Stephan Foldes, Russ Woodroofe
We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our characterization replaces the condition of gradedness with a second modularity condition on the maximal chain of left-modular elements.
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We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [McNamara-Thomas] by combining results of McNamara and Liu. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.
Modular flats of oriented matroids and poset quasi-fibrations
本文研究有向拟阵模平坦的组合学，证明到余秩一模平坦的局部化Salvetti复形的自然映射是偏序集拟纤维化，从而将超可解超平面排列的K(π,1)结果推广到超可解有向拟阵，并构造了许多不可实现的超可解有向拟阵。
The purpose of this article is to investigate the combinatorial properties of the cross section lattice of a $J$-irreducible monoid associated with a semisimple algebraic group of one of the types $A_n$, $B_n$, or $C_n$. Our main tool is a theorem of Putcha and Renner which identifies the cross section lattice in the Boolean lattice of subsets of the nodes of a Dynkin diagram. We determine the join irreducibles of the cross section lattice. Exploiting this we find characterizations of the relatively complemented intervals. By a result of Putcha, this determines the Möbius function for $Λ$. We …
Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that every supersolvable arrangement is inductively free, a notion due to Terao. So this is a natural subclass of this particular class of free arrangements. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = (A(W), V) be the associated hyperplane arrangement of W. In a recent paper, we determined all inductively free r…
Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable. For X in L(A), it is known that the restriction A^X is supersolvable provided A is. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A(W) = (A(W), V) be its associated hyperplane arrangement. In earlier work by the last two authors, we classified all supersolvable reflection arrangements. Extending this work, the aim of this note is to determine all supersolvable restr…
Combinatorial generation via permutation languages. VII. Supersolvable hyperplane arrangements
本文研究超可解超平面构形的区域图及其区域偏序商格的组合生成，证明区域图存在 Hamilton 回路，且规范基区域下任意格同余商的覆盖图存在 Hamilton 路径。
Supersolvability of built lattices and Koszulness of generalized Chow rings
本研究利用几何格的操作子结构得到超可解构造格广义Chow环的显式二次格罗布纳基，证明了对应极小构造集的广义Chow环具有Koszul性，还得到亏格0的扩展模操作子分量上同调代数的Koszul性。
Let $(W,S)$ be an arbitrary Coxeter system. For each word $ω$ in the generators we define a partial order--called the {\sf $ω$-sorting order}--on the set of group elements $W_ω\subseteq W$ that occur as subwords of $ω$. We show that the $ω$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the $ω$-sorting order is a "maximal lattice" in the sense that the addition of any collection of Bruhat covers results in a nonlattice. Along the way we define a class of structures called {\sf supersolvable antimatro…
In this paper we show that the set of closure relations on a finite poset P forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison). We also characterize the modular elements of this lattice and compute its characteristic polynomial.
We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To start with, t…
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…
In this paper, we present a simple lattice-theoretic characterization for affine buildings of type A. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and affine buildings of type A constitute the same object. This is an affine counterpart of the well-known equivalence between projective geometries ($\simeq$ complemented modular lattices) and spherical buildings of type A.
Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that $2^{n-3}$ is a lower bound for the number of nonisomorphic modular lattices of size $n$.
An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some optimizations that handle simple cases directly. For modular and semimodular lattices, the algorithm prunes the search tree much earlier than the method of Jipsen and Lawless, leading to a speedup of several orders of magnitude. With this new algorithm modular lattices are counted up to 30 elements, semimodular lattices up to 25 elements, graded lattices up to 21 ele…
Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective geometries describe complemented modular lattices. This e.g. yields the result that each modular lattice L of finite length d(L), and having s(L) many maximal congruences, has at least 2d(L)-s(L) many join-irreducible elements. Furthermore, an algorithm is proposed that calculates, in a compressed way, the (full) submodule lattice Sub(W) of any (sufficiently…
Left-modularity is a concept that generalizes modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial of a lattice with such an element, one of which generalizes Stanley's Partial Factorization Theorem for a geometric lattice with a modular element. Both formulae provide us with inductive proofs of Blass and Sagan's Total Factorization Theorem for LL lattices. The characteristic polynomials and Mobius functions of non-crossing partition lattices and shuffle posets are computed as examples.
Consider the noncrossing set partitions of an $n$-element set which either do not contain the block $\{n-1,n\}$, or which do not contain the singleton block $\{n\}$ whenever $1$ and $n-1$ are in the same block. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element. This work is motivated by a rec…
We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative inte…
Representing lattices L by equivalence relations amounts to embed them into the lattice Part(V) of all partitions of a set V, and has a long history. Here we are concerned with MODULAR lattices L and aim for sets V as small as possible, i.e. |V| = d(L)+1 where d(L) is the length of L. In other words, we strive for a tight (=cover-preserving) lattice homomorphism from L into Part(V). After a 24 year break the author offers progress, and outlines a program to finally fully characterize the lattices L that admit a tight embedding. Not just 'modular latticians' but also combinatorists are encourag…
On the structure of modular lattices -- Unique gluing and dissection
本文研究模的局部有限格的结构，证明在有限覆盖条件下有限格的粘合具有唯一性，且粘合与剖分互为逆过程。
Extremality, Left-Modularity and Semidistributivity
本文研究并半分配格、并极值格与左模格之间的关系，证明了每个并半分配且左模的格都是并极值的，并联系了字典序可壳性。
ST-Distributive and ST-Modular Lattices
本文定义格上相对于子集 S、T 的 ST-分配性与 ST-模性，研究其基本性质、极大子集对刻画，并给出在凸集和多面体中的应用。
A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation theory, particularly in the study of poset models for semisimple Lie algebra representations and their companion Weyl group symmetric functions. One of our goals is to gather in one place some elementary but foundational results concerning these lattice structures; this includes some new results as well as some new interpretations of classical results. We t…
Move-minimizing puzzles and diamond-colored modular/distributive lattices
本文研究一类单人组合博弈——最小化移动谜题，证明当其可编码为钻石着色的模格或分配格时存在显式解，并基于序论组合学的新解释建立理论基础。
It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in {1,2,3,5,6,7,11,14,15,23} ... (*), and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N = 1 and 2). For N > 1 in (*), lattices meeting the new bound are constructed that are analogous to the ``shorter'' and ``o…
On the structure of modular lattices -- Axioms for gluing
本文针对模格、局部有限且具有有限覆盖但可能无限高度的格，提出极大和极小两组粘合公理，其中极小公理仅涉及局部区间内的骨架元素关系，却能保证所得模格的全局结构。
A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations.This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barthélemy and Constantin for establishing Birkhoff-type theore…
Modular lattices of finite length (Part A)
本文是有限长度模格四部分研究的A部分，整理作者1992年笔记，涵盖同余格分配性的简短证明与点分裂概念，并预告后续将首次英文介绍Herrmann和Wille的深刻结果。
Modular lattices and algebras with straightening laws
研究证明并非所有模棱格都是积分的，通过直化定律的代数结构分析得出结论。
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.
A survey of lattice properties: modular, Arguesian, linear, and distributive
本文针对格的模性、Arguesian性等多种性质展开综述，梳理其特征与关联，侧重有限及有穷格，弱化有补格相关内容。
Simplifying modular lattices by removing doubly irreducible elements
本文通过移除双既约元得到更小的rack以简化模格，列出n≤40的竖直不可分解模rack、用波利亚计数得n≤40模格数量并提供SageMath代码，实现超3000倍存储节省。
We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi-Perron type algorithms.
It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.
Some years ago, Borcherds described in [Bo1] two methods for constructing modular forms on modular varieties related to the orthogonal group $Ø(2,n)$. They are the so called Borcherds' additive and multiplicative lifting. The multiplicative lifting has been used by Borcherds himself and other authors to construct modular forms with known vanishing locus and interesting properties. The additive lifting has been used to construct explicit maps from some modular varieties $Ø(2,n)$. In this paper we try a more systematic treatment in certain level 2 cases giving connections with Kondo's works
A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction of linear independence. As an important property of a matroid, it can be characterized by several different (but equivalent) axioms, such as the augmentation, the base exchange, or the rank axiom. A supermatroid is a generalization of a matroid defined on lattices. Here, the central question is whether a supermatroid can be characterized by several equivalent axioms similar to a matroid. Barnabei, Nicoletti, and Pezzoli characterized supermatroids on distributive lattices,…
A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovic 1998) that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem for a semimodular lattice. We consider the notion of a modular convex subset, which is a subset closed under the join and meet only for modular pairs, and show that the modular convex hull of t…
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…
On the $E$-base of Finite Lattices: Semidistributive, Modular, and Geometric Lattices
本文研究有限闭包格中 $E$-基的有效性，证明其在半分配格上有效且最小，刻画具有有效 $E$-基的模格与几何格，并证明任意格可嵌入此类格。
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