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.     

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．     

Partitioning Posets

Given a poset P = (X, ≺ ), a partition X ₁, ..., X _k of X is called an ordered partition of P if, whenever x ∈ X _i and y ∈ X _j with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X, ≺ ) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m − 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P. We show that this bound is best possible for k = 2, but we give an improved bound, , for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X, ≺ ) and an integer 2 ≤ k ≤ |X|, we can find an ordered partition of P into k parts that minimises the total number of comparable pairs within parts in time polynomial in the size of P. We prove more general, weighted versions of these results. Supported by an EPSRC doctoral training grant.     

Tchebyshev Posets

We construct for each $n$ an Eulerian partially ordered set $T_n$ of rank $n+1$ whose $ce$-index provides a non-commutative generalization of the $n$th Tchebyshev polynomial. We show that the order complex of each $T_n$ is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression $\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j|$ among the $f$-vectors of all $(n-1)$-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanleys conjecture on the non-negativity of the $cd$-index of all Gorenstein$^*$ posets.     

Posets and planar graphs

A k-decomposition (G₁,…,G_k) of a graph G is a partition of its edge set to form k spanning subgraphs G₁,…,G_k. The classical theorem of Nordhaus and Gaddum bounds χ(G₁) + χ(G₂) and χ(G₁)χ(G₂) over all 2-decompositions of K_n. For a graph parameter p, let p(k;G) denote the maximum of over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χℓ, and Szekeres–Wilf number σ satisfy ω(2;K_n) = χ(2;K_n) = χℓ(2;K_n) = σ(2;K_n) = n + 1. We obtain lower and upper bounds for ω(k;K_n), χ(k;K_n), χℓ(k;K_n), and σ(k;K_n). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Hypercontinuous Posets

The concepts of hypercontinuous posets and generalized completely continuous posets are introduced. It is proved that for a poset P the following three conditions are equivalent:(1) P is hypercontinuous;(2) the dual of P is generalized completely continuous;(3) the normal completion of P is a hypercontinuous lattice. In addition, the relational representation and the intrinsic characterization of hypercontinuous posets are obtained.     

k-Eulerian Posets

A poset P is called k-Eulerian if every interval of rank k is Eulerian. The class of k-Eulerian posets interpolates between graded posets and Eulerian posets. It is a straightforward observation that a 2k-Eulerian poset is also (2k+1)-Eulerian. We prove that the ab-index of a (2k+1)-Eulerian poset can be expressed in terms of c=a+b, d=ab+ba and e ^2k+1=(a–b)^2k+1. The proof relies upon the algebraic approaches of Billera-Liu and Ehrenborg-Readdy. We extend the Billera-Liu flag algebra to a Newtonian coalgebra. This flag Newtonian coalgebra forms a Laplace pairing with the Newtonian coalgebra ka,b studied by Ehrenborg-Readdy. The ideal of flag operators that vanish on (2k+1)-Eulerian posets is also a coideal. Hence, the Laplace pairing implies that the dual of the coideal is the desired subalgebra of ka,b. As a corollary we obtain a proof of the existence of the cd-index which does not use induction.     

伪超连续偏序集

引入了伪超连续偏序集和伪超代数偏序集的概念,讨论了它们的一些基本性质,给出了它们的若干刻画,特别是基于区间拓扑、某种特殊二元关系的刻画和内蕴式刻画.     

Z-Semicontinuous Posets

Z-semicontinuous posets include semicontinuous lattices and Z-continuous posets as special cases. We characterized when the associated Z-waybelow relation is multiplicative and also make a topological connection.     

L—Fuzzy偏序集

借助L-fuzzy关系在L-fuzzy中集中引入L-fuzzy偏序,自然地有了L-fuzzy偏序集,进一步借助水平截集刻画了L-fuzzy偏序集。     

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.     

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.     

Order-Convergence in Posets

The kind of convergence induced in subsets of posets by order-convergence is characterized: it is separated regular Fréchet filter convergence. An order-convergence completion for posets and progroups is presented.     

Reversible and Bijectively Related Posets

Call a poset reversible if every of its order-preserving self-bijections is an automorphism. Call two posets bijectively related if from each of the two posets exists an order-preserving bijection to the other. We present two examples of pairs of non-isomorphic, bijectively related posets and an example of a non-reversible poset that is bijectively related only to itself. Also, three classes of reversible posets are described and a sufficient condition for an order-preserving bijection to be an isomorphism is presented.     

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.     

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem–Mahler–Lech theorem, the ab-series of a level poset is shown to be a rational generating function in the non-commutative variables a and b. In the case the poset is also Eulerian, the analogous result holds for the cd-series. Using coalgebraic techniques a method is developed to recognize the cd-series matrix of a level Eulerian poset.     

Varieties of Posets

We introduce and investigate the notion of a homomorphism, of a congruence relation, of a substructure of a poset and consequently the notion of a variety of posets. These notions are consistent with those used in lattice theory and multilattice theory. There are given some properties of the lattice of all varieties of posets.     

Strongly Signable and Partitionable Posets

The class ofStrongly Signablepartially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs' recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flagh-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flagh-polynomial of a simplicial complex in terms of itsh-vector is given, and is used to demonstrate that the flagh-vector is symmetric or non-negative whenever theh-vector is.