Andrey O. Matveev
Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.
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An order-theoretic generalization of Seymour relations describing the connection between the set-theoretic blocker, deletion, and contraction maps on clutters, is presented.
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.
In 2015, Felsner, Trotter, and Wiechert showed that posets with outerplanar cover graphs have bounded dimension. We generalise this result to posets with $k$-outerplanar cover graphs. Namely, we show that posets with $k$-outerplanar cover graph have dimension $\mathcal{O}(k^3)$. As a consequence, we show that every poset with a planar cover graph and height $h$ has dimension $\mathcal{O}(h^3)$. This improves the previously best known bound of $\mathcal{O}(h^6)$ by Kozik, Micek and Trotter.
On ranked and bounded Kohnert posets
本文研究了与Kohnert多项式相关的Kohnert偏序集的组合性质，给出了其有界的充分条件和有秩的必要条件，并在特殊情况（如Demazure特征标）下完成了完整刻画。
Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of $P$ being homotopy equivalent to the suspension of the proper part of~$Q$. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders under both inclusion order (a known proved earlier by Ziegler) and single step inclusion order (which was not previously known).
In the early 1970's, Richard Stanley and Kenneth Johnson introduced and laid the groundwork for studying the order polynomial of partially ordered sets (posets). Decades later, Hamaker, Patrias, Pechenik, and Williams introduced the term "doppelgangers": equivalence classes of posets under the equivalence relation given by equality of the order polynomial. We provide necessary and sufficient conditions on doppelgangers through application of both old and novel tools, including new recurrences and the Ur-operation: a new generalized poset operation. In addition, we prove that the doppelgangers …
Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant $d$ such that if $P$ is a poset with cover graph of $P$ of pathwidth at most $2$, then $\dim(P)\leq d$. We answer this question in the affirmative by showing that $d=17$ is sufficient. We also show that if $P$ is a poset containing the standard example $S_5$ as a subposet, then the cover graph of $P$ has treewidth at least $3$.
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounde…
The Bounded-Linear Dichotomy Holds for all 4 Point Posets
本文证明了所有至多4个点的偏序集的饱和数要么有界要么线性，并特别解决了由V形加孤立点构成的偏序集的线性饱和问题，揭示了其饱和族的结构。
The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset $P$, the Boolean dimension of $P$ and the local dimension of $P$ are both bounded from above by the dimension of $P$ and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height…
We introduce a new combinatorial condition on a subinterval of a poset P (a clamped subinterval) that allows us to relate the Auslander-Reiten quiver of the bounded derived category of P to that of the subinterval. Applications include the determination of when a poset is fractionally Calabi-Yau and the computation of the Auslander-Reiten quivers of both the bounded derived category and of the module category.
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。
Variance vs. range for linear extensions, and balancing extensions in posets of bounded width
本研究证明Kahn-Saks关于大宽度偏序集存在平衡元素对的猜想，可推广至仅存在大量与某元素不可比元素的情况，还借此简化证明了Young图排序概率相关结果及其高维推广。
Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width $2$ has queue-number at most $2$, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width $w$ have queue-number at most $3w-2$ while any planar poset with $0$ and $1$ has queue-number at m…
Rowmotion on the chain of V's poset and whirling dynamics
该研究建立了$k$-有界$P$-划分与$P\times [k]$序理想间的等变双射，将旋转作用对应到rowmotion算子，推导了V型偏序集链上rowmotion的周期性与同均性结果，并部分推广到爪形偏序集的情形。
Saturation for Sums of Posets and Antichains
本文研究了偏序集和反链的线性和的饱和数，证明其至少为线性，扩展了钻石偏序集的结论。
We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result …
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension …
We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.
It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et …
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomasse and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation…
Cliquewidth and dimension
本文证明有界团宽且维数足够大的偏序集必含标准例子或Kelly例子，并据此完整刻画了覆盖图在子图封闭图类中时偏序集维数有界的条件，同时证明有界团宽蕴含布尔维数有界。
Generalized snake posets, order polytopes, and lattice-point enumeration
本文研究了广义蛇偏序集的顺序聚类的Ehrhart理论，推导了其$h^*$-多项式的显式公式和递推关系，并证明了其$h^*$-向量的界性。
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
In this article we introduce the $m$-cover poset of an arbitrary bounded poset $\mathcal{P}$, which is a certain subposet of the $m$-fold direct product of $\mathcal{P}$ with itself. Its ground set consists of multichains of $\mathcal{P}$ that contain at most three different elements, one of which has to be the least element of $\mathcal{P}$, and the other two elements have to form a cover relation in $\mathcal{P}$. We study the $m$-cover poset from a structural and topological point of view. In particular, we characterize the posets whose $m$-cover poset is a lattice for all $m>0$, and we cha…
We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.
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…
Extending the ab-index
该研究针对有限分次有界偏序集，证明Poincaré扩展的ab-index可通过Ω变换从ab-index得到，验证了相关猜想，为R-标记偏序集外的ab-index和Chow多项式提供了更具概念性的研究路径。
Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets
本文提出在有限维偏序的不可比图中，通过flashlight搜索方法实现输出多项式时间算法，并改进翻转方法以减少空间需求，从而提升极小支配集的枚举效率。
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x<y$?". This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a v…
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D(X) and D(Y) are equivalent as triangulated categories. We give explicit combinatorial properties of a poset which are invariant under derived equivalence, among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. Then we construct, for any closed subset Y of X, a strongly exceptional collection in D(X)…
Boolean dimension and dim-boundedness: Planar cover graph with a zero
本文针对带唯一极小元的平面覆盖图偏序集，证明其布尔维数至多为10，进而推导此类平面有向图存在O(log n)位的可达性标记方案，同时补充了相关维数理论的研究结果。
It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1,2,...,n without repetition. These labellings are called S_n EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a natural extension of S_n EL-labellings, called interpolating labellings. We show that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements. Furthermor…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the $d$ linear extensions of a witnessing realizer. Focusing on the encoding aspect Nešetřil and Pudlák defined a more expressive version of dimension. A poset $P$ has boolean dimension at most $d$ if it is possible to decide whether $x \leq y$ in $P$ by looking at the relative position of $x$ and $y$ in only $d$ permutations of the elements of $P$. We prove th…
We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the botto…
We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Nešetřil and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with bounded book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as …
Rowmotion and Echelonmotion
本文定义偏序集上的 echelonmotion，证明其在半分配格上与 rowmotion 一致，并刻画 echelon-independent 偏序集及 trim、Eulerian 情形。
Stabilization of the Spread-Global Dimension
本文证明了固定有限偏序集与任意有限全序的笛卡尔积上spread-全局维数一致有界，从而肯定回答了[AENY23]中的猜想，并建立了网格偏序集有限展示表示的有限spread-预解式存在性。
Segre powers of posets preserve EL-shellability
本文研究了格 poset 的格雷厄姆幂保持 EL-shellable 性，并推导了其等级选择不变量的公式。
Fractionally Calabi-Yau lattices that tilt to higher Auslander algebras of type A
本文证明两个有序链乘积的序理想格具有分数Calabi-Yau性质，且与A型高Auslander代数导出等价，从而证实了Chapoton关于偏序集与Fukaya-Seidel范畴的猜想。
For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex geometry, then Reg(P,f) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P,f) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For …
For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in this poset. Our main result states that C(L,g) is a semidistributive lattice if L is semidistributive, and that C(L,g) is a bounded lattice if L is bounded. Let S_n be the permutohedron on n letters and T_n be the associahedron on n+1 letters. Explicit computations show that C(S_n,a) = S_{n-1} and C(T_n,a) = T_{n-1}, up to isomorphism, whenever a is an atom. …
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