Related papers: Lattice structure of Grid-Tamari orders

We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for…

Motivated by Gr\"obner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant…

This paper has been withdrawn by the authors due to a crucial computational error. In this paper we deal with the finite case. We prove that a finite bounded ordered set can be represented as the order of principal congruences of a finite…

This paper presents a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal…

Let \Gamma be a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of \Gamma in X…

We construct a distributive algebraic lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth, from the forties. The lattice D has…

This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a…

Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and…

The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$, introduced by Bergeron and Pr{\'e}ville-Ratelle, are defined as a poset of $m$-Dyck paths equipped with the generalized rotation order, and constitute a Fuss-Catalan generalization of the…

Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in $G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of all lattice…

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called…

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly…

A magma (or groupoid) is a set with a binary operation $(A,f)$. Roughly speaking, a magma is said to be lazy if compositions such as $f(x,f(f(y,z),u))$ depend on at most two variables. Recently, Kaprinai, Machida and Waldhauser described…

We discuss a proposal for the construction of lattice QCD with gauge action, fermionic action, theta-term, and the operators all based on the lattice Dirac operator D with exact chiral symmetry. The simplest regularization of this type uses…

Quasi-lattices are introduced in terms of 'join' and 'meet' operations. It is observed that quasi-lattices become lattices when these operations are associative and when these operations satisfy 'modularity' conditions. A fundamental…

The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a…

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some…

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle…

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order}…