Related papers: The Shard Intersection Order on Permutations
We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems…
In this article we give a simple, almost uniform proof that the lattice of noncrossing partitions associated with a well-generated complex reflection group is lexicographically shellable. So far a uniform proof is available only for Coxeter…
The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of Sets where the Green's relation D is a congruence describing an adjunction to the category of Lattices. In this paper we will…
In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the…
Lehmer's code defines a bijection between the symmetric group and the set of staircase compositions. In this paper, we characterize a poset structure on these compositions that is equivalent to the strong Bruhat order on the symmetric…
Noncrossing arc diagrams are combinatorial models for permutations that encode information about lattice congruences of the weak order and about the associated discrete geometry. In this paper, we consider two related, analogous models for…
We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of…
This thesis finds its place in the interplay between algebraic and geometric combinatorics. We focus on studying two different families of lattices in relation to the weak order: the permutree lattices and the $s$-weak order. The first part…
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…
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…
We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, $\Delta$, between almost degenerate ground states, is simply related to a novel order parameter $\tilde W$, which is…
We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in…
The paper presents geometric models for the set WO of weak orders on a finite set. In particulary, WO is modeled as a set of vertices of a cubical subdivision of a permutahedron. This approach is an alternative to the usual representation…
A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated…
A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and…
A linear-interval order is the intersection of a linear order and an interval order. For this class of orders, several structural results have been known. This paper introduces a new subclass of linear-interval orders. We call a partial…
Let $(W,S)$ be a Coxeter system of type $A$, so that $W$ can be identified with the symmetric group $\mathrm{Sym}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions $\{\,(i,i+1)\mid 1\leqslant i\leqslant…
This paper pursues an investigation on groups equipped with an $L$-ordered relation, where $L$ is a fixed complete complete Heyting algebra. First, by the concept of join and meet on an $L$-ordered set, the notion of an $L$-lattice is…
A structure of a complete lattice (in the sense of a poset) is defined on the underlying set of the orhtogonal group of a real Euclidean space, by a construction analogous to that of the weak order of a Coxeter system in terms of its root…
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.