A note on random k-dimensional posets

In 2009, Janson [Poset limits and exchangeable random posets, Institut Mittag-Leffler preprint, 36pp, arXiv:0902.0306] extended the recent theory of graph limits to posets, defining convergence for poset sequences and proving that every such sequence has a limit object. In this paper, we focus on k-dimensional poset sequences. This restriction leads to shorter proofs and to a more intuitive limit object. As before, the limit object can be used as a model for random posets, which generalizes the well known random k-dimensional poset model. This investigation also leads to a definition of quasirandomness for k-dimensional posets, which can be captured by a natural distance that measures the discrepancy of a k-dimensional poset.     

On forbidden configurations for strong posets

The concept of strong elements in posets is introduced. Several properties of strong elements in different types of posets are studied. Strong posets are characterized in terms of forbidden structures. It is shown that many of the classical results of lattice theory can be extended to posets. In particular, we give several characterizations of strongness for upper semimodular (USM) posets of finite length. We characterize modular pairs in USM posets of finite length and we investigate the interrelationships between consistence, strongness, and the property of being balanced in USM posets of finite length. In contrast to the situation in upper semimodular lattices, we show that these three concepts do not coincide in USM posets.     

To the Spectral Theory of Partially Ordered Sets

Siberian Mathematical Journal - We suggest an approach to advance the spectral theory of posets. The validity of the Hofmann-Mislove Theorem is established for posets and a characterization is...     

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.     

Rudin性质与拟Z-连续Domain

对一般子集系统 Z,引入了 Rudin性质,给出了它的映射式刻划,作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了 Rudin性质及其映射式刻划在拟 Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟 Z-连续 Domain情形。     

Dynkin Diagram Classification of λ-Minuscule Bruhat Lattices and of d-Complete Posets

d-Complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to -minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to -minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.     

Topologies on products of partially ordered sets II: Ideal topologies

Within the theory of ideals in partially ordered sets, several difficulties set in which do not occur in the special case of lattices (or bidirected posets). For example, a finite product of ideals in the factor posets need not be an ideal in the product poset. The notion ofstrict ideals is introduced in order to remedy some deficiencies occurring in the general case of an arbitrary product of posets. Besides other results, we show the following main theorem: The ideal topology (cf. [2]) of a product of non-trivial posets coincides with the product topology if and only if the number of factors is finite (4.19.). Presented by L. Fuchs     

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.     

AZ-Identities and Strict 2-Part Sperner Properties of Product Posets

One of central issues in extremal set theory is Sperner’s theorem and its generalizations. Among such generalizations is the best-known LYM (also known as BLYM) inequality and the Ahlswede–Zhang (AZ) identity which surprisingly generalizes the BLYM into an identity. Sperner’s theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.     

Finite Eulerian posets which are binomial, Sheffer or triangular

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:

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We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets.

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We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases.

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In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.

We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.     

Reverse mathematics and order theoretic fixed point theorems

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within \(\mathsf {RCA_0}\) give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over \(\mathsf {RCA_0}\). Then we within \(\mathsf {RCA_0}\) give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over \(\mathsf {RCA_0}\). Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.     

The Face Structure and Geometry of Marked Order Polyhedra

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand–Tsetlin polytopes and cones, as well as Berenstein–Zelevinsky polytopes—all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.     

Homotopy type of posets and lattice complementation

This paper is concerned with homotopy properties of partially ordered sets, in particular contractibility. The main result is that a noncomplemented lattice with deleted bounds is contractible. The paper also presents (i) the homology of final sets and cutsets, (ii) a generalization to posets of Rota's crosscut theorem, (iii) contractibility proofs for some classes of posets of interest in fixed point theory, and (iv) a simple characterization of the Cohen-Macaulay property for dismantlable lattices.     

Some Remarks on Pseudotrees

This paper provides new results on pseudotrees. First, it is shown that pseudotrees are precisely those posets for which consistent sets, directed sets, and nonempty chains coincide. Second, we show that chain-complete pseudotrees yield complete meet-semilattices. Third, we prove that pseudotrees are precisely those posets that admit a set representation by sets of appropriate chains. This latter result generalizes results needed for applications in game theory and economics.     

Unimodality and Lie Superalgebras

It is well-known how the representation theory of the Lie algebra sl(2, ?) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.     

Effect algebras with the maximality property

The maximality property was introduced in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch–Piron states on an effect algebra with the maximality property is strongly order determining.     

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.     

Fixed point theorems and their applications to theory of Nash equilibria

In this paper we prove fixed point theorems for set-valued mappings in products of posets. Applications to the theory of Nash equilibria are presented.     

Order varieties generated by finite posets

In a 1981 paper, Duffus and Rival define an order variety as a class of posets that is closed under the formation of products and retracts. They also introduce the notion of an irreducible poset. In the present paper we define nonextendible colored posets and certain minimal nonextendible colored posets that we call zigzags. We characterize via nonextendible colored posets the order varieties generated by a set of posets. If the generating set contains only finite posets our characterization is via zigzags. By using these theorems we give a characterization of finite irreducible posets.As an application we show that two different finite irreducible posets generate two different order varieties. We also show that there is a poset which has two different representations by irreducible posets. We thereby settle two open problems listed in the Duffus and Rival paper.     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.