Related papers: Lattice properties of the sharp partial order
In ab-initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. "Ab-initio" means a process performed without any good…
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…
For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let $L_0$ be a lattice with largest element $u_0$, $L_1$ be a lattice disjoint from $L_0$ with smallest element $v_1$, and $a \in…
The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a…
We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible…
In recent years, persistence modules have been viewed as graded modules with gradation over a preordered set serving as the indexing set. We provide sufficient criteria for a projective module over a PID to be free when the indexing set is…
Ultracold atoms in optical lattices provide clean, tunable, and well-isolated realizations of paradigmatic quantum lattice models. With the recent advent of quantum-gas microscopes, they now also offer the possibility to measure the…
This paper analyzes the representation theoretic stability, in the sense of Thomas Church and Benson Farb, of the rank-selected homology of the Boolean lattice and the partition lattice, proving sharp uniform representation stability bounds…
The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity ({\bf LR}) is a form of ideal regularity of aperiodic patterns. Recently,…
In this paper we investigate multivariate integration in weighted unanchored Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we employ higher order polynomial lattice point sets over $\mathbb{F}_{2}$ which are…
This paper has been withdrawn by the authors due to a crucial computational error.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…
We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a…
Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a…
Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an…
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 study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and…
We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we…
The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple…
This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice…
We exhibit algorithms for calculating Tits' buildings and orbits of vectors in a lattice $L$ for certain subgroups of $\operatorname{O}(L)$. We discuss how these algorithms can be applied to understand the configuration of boundary…