Andrey O. Matveev
Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.
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The blocker $A^{*}$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done: 1. The posets $P$ for which $A^{**}=A$ for all antichains are characterized. 2. The blocker $A^*$ of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed.
Relative Dimension of Posets
本文细化偏序集维数定义得到相对维数概念，讨论其性质并给出部分知名偏序集族的相对维数界。
It is elementary and well-known that if an element x of a bounded modular lattice L has a complement in L then x has a relative complement in every interval [a,b] containing x. We show that the relatively strong assumption of modularity of L can be replaced by a weaker one formulated in the language of so-called modular triples. We further show that, in general, we need not suppose that x has a complement in L. By introducing the concept of modular triples in posets, we extend our results obtained for lattices to posets. It should be remarked that the notion of a complement can be introduced a…
The consecutive pattern poset is the infinite partially ordered set of all permutations where $σ\leτ$ if $τ$ has a subsequence of adjacent entries in the same relative order as the entries of $σ$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Möbius function equal to zero.
We show that with any finite partially ordered set one can associate a matrix whose determinant factors nicely. As corollaries, we obtain a number of results in the literature about GCD matrices and their relatives. Our main theorem is proved combinatorially using nonintersecting paths in a directed graph.
Relative poset polytopes and semitoric degenerations
本文提出相对偏序集多胞形这一新族，系统研究其环面簇性质，并用于刻画旗簇中介于环面与单项式之间的半环面退化，同时推广了Fang等人的多胞形族。
Previously, Erdős, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0<p\le 1/2$. While bounds were given for the range $1/2\le p <1$, the relative accuracy of the results in the original paper deteriorated as $p$ approa…
Let $Δ$ be a root system with a subset of positive roots, $Δ^+$. We consider edges of the Hasse diagrams of some posets associated with $Δ^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $Δ^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $Δ$. We prove that the …
In 2022, Defant and Kravitz introduced extended promotion (denoted $\partial$), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger's promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if $L$ is a labeling of an $n$-element poset $P$, then $\partial^{n-1}(L)$ is a linear extension. This allows us to regard $\partial$ as a sorting operator on the set of all labelings of $P$, where we think of the linear extensions of $P$ as the labelings which have been sorted. The labelings requ…
Blocking codimension-one simplices on the moment curve
本研究探讨阻断一般位置点集张成的余维一单形所需最少点数，改进平面下界，给出偶数维矩曲线的下界及奇数维的最优构造，推进阻断猜想相关研究。
Cobweb posets uniquely represented by directed acyclic graphs are such a generalization of the Fibonacci tree that allows joint combinatorial interpretation for all of them under admissibility condition. This interpretation was derived in the source papers ([6,7] and references therein to the first author).[7,6,8] include natural enquires to be reported on here. The purpose of this presentation is to report on the progress in solving computational problems which are quite easily formulated for the new class of directed acyclic graphs interpreted as Hasse diagrams. The problems posed there and …
We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising fro…
To each lattice simplex $Δ$ we associate a poset encoding the additive structure of lattice points in the fundamental parallelepiped for $Δ$. When this poset is an antichain, we say $Δ$ is antichain. To each partition $λ$ of $n$, we associate a lattice simplex $Δ_λ$ having one unimodular facet, and we investigate their associated posets. We give a number-theoretic characterization of the relations in these posets, as well as a simplified characterization in the case where each part of $λ$ is relatively prime to $n-1$. We use these characterizations to experimentally study $Δ_λ$ for all partiti…
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 New…
An Overlap Construction for Relative Linear Extension Ratios
本文通过重叠构造方法，证明了在d≥(1+ε)c条件下，相对线性扩展比的最小元素数目的上界，并去除了陈和帕克结果中的因子3。
Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory
本文研究有限连通偏序集的区间端态射代数，证明其Reedy代数结构，给出标准模的显式投射分解，将Ext群计算化为单纯复形上同调，并推导全局维数公式，解决网格猜想。
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