Related papers: Lattice homomorphisms between weak orders

Let $(X,o)$ be a complex analytic normal surface singularity with rational homology sphere link $M$. The `topological' lattice cohomology ${\mathbb H}^*=\oplus_{q\geq 0} {\mathbb H}^q$ associated with $M$ and with any of its spin$^c$…

Suppose $G$ is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on $G$ which have topological structures. In this paper, our attempt is to assign lattice structures on them. More…

Skew lattices are non-commutative generalizations of lattices. The coset structure decomposition is an original approach to the study of these algebras describing the relation between its rectangular classes. In this paper we will look at…

This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler…

We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with $(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular, the non-trivial isomorphisms fall into a small number of cases which are highly…

We call a lattice crosscut-simplicial if the crosscut complex of every atomic interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and…

Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at…

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…

In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely…

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an "orthogonal" basis…

We consider the problem of covering $\mathbb{Z}^2$ with a finite number of sublattices of finite index, satisfying a simple minimality or non-degeneracy condition. We show how this problem may be viewed as a projective (or homogeneous)…

There exist as many index-$k$ sublattices of the hexagonal lattice up to isometry as there exist lattice triangles with normalized volume $k$ up to unimodular equivalence, which can be explained using orbifolds. In dimension 3, it was noted…

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps.…

We show that the congruence lattice of a semilattice satsifies a form of distributivity relative to principal congruences of the form $ \Theta_{t \odot s, s}$. Particularly, we establish that semilattice congruences obey the ``pairwise…

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$.…

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and…

It is shown that a surjective monotone map $X\to Y$ between finite $T_0$-spaces induces a surjective map on homology. As such a map turns out to be a sequence of edge contractions in the Hasse diagram of $X$, followed by a homeomorphism,…

In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. This means that if such an…

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take…

We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter…