Maurice Pouzet, Imed Zaguia
We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain.
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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…
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.
Planar posets can have arbitrarily large dimension. However, a planar poset of height $h$ has dimension at most $192h+96$, while a planar poset with $t$ minimal elements has dimension at most $2t+1$. In particular, a planar poset with a unique minimal element has dimension at most $3$. In this paper, we extend this result by showing that a planar poset has dimension at most $6$ if it has a plane diagram in which every minimal element is accessible from below.
Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension
作为Kunz多面体$P_m$组合学系列研究的第三篇，本文刻画数值半群极小表示的Kunz偏序集可恢复性，证明同面内部半群的极小表示基数一致，并给出由Kunz偏序集求面维度的组合方法。
Minimal Posets with Prescribed Maximal Chain Cardinalities
本文研究如何构造基数最小的偏序集，使其极大链基数多重集等于给定多重集S，并证明了|P|的紧上下界。
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.
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。
An unlabeled poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Let $p_n$ denote the number of (2+2)-free posets of size $n$. In a recent paper, Bousquet-Mélou et al.\cite{BCDK} found, using so called ascent sequences, the generating function for the number of (2+2)-free posets of size $n$: $P(t)=\sum_{n \geq 0} p_n t^n = \sum_{n\geq 0} \prod_{i=1}^{n} (1-(1-t)^i)$. We extend this result in two ways. First, we find the generating function for (2+2)-free posets when four statistics are taken into accou…
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。
We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.
Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun {\em et al.} We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the…
In this paper we study some of the basic properties of a graph which is constructed from the equivalence classes of non-zero zero-divisors determined by annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.
Minimal elements in the skew extended 0-Hecke poset
本文研究斜扩展0-Hecke偏序集的极小元素，对“龙虾”族斜形状完成分类，并证明连通斜形状下偏序集有唯一极小元素，进而表明相关0-Hecke模为循环模。
We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verb…
Harding showed that the direct product decompositions of many different types of structures, such as sets, groups, vector spaces, topological spaces, and relational structures, naturally form orthomodular posets. When applied to the direct product decompositions of a Hilbert space, this construction yields the familiar orthomodular lattice of closed subspaces of the Hilbert space. In this note we consider orthomodular posets Fact X of decompositions of a finite set X. We consider the structure of these orthomodular posets, such as their size, shape, and connectedness, states, and begin a study…
Let $P$ be a finite partially ordered set with unique minimal element $\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\in P$ for which the open interval $(\hat{0}, q)$ is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form $(\hat{0},p)\subset P$ and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its $\mathbb{Z}^{d}$-graded Betti numbers and the structure of its minimal free resolution. In the case of ri…
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.
Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions: (1) the size of $P$ is a power of $2$, (2) $P$ has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if $P$ only satisfies condition (2), we can still almost partition $2^{[n]}$ into copies of $P$. We prove that if $P$ has a unique maximal and minimal elem…
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is crucially important to phylogenetic applications. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Near…
Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties
本文研究点线构型及其极小拟阵与环路簇，提出一种在依赖序下识别极小拟阵的算法，用于计算环路簇的不可约分解，并应用于 Fano 等经典构型及探讨极小拟阵唯一性猜想。
We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also …
Optimal and minimal $p$-ary linear codes from generalized order ideals of hierarchical posets
本文引入分层偏序集的广义序理想，构造了最优和极小的 $p$ 元线性码，确定了补码的重量分布，并获得了违反 Ashikhmin--Barg 条件的无限码族。
Subdivisions of lower Eulerian posets
本文研究了下欧拉 poset 的强形式细分，并建立了与三元组的自然双射，同时应用于计算 $cd$-指数和 Kazhdan-Lusztig-Stanley 理论。
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。
In this paper, we are interested in the combinatorial analysis of the whole genome duplication - random loss model of genome rearrangement initiated in a paper of Chaudhuri, Chen, Mihaescu, and Rao in SODA 2006 and continued by Bouvel and Rossin in 2007. In this model, genomes composed of n genes are modeled by permutations of the set of integers [1..n], that can evolve through duplication-loss steps. It was previously shown that the class of permutations obtained in this model after a given number p of steps is a class of pattern-avoiding permutations of finite basis. The excluded patterns we…
Given a monomial ideal $I$ with minimal free resolution $\mathcal{F}$ supported in characteristic $p>0$ on a CW-complex $X$ with regular $2$-skeleton, we construct a CW-complex $Y$ that also supports~$\mathcal{F}$ and such that the face poset $P(Y)$ also supports $\mathcal{F}$ in the sense of Clark and Tchernev.
In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal ``ideal solutions'' are to be generated. This in particular allows us to study the complexity of the problem un…
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the …
Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs
本文刻画了有限偏序集可保维饱和嵌入局部诺特UFD谱的充分必要条件，并进一步构造了任意有限偏序集到半局部拟优环谱的饱和嵌入，使其极小元对应极小素理想且余高与最长链长度一致。
In finite group theory, studying the prime graph of a group has been an important topic for almost the past half-century. Recently, prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs without any knowledge of the group theoretical background. In this paper we study prime graphs from a linear algebra angle and focus on the class of minimally connected prime graphs introduced in earlier work on the subject. As our main results, we determine the determinants of the adjacency matrices and the spectra of some important f…
Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ is the ground set and $P$ is a reflexive, anti-symmetric and…
Degree 2 vertices in minimal prime graph complements
本文研究极小素图补图中二度顶点的结构，证明其结构为有限重复的5-cycle，属于reseminant图类。
Forcing the Wheel
本文通过证明具有平面覆盖图的偏序集维数满足 dim(P) ≤ 2 wheel(P) + 2，肯定解决了 Joret 和 Micek 关于用极小元素个数与树宽界定维数的猜想。
5-cycles in the complement of minimal prime graphs
本文研究了最小素图补图的循环结构，证明了补图中每条边都包含在5环中，强化了关于诱导5环的存在性结论。
Posets of Geometric Graphs
本文用保持邻接与交叉的几何同态定义抽象图几何实现之间的偏序，给出可比较性判定工具，并刻画 P_n、C_n、K_n 的同构类、Hasse 图及一般结构性质。
On minimal collections of sequences for testing continuity
研究了测试集：收敛到点P的序列子家族，足以检测实值函数在P处的所有不连续性。证明了在自然假设下，这些测试集构成的偏序集有最小元素，并分析了其最大链。
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y \in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= \frac{p-1}{2}$, then $S=g^{\frac{p-11}{2}}(\frac{p+3}{2}g)^4(\frac{p-1}{2}g)$ or $g^{\frac{p-7}{2}}(\frac{p+5}{2}g)^2(\frac{…
Antichains among Divisor Sums of Divisors
本文研究正整数除数在映射到其因子和函数后，原除数格中反链宽度的保留情况，通过整除偏序集乘积宽度刻画最大值，并利用Engel极限理论及素数方向2和3确定精确方差，给出极限公式与渐近结果。
Connectivity of $p$-subgroup posets with irreducible characters
本文研究了有限群中p-子群与不可约特征标之间的连通性，证明了特定条件下偏序集的连通分量结构。
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete description of such classes. In fact, each one of these classes is a well-quasi-ordered (w.q.o) age and there are uncountably many of them. Eleven of these ages are almost multichainable; they remain w.q.o when labels in a w.q.o are added, he…
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary f…
We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions --- the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symme…
In this paper we discuss minimal primes over permanental ideals of generic matrices. We give a complete list of the minimal primes over ideals of 3 x 3 permanents of a generic matrix, and show that there are monomials in the ideal of maximal permanents of a d x 2d-1 matrix if the characteristic of the ground field is sufficiently large. We also discuss the Alon-Jaeger-Tarsi Conjecture, using our results and techniques to strenthen the previously known results.
In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We characterize regular reseminant graphs and study t…
We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to ten crossings.
Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincaré duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology class in a simplicial poset the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation. As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$ Higher dimensional analogs of discrete normal surfaces …
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。
Minimal primes and radicality of ideals generated by adjacent 2-minors
本文研究了由相邻2-余子式生成的理想极小素理想及其根性，证明了若干代数性质等价，并给出了相关组合特征。
In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.
Given a density t in (0,1], and a prime p, let S be any subset of F_p having at least tp elements, and having the least number of three-term arithmetic progressions mod p among all subsets of F_p with at least tp elements. Define N(t,p) to be 1/p^2 times the number of three-term arithmetic progressions in S modulo p. Note that N(t,p) does not depend on S -- it only depends on t and p. An old result of Varnavides shows that for fixed t, N(t,p) > c(t) > 0 for all primes p sufficiently large. But, does N(t,p) converge to a limit as p -> infinity? We prove that it does.
Sizes of witnesses in Covtree
本文研究了在Covtree中见证体的大小上限，通过引入下集交换图工具，证明了最小见证体的大小上限为nk-n，并在k=3时给出具体结果。
Let $G$ be a graph on $[n]$ and $J_G$ be the binomial edge ideal of $G$ in the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of $J_G$. We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs $G$ for which $\mathrm{depth}\hspace{1.2mm} S/J_G=4$.
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