Related papers: The canonical join complex
We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In…
The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient…
The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet $\Sigma$ = {x,y,z,...}, where each letter has a fixed…
Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} = {\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in \mathbb N$…
Let $\mathbf{F}=\left\langle F,R\right\rangle $ be a finite Kripke frame. A congruence of $\mathbf{F}$ is a bisimulation of $\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\mathbf{F}$ is a lattice…
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…
We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in…
We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices,…
To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of…
Brewer and Heinzer studied the (integral) domains D having the property that each proper ideal A of D has a comaximal ideal factorization with some additional property. They proved that for a domain D, the following are equivalent: (1) Each…
In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq…
Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set.…
Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works…
We study canonical filtrations of finite-dimensional associative algebras and Lie algebras. These filtrations are defined via optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT), and appear to be a…
We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between…
Let $\Lambda$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $\tau$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated…
In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…
In this paper we consider a general way of constructing profinite struc- tures based on a given framework - a countable family of objects and a countable family of recognisers (e.g. formulas). The main theorem states: A subset of a family…
We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…
We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice…