Maps on posets, and blockers

Collapsing along monotone poset maps

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $ϕ:P\to P$, the simplicial complex $Δ(P)$ {\tt NE}-reduces to $Δ(Q)$, for any $Q\supseteq{\text{\rm Fix}}ϕ$. As a corollary, we prove that for any order-preserving map $ϕ:P\to P$ satisfying $ϕ(x)\geq x$, for any $x\in P$, the simplicial complex $Δ(P)$ collapses to $Δ(ϕ(P))$. We also obtain a generalization of Crapo's closure theorem.

Maps between higher Bruhat orders and higher Stasheff-Tamari posets

We make explict a description in terms of convex geometry of the higher Bruhat orders B(n,d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S_1(n,d). We show that the map f sketched by Kapranov and Voevodsky from B(n,d) to S([0,n+1],d+1) coincides with the map constructed by Rambau, and is a surjection for d<=2. We construct a map analogous to f from S_1(n,d) to B(n-1,d), and show that it is always a poset embedding. We also give an explicit criterion to determine if an element of B(n-1,d) is in the image of this map.

An inverse Fraïssé limit for finite posets and duality for posets and lattices

We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fraïssé sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of lattices associated with it. Each of these two sequences has a limit object -- an inverse limit, which is an object of our interest as well. In the first chapter there are some preliminaries considering partial orders, lattices, topology, inverse limits, category theory and Fraïssé theory, which are used later. In the second chapter there are our results consider…

Poset Hopf Monoids

We initiate the study of a large class of species monoids and comonoids which come equipped with a poset structure that is compatible with the multiplication and comultiplication maps. We show that if a monoid and a comonoid are related through a Galois connection, then they are dual to each other. This duality is best understood by introducing a new basis constructed through Möbius inversion. We use this new basis to give uniform proofs for cofreeness and calculations of primitives for the Hopf monoids of set partitions, graphs, hypergraphs, and simplicial complexes. Further, we show that the…

双有理映射、PBW退化旗与偏序集多面体

Birational maps, PBW degenerate flags and poset polytopes

本研究将射影空间与格拉斯曼流形间双有理映射的图闭包结果推广至PBW退化旗簇，分析其组合、代数与几何性质，验证了环面退化的普适性，并通过箭图表示刻画旗簇图闭包。

Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $δ$-cliffs, a generalization of permutations and increasing trees depending on a range map $δ$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $δ$, have underlying sets enumerated by the Fuss-Catala…

On the Order Dimension of Outerplanar Maps

Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertex-edge-face poset $\Pvef{M}$ of a planar map $M$ is at most four. In this paper we investigate cases where $\dim(\Pvef{M}) \leq 3$ and also where $\dim(\Qvf{M}) \leq 3$; here $\Qvf{M}$ denotes the vertex-face poset of $M$. We show: - If $M$ contains a $K_4$-subdivision, then $\dim(\Pvef{M}) = \dim(\Qvf{M}) = 4$. - If $M$ or the dual $M^*$ contains a $K_{2,3}$-subdivision, then $\dim(\Pvef{M}) = 4$. Hence, a map $M$ with $\dim(\Pvef{M}) \leq 3$ must be outerplanar and h…

到 Grassmann 簇的双有理映射、表示与偏序集多胞体

Birational maps to Grassmannians, representations and poset polytopes

本文研究射影空间到 Grassmann 簇的双有理映射的图闭包，给出其显式描述、纤维计算及到退化李代数循环表示射影化的嵌入，并利用新偏序集多胞体推广 FFLV 基，得到平坦环状退化。

有限偏序集上的对偶与典范复形 II：正则性

Dualizing and canonical complexes on finite posets II: properness

本文在有限偏序集语境下发展了正则空间与正则映射的理论。

An Introduction to the Moebius Function

This is an introduction to the Möbius function of a poset. The chief novelty is in the exposition. We show how order-preserving maps from one poset to another can be used to relate their Möbius functions. We derive the basic results on the Möbius function, applying them in particular to geometric lattices.

What is $-Q$ for a poset $Q$?

In the context of combinatorial reciprocity, it is a natural question to ask what "$-Q$" is for a poset $Q$. In a previous work, the definition "$-Q:=Q\times\mathbb{R}$ with lexicographic order" was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley's reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of $Q$ is me…

关于高阶量子映射的结构

On the structure of higher order quantum maps

本文研究了在*自主范畴中 affine 子空间的高阶量子映射结构，通过布尔函数类型函数识别高阶映射类型，并利用 Möbius 变换将类型函数与 poset 相关联，探讨了类型函数与 comb 类型的关系及 poset 分解方法。

Differential posets and restriction in critical groups

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a different…

有限偏序集的迭代性质

Iterative properties of birational rowmotion

本文研究了有限偏序集的birational rowmotion映射，证明其在(p,q)-矩形偏序集上有有限阶，并探讨了其在三角形和skeletal类偏序集上的性质，揭示了其与Zamolodchikov周期猜想的类比关系。

Some properties of a new partial order on Dyck paths

We introduce and study a new partial order on Dyck paths. We prove that these posets are meet-semilattices. We show that their numbers of intervals are the same as the number of bicubic planar maps. We describe an unexpected connection with the Hochschild polytopes of Saneblidze.

贪婪 Tamari 偏序集中的区间

Intervals in the greedy Tamari posets

本文研究 m-Dyck 路径上贪婪 Tamari 序的区间计数，通过递归方法和“催化”参数得到简单公式，并证明该公式同时计数 (m+1)-星座平面地图，还恢复了普通 m-Tamari 格的已知区间计数结果。

ZZ Polynomials of Regular $m$-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets -- Part 1. Proof of Equivalence

In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\text{ZZ}(\boldsymbol{S},x)$ of a Kekuléan regular $m$-tier strip $\boldsymbol{S}$ of length $n$ and the extended strict order polynomial $\text{E}_{\mathcal{S}}^{\circ}(n,x+1)$ of a certain partially ordered set (poset) $\mathcal{S}$ associated with $\boldsymbol{S}$. The discovered equivalence is a consequence of the one-to-one correspondence between the set $\left\{ K\right\}$ of Kekulé structures of $\boldsymbol{S}$ and the set $\left\{ μ:\mathcal{S}\supset\mathcal{A}\rightarrow\lef…

有限Coxeter群弱序的格 congruence

Lattice congruences of the weak order

研究超平面排列区域偏序的格 congruence，重点在有限Coxeter群的弱序。通过几何描述，定义投影并给出弱序格 congruence的算法。

偏序集流的链替换

The chain replacement of a poset flow

研究偏序集流的链替换，通过考虑严格递增链偏序集的单纯神经按细化排序得到。它能将有限偏序集映射到q-余纤流等，利用其组合性质证明相关推出保持执行路径空间，引入特定模型结构后对q-余纤替换也有相同结论。

Barcode Posets: Combinatorial Properties and Connections

A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The combinatorial properties of barcodes have also been studied, mainly in the context of interval orders and interval graphs. In this paper, we define a new family of maps from the space of barcodes with $n$ bars to the permutation sets of various multisets, known as multipermutations. These multipermutations provide new combinatorial invariants on the space of …

The cylinder of a relation and generalized versions of the Nerve Theorem

We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps, and derive novel formulations of the classical Nerve Theorem for posets and simplicial complexes, suitable for covers with not necessarily contractible intersections.

Pop, Crackle, Snap（与 Pow）：碎片的一些面向

Pop, Crackle, Snap (and Pow): Some Facets of Shards

本文将超平面构型中的碎片解释为补集基本群的生成元，并构造纯碎片幺半群，进而通过Pop算子与嵌入Crackle、Snap建立碎片交序与纯碎片幺半群及正辫子幺半群之间的联系。

Equivariant closure operators and trisp closure maps

A trisp closure map is a special map on the vertices of a trisp T with the property that T collapses onto the subtrisp induced by the image of the map. We study the interaction between trisp closure maps and group operations on the trisp, and give conditions such that the quotient map is again a trisp closure map. Special attention is on the case that the trisp is the nerve of an acyclic category, and the relationship between trisp closure maps and closure operators on posets is studied.

Profunctors between posets and Alexander duality

We consider profunctors $f : P \promap Q$ between posets and introduce their {\em graph} and {\em ascent}. The profunctors $\Pro(P,Q)$ form themselves a poset, and we consider a partition $\cI \sqcup \cF$ of this into a down-set $\cI$ and up-set $\cF$, called a {\it cut}. To elements of $\cF$ we associate their graphs, and to elements of $\cI$ we associate their ascents. Our basic result is that this, suitable refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of $Q \times P$. Cuts in finite Booleans lattices correspond precisely to finite simp…

关于围栏和圆形围栏序多项式的贪婪记录与伯恩斯坦转移

Bernstein Transfers and Greedy Records for Fence and Circular-Fence Order Polynomials

研究关于围栏和圆形围栏序多项式，定义贪婪记录统计量，通过伯恩斯坦基证明相关等式，利用转移得到双射，构造循环记录，最终证明了卡哈内的圆形围栏猜想。

提升弱序在类型A和B中的三次实现

Lifting Cubic Realizations of Weak Orders in Types A and B

研究兼容投影映射的三次正则化结构，引入预Reeb图和增强预Reeb图以控制兼容的三次提升和兼容顺序嵌入的三次提升，应用于弱序类型A和B的删除塔。

嵌入在曲面上的图之间的同态

Homomorphisms between graphs embedded on surfaces

该研究将图同态扩展到胞腔嵌入图（地图），提出同时保持组合与拓扑结构的地图同态定义，构建相关纯组合框架，证明地图核偏序集的连通性等性质并给出典型构造示例。

非交换环上矩形的双有理行移动

Birational rowmotion on a rectangle over a noncommutative ring

本文证明非交换环上矩形偏序集双有理行移动的周期性，解决2014年猜想，并给出对径互反公式。

Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).

Maximal chain descent orders

This paper introduces a partial order on the maximal chains of any finite bounded poset $P$ which has a CL-labeling $λ$. We call this the maximal chain descent order induced by $λ$, denoted $P_λ(2)$. As a first example, letting $P$ be the Boolean lattice and $λ$ its standard EL-labeling gives $P_λ(2)$ isomorphic to the weak order of type A. We discuss in depth other seemingly well-structured examples: the max-min EL-labeling of the partition lattice gives maximal chain descent order isomorphic to a partial order on certain labeled trees, and particular cases of the linear extension EL-labeling…