Serkan Hosten, Diane Maclagan
We introduce a monomial ideal whose standard monomials encode the vertices of all fibers of a lattice. We study the minimal generators, the radical, the associated primes and the primary decomposition of this ideal, as well as its relation to initial ideals of lattice ideals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.
We give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.
Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G \cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G \cup W(S))$ have linear quotients. Moreo…
Three invariants of geometrically vertex decomposable ideals
本文研究几何顶点可分解理想的三个不变量，证明它们可通过几何顶点分解过程递归计算，并应用于证明$a$-不变量非正及二部图toric理想的相关结果。
We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.
Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees algebra and the Simis cone of the edge ideal of G.
Bipartite Determinantal Ideals and concurrent vertex maps
本文引入并发顶点映射模型，描述 bipartite determinantal ideals 初始理想的 Stanley-Reisner 复形，并研究其顶点可分解性、壳化顺序、Hilbert 级数和 $h$-多项式等性质与应用。
On The Toric Ideals of the Coloured Graphs of Reduced Words
本文研究由阶梯分拆索引的伪多部图，给出其一般色多项式，并利用分拆恒等式构造与顶点正常着色相关的仿射环面理想，证明其射影版本可由卡通图实现。
In this work we study how some elementary graph operations (like the disjoint union) and the collapse of two vertices modify the cut ideal of a graph. They pave the way for reducing the cut ideal of every graph to the cut ideal of smaller ones. To deal with the collapse operation we generalize the definition of cut ideal given in literature, introducing the concepts of edge labeling and edge multiplicity: in fact we state the \emph{non-classical behavior} of the cut ideal. Moreover we show the transformation of the toric map hidden behind these operations.
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with some other results
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equ…
An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and different. Here, we provide an overview on codes for the $n$-dimensional lattice, discussing the case of 1-identifying codes, constructing a sparse code for the 4-dimensional lattice as well as showing that for fixed $n$, the minimum density of an $r$-identifying code is $Θ(1/r^{n-1})$.
Vertex cover ideals of simplicial complexes
本文构造新单纯复形并给出超图顶点可分解性检验程序，证明所得复形的关联超图可顶点分解且覆盖理想的所有符号幂分量线性，推广了已有图论结果。
Motivated by the problem of bounding the number of rays of plane tropical curves we study the following question: Given $n\in\mathbb{N}$ and a unimodular $2$-simplex $Δ$ what is the maximal number of vertices a lattice polytope contained in $n\cdot Δ$ can have? We determine this number for an infinite subset of $\mathbb{N}$ by providing a family of vertex-maximal polytopes and give bounds for the other cases.
In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of 3-disjoint edges in $G$. Moreover for this class of graphs we characterize $\T{pd}(R/I(G))$ and $\T{depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.
In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph $G$, $\reg(R/I(G))\leq \min\{a'(G),n(G)\}$ and for a shellable graph $G$, $\reg(R/I(G))\leq n(G)$. Moreover it is shown that for a graph $G$, where $G^c$ is a $d$-tree, we have $\pd(R/I(G))=\max_{v\in V(G)} \{°_G(v)\}$.
Powers of vertex cover ideals of Simplicial Trees
针对Hà与Tuyl关于单纯树覆盖理想所有幂是否分量线性的问题，本文证明其答案为肯定，推广了弦图覆盖理想的相关猜想。
Geometric vertex decomposition and liaison
本文建立了几何顶点分解与联络两种框架间的显式联系，证明几何顶点可分解理想通过高度为1的初等G-联络与不定元理想相连，并由此推出多个已知理想族为glicci。
Ideally Connected Cographs and Chordal Graphs
该研究刻画了余图和弦图两类图中的理想连通图，证明理想连通余图为无$2K_2$余图，理想连通弦图为阈值图，完善了特殊图类的连通性理论。
We introduce new methods to describe admissible states of the six-vertex and the eight-vertex lattice models of statistical mechanics. For the six-vertex model, we view the admissible states as differential forms on a grid graph. This yields a new proof of the correspondence between admissible states and 3-colorings of a rectangular grid. For the eight-vertex model, we interpret the set of admissible states as an $\mathbb{F}_2$-vector space. This viewpoint lets us enumerate the set of admissible states. Finally, we find necessary conditions for a Yang-Baxter equation to hold for the general ei…
A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to investigate prime ideals, minimal prime ideals and special ideals of a decomposable lattice. These are keys to understand the algebraic structure of decomposable lattices.
In this article, we completely determine the isomorphism classes of lattice vertex operator algebras and the vertex operator subalgebras fixed by a lift of the -1-isometry of the lattice. We also provide similar results for certain even lattices associated with doubly-even binary codes.
This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of $\G(R)$ is a union of triangles and rectangles, while a vertex in $\G(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\G(R)$ are identical with the number of maximal ideals of $R$. A graph $\G_r(R)$ is also introduced on the vertex set $\{Rx|\,x\in R\setminus (U(R)\cup J(R…
On basic double G-links of squarefree monomial ideals
本文研究无平方单项理想的基本双重G-链，给出边理想作为基本双重G-链时被链理想的生成元集并证明其次数为1，同时证明文献中Cohen-Macaulay但非弱顶点可分解的复形不能作为其他无平方单项理想的基本双重链。
Vertex decomposability and weakly polymatroidal ideals
本文研究单纯复形的顶点可分解性，证明拟阵理想情形下序列 Cohen-Macaulay、可剥壳与顶点可分解等价，并刻画二次单项式理想的弱拟阵性、线性商与顶点可分裂性。
The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.
In this paper, we study certain Virasoro frames for lattice vertex operator algebras and their $\Z_2$-orbifolds using linear codes over $\Z_4$. We also compute the corresponding frame stabilizer from the view point of binary codes and $\Z_4$-codes. As an application, we determine the frame stabilizers of several Virasoro frames of the vertex operator algebra $V_{E_8}$ and the moonshine vertex operator algebra $V^\natural$.
In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.
The free resolution and the Alexander dual of squarefree monomial ideals associated with certain subsets of distributive lattices are studied.
It is proved that all vertex cover algebras of a hypergraph are standard graded if and only if the hypergraph is unimodular. This has interesting consequences on the symbolic powers of monomial ideals.
Finite distributive lattices whose join-meet ideals are of König type will be classified. Furthermore, a class of polyominoes whose polyomino ideals are of König type will be studied.
Depth of Binomial Edge Ideals in terms of Diameter and Vertex Connectivity
本文基于图的直径和顶点连通度等不变量，对满足深度上下界关系$L(G)+1=U(G)$的简单连通非完全图进行了结构分类，并计算了此类图二项边理想$S/J_G$的深度。
Radical binomial ideals associated with finite lattices are studied. Gröbner basis theory turns out to be an efficient tool in this investigation.
Properties of LCM Lattices of Monomial Ideals
研究单项式理想LCM格的格性质，包括布尔格、模格等，并证明极小单项式理想在模格下的Cohen-Macaulay性质，以及项目维数与LCM格高度的关系。
A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is a graph with the vertex set $I(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, it is shown that $G(\mathbb{Z}_n)$, for every positive integer $n$, is a weakly perfect graph. Also, for some values of $n$, we give an explicit formula for the vertex chromatic number of $G(\mathbb{Z}_n)$. Furthermore, it i…
There are many analogies between codes, lattices, and vertex operator algebras. For example, extremal objects are good examples of combinatorial, spherical, and conformal designs. In this study, we investigated these objects from the aspect of design theory.
Let $\mathscr{L}\subset \mathbb{Z}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $\mathscr{L}$. We produce infinitely many examples, in every codimension, of pairs $I,J$ where one of these ideals is Cohen--Macaulay but the other is not.
For a simple graph $G$, assume that $J(G)$ is the vertex cover ideal of $G$ and $J(G)^{(s)}$ is the $s$-th symbolic power of $J(G)$. We prove that $(J(C)^{(s)})=(J(C)^s)$ for all $s\geq 1$ and for all odd cycle $C$. For a simplicial complex $Δ$, we show that $Δ$ is vertex decomposable if $I_Δ^{\vee}$ is weakly polymatroidal. Let $W=G^π$ be a fully clique-whiskering graph, we prove that $J(W)^s$ is weakly polymatroidal for all $s\geq 1$.
We study join-meet ideals associated with modular non-distributive lattices. We give a lower bound for the regularity and show that they are not linearly related.
Algebraic Invariants of Edge Ideals Under Suspension
本文研究了边理想在悬垂操作下的代数不变量变化，发现正则性和项目维数的变化规律，以及独立多项式和a-不变量的特性。
We compute the reduced Gröbner basis of the toric ideal with respect to a suitable monomial order and we study the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is an unmixed bipartite graph without isolated vertices.
Fröberg's Theorem, vertex splittability and higher independence complexes
该研究将Fröberg定理推广至高次无平方单项式理想，通过r-独立构造r-均匀超图，证明共弦图对应边理想具顶点可分裂性，计算了分次Betti数，并给出线性消解存在性的拓扑证明。
We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gröbner basis. We then obtain an analogous result for lattice path matroids.
Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras
本文研究了无根偶格顶点算子代数在奇素数阶无不动点等距提升下的循环轨道折叠自同构群，证明了额外自同构存在的充要条件是该格由 Construction B 构造或同构于 Leech 格的余不变格。
We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertex-facet incidences in time O(min{n,m}*a*f), where n is the number of vertices, m is the number of facets, a is the number of vertex-facet incidences, and f is the total number of faces of P. This improves results of Fukuda and Rosta (1994), who described an algorithm for enumerating all faces of a d-polytope in O(min{n,m}*d*f^2) steps. For simple or simplicial d-polytopes our algorithm can be specialized to run in time O(d*a*f). Furthermore, applications of the algorithm to other ato…
We show that real tight frames that generate lattices must be rational, and use this observation to describe a construction of lattices from vertex transitive graphs. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. This is the case for the more restrictive class of distance transitive graphs. We show that such lattices exist in arbitrarily large dimensions and demonstrate examples arising from some notable families of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。