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.     

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.     

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.     

The Involutory Dimension of Involution Posets

The involutory dimension, if it exists, of an involution poset P:=(P,,) is the minimum cardinality of a family of linear extensions of , involutory with respect to , whose intersection is the ordering . We show that the involutory dimension of an involution poset exists iff any pair of isotropic elements are orthogonal. Some characterizations of the involutory dimension of such posets are given. We study prime order ideals in involution posets and use them to generate involutory linear extensions of the partial ordering on orthoposets. We prove several of the standard results in the theory of the order dimension of posets for the involutory dimension of involution posets. For example, we show that the involutory dimension of a finite orthoposet does not exceed the cardinality of an antichain of maximal cardinality. We illustrate the fact that the order dimension of an orthoposet may be different from the involutory dimension.     

On Jump-Critical Posets with Jump-Number Equal to Width

We investigate the structure of m-jump-critical posets P with w(P) = m. We prove that the size of such posets satisfies |P| 3m ². For the special when the maximum antichain occurs as the maximal (or minimal) elements, we have the sharp upper bound |P| 3m - k; where k = min {|{Max}(P)|, |{Min}(P)|}. We give examples of posets which illustrate the explored structure of these m-jump-critical posets P with width m.     

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.     

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.     

Cardinal ECT-splines

Cardinal ECT-spline curves are generated from one ECT-system of order n which is shifted by integer translations via one connection matrix. If this matrix is nonsingular, lower triangular and totally positive, there exists an ECT-B-spline function N₀ⁿ(x) having minimal compact support [0,n] whose integer translates span the cardinal ECT-spline space. This B-spline is computed explicitly piece by piece. Involved is the characteristic polynomial of a certain matrix which is the product of a matrix related to the connection matrix and of the generalized Taylor matrix of the basic ECT-system. This approach extends results for polynomial cardinal splines via connection matrices [6] to the more general setting of cardinal ECT-splines. The method is illustrated by two examples based on ECT-systems of rational functions with prescribed poles. Also, a Greens function involved is expressed explicitly as an ECT-B-splines series. AMS subject classification 41A15, 41A05     

A Cardinal Pattern Inspired by AD

Assuming Con(AD), a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con(AD), we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals. Mathematics Subject Classification: 03E55, 03E60.     

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.     

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.     

Jónsson Terms and Near-Unanimity Functions in Finite Posets

We construct monotone Jónsson terms and near-unanimity functions based on combinatorial properties of finite posets.     

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.     

The Pointwise Characterizations of Fuzzy Mappings and Fuzzy Cardinal Numbers

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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.     

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.     

On the Cofinality of Infinite Partially Ordered Sets: Factoring a Poset into Lean Essential Subsets

We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable cofinality remains open.