的考查,证明了Plotkin在1978年提出的一个猜想,并部分回答了Mislove和Lawson在"Open Problems in Topology"中提出的一个公开问题.     

一个Cluster-Tilted代数的Hochschild同调与循环同调

基于Furuya构造的一个Cluster-Tilted代数的极小投射双模分解,用组合的方法计算了Cluster-Tilted代数的Hochschild同调空间的维数与基.当基础域的特征为零时,也计算了代数的循环同调群的维数.     

On Uniquely Complemented Posets

In this paper, some classical results of uniquely complemented lattices are extended to uniquely complemented posets (with 0 and 1) like Peirce's Theorem, the Birkhoff–von Neumann Theorem, the Birkhoff–Ward Theorem. Further, it is shown that a section semi-complemented pseudocomplemented poset is a Boolean poset. Mathematics Subject Classification (2001) 06A06, 06A11, 06C15, 06C20, 06D15     

First Observations on Prefab Posets’ Whitney Numbers

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers’ triangular array — are then calculated and the explicit formula for them is provided. Next — in the second construction — we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F-sequence independent while the second partial order is F-sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F-determined cobweb poset’s Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section. Presentation (November 2006) at the Gian-Carlo Rota Polish Seminar .     

Posets on up to 16 Points

In this article we describe a very efficient method to construct pairwise non-isomorphic posets (equivalently, T ₀ topologies). We also give the results obtained by a computer program based on this algorithm, in particular the numbers of non-isomorphic posets on 15 and 16 points and the numbers of labelled posets and topologies on 17 and 18 points.     

Hereditary properties of combinatorial structures: Posets and oriented graphs

A hereditary property of combinatorial structures is a collection of structures (e.g., graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g., induced subgraphs), and contains arbitrarily large structures. Given a property , we write for the collection of distinct (i.e., non‐isomorphic) structures in a property with n vertices, and call the function the speed (or unlabeled speed) of . Also, we write for the collection of distinct labeled structures in with vertices labeled , and call the function the labeled speed of . The possible labeled speeds of a hereditary property of graphs have been extensively studied, and the aim of this article is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labeled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least . We also show that there is an initial jump in the possible unlabeled speeds of hereditary properties of posets, tournaments, and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 311–332, 2007