Jonathan A. Barmak
We prove that every finite group $G$ can be realized as the automorphism group of a poset with $4|G|$ points. We also provide bounds for the minimum number of points of a poset with cyclic automorphism group of a given prime power order.
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For each $n\ge 1$ we determine the minimum number of points in a poset with cyclic automorphism group of order $n$.
Generating naturally labeled posets through matrix extensions, order ideals and automorphism groups
本文提出了一种矩阵方法，通过将每个偏序集P表示为布尔偏序矩阵A，生成自然标记的偏序集。该方法利用矩阵扩展A^v系统地处理偏序关系，并通过序理想和固定点方程定义有效的偏序矩阵。进一步探索了A的双胞分解，为Birkhoff关于通过自同构群计数非同构偏序集的问题提供了代数基础。
Minimum Size of a Poset Realizing $\Z_{2}\times\Z_{2^{n}}$ as its Automorphism Group
研究有限群作为有限偏序集自同构群的实现问题，证明对每个 $n\ge 3$，$\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$。
The Minimum Size of a Poset Realizing $\mathbb{Z}_2 \times \mathbb{Z}_4$ as its Automorphism Group
本文证明 β(Z₂ × Z₄) = 14，通过轨道分解和高度函数论证，构造了一个14元偏序集并分析其覆盖关系的刚性。
The automorphism group of the random poset does not admit a generic pair
该研究针对Truss提出的问题，证明随机偏序集的每一对自同构的共轭类都是贫集，同时说明当前定理2的证明存在缺漏，研究仍在推进中。
We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an infinite sequence of binary relations which encode the action of the automorphism. We then explicitly characterize generic automorphisms in terms of properties of this structure. Two notable such properties are ultrahomogeneity, and universality for a certain class of finite structures in this language. As this auxiliary structure seems to be new, we also address…
Fixed points of involutive automorphisms of the Bruhat order
应用Smith的一个经典定理，证明了在对合偏序集自同构下的不动点的子集继承了$\mathbb{Z}_2$上的GorensteinPDMATHABMM的偏序集性质.作为应用，我们证明了在任意Coxeter群中的（扭曲）对合上的Bruhat阶中的每个区间都具有这个性质，并且我们找到了秩函数。这意味着F. Incitti.我们还表明，Bruhat秩序的不动点的对合自同构诱导的Coxeter图自同构的Bruhat秩序的固定子群视为一个Coxeter组在自己的权利。
Path Covering Spaces of Posets
本文拓展偏序集覆盖空间理论，针对具唯一极小元的非分层有限偏序集定义路径覆盖，证明其形成超可解格，探究甲板变换与相关自同构的联系。
Automorphisms of decompositions
哈丁证明了许多不同类型的结构，如集合、群、向量空间、拓扑空间和关系结构的直积分解自然地形成正交模偏序集。当应用于希尔伯特空间的直积分解时，这种构造产生了希尔伯特空间的闭子空间的熟悉的正交模格。本文考虑有限集X的分解的正交模偏序集Fact X。我们考虑这些正交模偏序集的结构，如它们的大小、形状、连通性、状态，并在从X的置换群到事实X的自同构群的自然映射Γ的背景下开始研究它们的自同构群。我们证明了除了|X|是素数或4时，Γ是一个嵌入，并完整地描述了当|X|有两个或更少的素因子，当|X| =8和当|X| =27时的情形.我们的大部分努力在于一系列的组合论证，以证明当|X| =27时，Γ是同构。我们猜想，只要|X|有足够多的具有足够大小的素因子，情况就是这样，并希望我们这里的论证可以适用于一般情况。
Let $X$ be a finite connected poset and $K$ a field. We study the question, when all Lie automorphisms of the incidence algebra $I(X,K)$ are proper. Without any restriction on the length of $X$ we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of $X$. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns and ordinal sums of two antichains we give a complete answer.
For an integer $n\geq 1$, we display a poset $Φ_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat Φ_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat Φ_n$ as a group of automorphisms.
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense …
Fixed points of zircon automorphisms
锆石是一个偏序集，其中每个主阶理想都是有限的，并配备了所谓的特殊匹配。证明了锆石的任何自同构的不动点诱导的子集本身就是锆石。这提供了一个自然的背景下，查看最近的结果Bruhat订单扭曲对合在考克斯特群。
The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
A new operation on partially ordered sets
最近证明了所有包含随机偏序集的自同构群的非平凡闭置换群都是由两种类型的置换生成的：第一种是颠倒次序的置换，第二种是由所谓的旋转引起的置换.本文引入了有限偏序集的旋转，它可以看作是有限图的Seidel-开关的偏序集对应。我们分析了它们的一些组合性质，并特别研究了两个有限偏序集何时是旋转等价的问题。此外，我们给出了一个明确的组合结构的旋转的随机偏序集的图像是同构的随机偏序集。作为有限偏序集旋转结果的推论，我们得到随机偏序集的旋转置换群是有限语言中齐次结构的自同构群。
We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group $(\mathbb{Z}_2^4 \times S_4)\rtimes_α \mathbb{Z}_2$. Though unable to prove it, we solidify this conjecture by showing that the automorphism group behaves as expected by the conjecture on the first few levels of the poset in question. With the use of computer calculation we analyze the first four levels holding 3160 directed graphs.
The Minimal Automorphism-Free Tree
一个有$|V(T)| \geq 2$的有限树$T$称为{ \it自同构自由}，如果没有$T$的非平凡自同构。设$\mathcal{AFT}$是所有有限自同构自由树（直到图同构）的元素集按$T_1 \preceq T_2$排序的偏序集，如果$T_1$可以从$T_2$通过每次连续删除一个叶子得到，使得每个中间树也是自同构自由的。本文证明了$\mathcal{AFT}$有唯一的极小元.这个结果对Rupinski提出的问题给出了肯定的回答。
Characterization of hereditarily reversible posets
偏序集P称为可逆的，如果P的每个保序双射自映射是序自同构。P称为遗传可逆的，如果P的每个子集都是可逆的。我们用禁止子集的概念给出了遗传可逆偏序集的一个完整刻画。一个类似的结果也表示为preordered集。作为推论，我们扩展了已知的遗传可逆拓扑空间的例子。
In 1978, Dwight Duffus---editor-in-chief of the journal "Order" from 2010 to 2018 and chair of the Mathematics Department at Emory University from 1991 to 2005---wrote that "it is not obvious that $P$ is connected and $P^P$ isomorphic to $Q^Q$ implies that $Q$ is connected," where $P$ and $Q$ are finite non-empty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.
A Non-Multiplicable Upho Poset Constructed from the Petersen Graph
通过从Petersen图构造一个有限上同调偏序集，并证明其不可乘，反驳了Fu-Peng-Zhang猜想。
On the inverse limits of finite posets
该研究推广Clader的结果，证明任意有限单纯复形都同胚于有限偏序集序列的逆极限，拓展了有限偏序集逆极限相关理论的应用范围。
Automorphism Groups of Comparability Graphs
可比较图是具有传递方向的图。偏序集的维数是其交给出这个偏序集的线性阶的最小数目。可比性图$X$的维度${\rm dim}(X)$是X的任何传递方向的维度，并且通过$k$，我们用${\rm dim}(X) \le k$表示可比性图$X$的类别。已知可比较图的补图是函数图，置换图是2-D图，本文给出置换图的自同构群的刻画，类似于Jordan（1869）对树的刻画.对于置换图来说，有一个额外的操作，所以有一些额外的群不是由树实现的。对于$k \ge 4$，我们证明了每个有限群都可以实现为$k$ -DIM中某个图的自同构群，并且证明了$k$ -DIM的图同构检验是GI-完全的.
Discrete homotopy and homology theories for finite posets
本文研究了有限偏序集的离散同调与同源理论，揭示了其同调群的同构性及通过Hurewicz映射的关联。
On scattered posets with finite dimension
本文讨论了可嵌入到多个离散链的乘积中的偏序集的一个可能的特征，即通过禁用配置。
Finite posets and Ferrers shapes
一个调查的格林-克莱特曼对应，完整的证明，其中许多是新的。
Finite Eulerian posets which are binomial, Sheffer or triangular
本文研究了二项式、Sheffer和三角形的有限欧拉偏序集。这些重要的偏序集类与生成函数理论和几何学有关。本文的主要结果如下：完全确定了Eulerian二项式偏序集的结构，从而可以对Eulerian二项式偏序集的阶乘函数进行分类：通过将原问题分成几种情况，给出了Eulerian Sheffer偏序集的阶乘函数的几乎完全分类;在上面的大多数情况下，我们完全确定了Eulerian Sheffer偏序集的结构，这一结果比仅仅对这些Eulerian Sheffer偏序集的阶乘函数进行分类更强。我们还研究了欧拉三角偏序集。本文回答了R. Ebergborg和M.准备好了本研究也是受R. Stanley通过观察更小的间隔来识别布尔格。
Group Actions on Posets
本文利用群作用研究偏序集的群元。为了正确地定义商，我们扩大考虑类的范畴从偏序集的loopfree类别：类别没有非平凡的自同构和逆。我们把群作用看作是某些函子，并把群作用定义为这些函子的余极限。这个定义相对于研究商偏序集（在我们的语言中是偏序集范畴中的上极限）的优点是，商无环范畴的实现更经常同胚于原偏序集的实现的商。我们给出的条件下，商与神经函子交换，以及条件，保证商再次偏序集。
Aharoni and Korman (Order 9 (1992) 245--253) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for $(3 + 1)$-free posets with no infinite antichains.
It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.
N-free posets and orthomodularity
本文研究有限偏序集的不可比正交集，证明其具有 Dacey 性质等价于偏序集无 N，并刻画了不可比正交集相容的有限偏序集。
We describe four natural operad structures on the vector space generated by isomorphism classes of finite posets. The three last ones are set-theoretical and can be seen as a simplified version of the first, the same way the NAP operad behaves with respect to the pre-Lie operad. Moreover the two first ones are isomorphic.
We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset P. Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points and inducing the poset P on the non- minimal points. It turns out that these counting functions are closely related to a collection of downset numbers of certain subposets. Since any function of that kind is an exponential sum (with the number of minimal points as exponent), we call it the exponential function of the poset. Some linear equations, divisibilit…
In this paper, we first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then we construct the corresponding nonassociative permutative coproduct, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and is a co-free non-associative permutative coalgebra. In the end, we give an explicit duality between the non-associative permutative product and the proposed non-associative permutative coproduct. Finally, we prove that the results in this paper remain true for the fini…
A famous theorem of Dilworth asserts that any finite poset of width $k$ can be decomposed into $k$ chains. We study the following problem: given a Borel poset $P$ of finite width $k$, is it true that it can be decomposed into $k$ Borel chains? We give a positive answer in a special case of Borel posets embeddable into the real line. We also prove a dual theorem for posets whose comparability graphs are locally countable.
Double posets and the antipode of QSym
本文研究双偏序集上的拟对称函数，证明其在对映下的公式，推广了 Malvenuto 和 Reutenauer 的结果，并进一步推广到群作用情形。
Let $S$ be a numerical semigroup and let $\left(\mathbb{Z},\leqslant\_S\right)$ be the (locally finite) poset induced by $S$ on the set of integers $\mathbb{Z}$ defined by $x \leqslant\_S y$ if and only if $y-x\in S$ for all integers $x$ and $y$. In this paper, we investigate the M{ö}bius function associated to $\left(\mathbb{Z},\leqslant\_S\right)$ when $S$ is an arithmetic semigroup.
Hockey-Stick Domination and Distributional Comparison on Finite Posets
通过区间计数测试函数定义曲棍球棒支配序，建立有限偏序集上概率测度的比较框架，给出精确量化刻画并证明该序在多种构造下保持。
On blockers in bounded posets
有限有界偏序集的反链被赋予一个反链，它扮演的角色类似于有限集的所有子集的布尔格中的阻断子所扮演的角色。讨论了广义Blocker格的一些性质。
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
Antichain cutsets of strongly connected posets
Rival和Zaguia证明了有限布尔格的反链割集就是水平集。我们发现，一个类似的反链割集的特征持有任何强连通偏序集的局部有限高度。作为推论，我们得到了半模格、超可解格、Bruhat序、局部可壳格等的一个刻画。我们还考虑了具有有限边的强连通超图的推广。
A contribution to the characterization of finite minimal automorphic posets of width three
本文研究了宽度为三的有限极小自偏序集的刻画问题，针对其子类$\mathfrak{N}_2$，分别刻画了以非平凡nice section塔和4-crown栈为收缩的偏序集，并提供了递归确定高度不超过六的此类偏序集的方法。
CW posets after the Poincare Conjecture
安德斯比约纳的特点是有限分次偏序集出现的偏序集的封闭关系的细胞有限，经常CW复杂。他对这些“CW偏序集”的刻画要求每一个开区间$(\hat{0},u)$都有一个与维度为$rk(u)-2$的球面同胚的序复体。Danaraj和Klee的工作证明了偏序集是薄可壳的充分条件。Poincare猜想的证明使得可壳性的要求可以被同伦Cohen-Macaulay性质所取代。这扩展了可以用来证明偏序集是CW偏序集的工具的范围。
Incomparable copies of a poset in the Boolean lattice
设$B_n$是由$[n]$的子集以包含为关系生成的偏序集，$P$是有限偏序集.我们希望尽可能多地将$P$嵌入到$B_n$中，以便不同副本中的子集是不可比较的。对于所有有限偏序集$P$，这种嵌入的最大数目渐近确定为$\frac{n \choose \lfloor n/2\rfloor}{M(P)}$，其中$M(P)$表示$P$的副本的凸包的最小大小。我们讨论弱嵌入和强（诱导）嵌入。
Fixed elements of pircon automorphisms
本文证明 pircon 自同构的不动元素仍构成 pircon，并将其应用于对称群，证明无不动点带符号对合在超八面体群 Bruhat 序对偶下形成 EL 可壳的 pircon。
We compute the (primary) equivariant Euler characteristics of the building for the general linear group over a finite field.
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that $N$-free chain complete posets and $N$-free posets with no infinite antichains are series-parallel. As a consequence, we obtain that every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order \textbf{9} (1992) 245--253) in this case.
Gluing Posets and the Dichotomy of Poset Saturation Numbers
本文通过定义偏序集的“粘合”操作，证明该操作保持饱和数的有界与无界性，并应用于证明任何偏序集最多添加3个元素后其饱和数增长至多为线性。
Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P \in \mathfrak{P}_r$, a one-to-one mapping $ρ_P : {\cal H}(P,R) \rightarrow {\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R \sqsubseteq_G S$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite posets $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. In consequence…
Block-transitive designs with a poset of imprimitive partitions
本文研究了具有特定偏序集的块设计，通过广义wreath产品理论，给出了块传递2-设计的条件，并提供了多个偏序集的显式例子。
A Non-Messing-Up Phenomenon for Posets
我们分类有限偏序集与一个特定的排序属性，推广的结果为矩形阵列。每个偏序集被两组不相交的饱和链覆盖，使得对于任何原始标签，在沿着两组链对标签进行排序之后，第一组中的链的标签保持排序。我们还刻画了偏序集的更严格的排序性质。
Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets
在序理论中，秩函数度量偏序集元素的垂直“水平”。它是偏序集上的一个整值函数，它随覆盖关系而增加，并且只在分次偏序集上可用。定义任意有限偏序集的垂直测度可以通过将秩函数扩展为区间值来实现。这建立了从基偏序集到实区间上的偏序集的序同态，并且标准（规范）特定区间秩函数可用作极端情况。各种顺序关系可在区间，我们开始在本文中考虑共轭秩序的“分区”空间的两两比较的顺序元素。对我们来说，这些元素是实区间，我们认为弱和子集区间顺序（近）共轭。我们也很自然地会问，区间秩函数自反地应用于我们所选择的任何区间偏序集，从而得到区间秩的一般迭代策略。我们探讨了标准和共轭区间秩的收敛性质，并以支持这项工作所需的实验数学的讨论结束。
The Kelly--Trotter product conjecture for posets of dimension three
该研究证明了三维偏序集下的Kelly–Trotter乘积猜想，否定了Trotter的相关猜想，还得出特定偏序集乘积维数的结论，所用方法涉及偏序集分类与图着色搜索。
Conclusive Posets and Torsion in First Homology
该研究通过反例推翻了所有偏序集都是确定性的猜想，并提供了有限偏序集确定性的组合特征化描述。
Polyunsaturated Posets and Graphs and the Greene-Kleitman Theorem
有限偏序集的链划分给k个反链的并的大小带来了一个自然的上界。一个链划分是k-饱和的，如果达到这个界限。Greene和Kleitman证明了，对于每个k，每个有限偏序集都有一个同时k-和k+1-饱和的链划分。West证明了Greene-Kleitman定理在强意义上是最好可能的，对每个c \ge 4，展示了一个偏序集，它具有基数为c的最长链，并且对任何不同的、非连续的k，l < c，没有k-和l-饱和链划分。我们称这种偏序集为多不饱和偏序集。给出了具有给定高度、宽度和基数的多不饱和偏序集存在的充要条件。我们证明这些结果在更一般的情况下，图满足类似的格林-克莱特曼定理。最后，我们讨论了反链分拆的类似结果。
An extension of Birkhoff's representation theorem to locally-finite distributive lattices
本文扩展了布劳尔表示定理，针对局部有限分配格，提出新的表示定理，通过素滤子的对称差与特定理想的关系来定义偏序。
We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent and derive some numerical results concerning 2-chains.
Frieze patterns as root posets and affine triangulations
frieze模式的元素可以解释为秩为2的有限Weyl广群的根的坐标。我们证明了根偏序集上的极大元的存在性，并对可以用来构造仿射单纯形排列的frieze模式进行了分类。
Graphical view on linear extensions of finite posets
本文通过图论视角揭示有限偏序集的隐喻定义，证明测地凸性与线性扩展的关系，并探讨其几何和组合特性。
EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets
我们证明了秩为n的有限分次格是超可解的当且仅当它有一个EL-标号，其中沿任意极大链的标号形成一个置换。我们称这样的标号为S_nEL-标号，并考虑具有唯一顶元和底元的秩为n的有限分次偏序集具有S_nEL-标号。我们描述了一类A0-Hecke代数在这类偏序集的极大链上的作用.这个作用是局部的，并给出了这些Hecke代数的一个表示，其特征具有与Escherborg的旗拟对称函数密切相关的特征。我们问什么其他类别的偏序集有这样的作用，特别是我们证明了有限分次格的秩n有这样的作用当且仅当它们有一个S_nEL-标签。
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