We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let η_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of η_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.
We investigate the alternate order on a congruence-uniform lattice $\mathcal{L}$ as introduced by N. Reading, which we dub the core label order of $\mathcal{L}$. When $\mathcal{L}$ can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general $\mathcal{L}$, however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the c…
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustive…
Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra
LISN (UMR CNRS 9015), Université Paris Saclay, INRIA, CNRS（巴黎萨克雷大学）；Universitat de Barcelona & Centre de Recerca Matemàtica（巴塞罗那大学）；LIGM, Université Gustave Eiffel（巴黎东马恩拉瓦莱大学）
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 $2^{n-5}$ congruences.
Separating trees and simple congruences of the weak order
DePaul University（德保罗大学）；LIGM, Université Gustave-Eiffel, CNRS, ENPC, ESIEE-Paris（高等信息与数学实验室，古斯塔夫·埃菲尔大学，法国国家科学研究中心，国立路桥学校，巴黎ESIEE学院）；Universitat de Barcelona & Centre de Recerca Matemàtica（巴塞罗那大学与数学研究中心）
We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.
We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$ both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. We call a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle provided that the set of its join-irreducible elements join-generates $\boldsymbol{S}$ and it satisfies DCC. We prove that the congruence lattice of a parti…
We study the congruence lattices of the multinomial lattices L(v) introduced by Bennett and Birkhoff. Our main motivation is to investigate Parikh equivalence relations that model concurrent computation. We accomplish this goal by providing an explicit description of the join dependency relation between two join irreducible elements and of its reflexive transitive closure. The explicit description emphasizes several properties and makes it possible to separate the equational theories of multinomial lattices by their dimensions. In their covering of non modular varieties Jipsen and Rose define …
We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.
A partial algebra construction of Grätzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered …
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Grünbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide addition…
The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. We consider a larger class of posets, the Grid-Tamari orders, which arise as an ordering on the facets of the non-kissing complex introduced by Pylyavskyy, Petersen, and Speyer. In addition to Tamari orders, some interesting examples of Grid-Tamari orders include the Type A Cambrian lattices and Grassmann-Tamari orders. We prove that the Grid-Tamari orders ar…
We give a combinatorial proof that congruence permutability is prime in the lattice of interpretability types of varieties. Thereby, we settle a 1984 conjecture of Garcia and Taylor.
We consider the closure space on the set of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows. Finite congruence-uniform …
In 1978, motivated by E. Hückel's work in quantum chemistry, I. Gutman introduced the concept of the energy of a finite simple graph $G$ as the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. At the time of writing, the MathSciNet search for "Title=(graph energy) AND Review Text=(eigenvalue)" returns 351 publications, most of which going after Gutman's definition. A congruence $α$ of a finite algebra $A$ turns $A$ into a simple graph: we connect $x\neq y\in A$ by an edge iff $(x,y)\inα$; we let En$(α)$ be the energy of this graph. We introduce the congruence energ…
For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.
