Related papers: Counting and Computing Join-Endomorphisms in Latti…
By a 1997 result of R. Freese, an $n$-element lattice has at most $2^{n-1}$ congruences. This motivates us to define the congruence density cd$(L)$ of a finite $n$-element lattice as $|$Con$(L)|/2^{n-1}$, where $|$Con$(L)|$ is the number of…
We present a systematic study of join-extensions and join-completions of ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from…
In this work, we study the numerical optimization of nearest-neighbor concurrence of bipartite one and two dimensional lattices, as well as non bipartite two dimensional lattices. These systems are described in the framework of a…
Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and…
This document is an extended abstract for two articles in preparation. Recently, framing lattices were introduced to generalize many classical lattices such as the Tamari lattice and the weak order on the symmetric group. We define bricks…
We consider a simple model of higher order, functional computation over the booleans. Then, we enrich the model in order to encompass non-termination and unrecoverable errors, taken separately or jointly. We show that the models so defined…
Let I be a dense linear order with a left endpoint but no right endpoint. We consider the lattice L(I) of finite unions of closed intervals of I. This lattice arises naturally in the setting of o-minimality, as these are precisely the…
We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with…
An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some…
In this article we introduce the study of the number of pairs of non-comparable elements in a distributive lattice $\L$. We give several tight lower and upper bounds for the number and give as an application the lattices precisely for which…
The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to $0,1$-monotone clones, as the main result we show that for any…
We study numerically Anderson localization on lattices that are tree-like except for the presence of one loop of varying length $L$. The resulting expressions allow us to compute corrections to the Bethe lattice solution on i)…
Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and…
Euclidean lattices occupy a central position in number theory, the geometry of numbers, and modern cryptography. In the present article, the theory of Euclidean lattices is employed to investigate normed $\mathbb{Z}$-modules of finite rank.…
We study different extended formulations for the set $X = \{x\in\mathbb{Z}^n \mid Ax = Ax^0\}$ in order to tackle the feasibility problem for the set $X_+=X \cap \mathbb{Z}^n_+$. Here the goal is not to find an improved polyhedral…
For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, --The class of all finite…
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…
Given an arbitrary graph $E$ we investigate the relationship between $E$ and the groupoid $G_E$. We show that there is a lattice isomorphism between the lattice of pairs $(H, S)$, where $H$ is a hereditary and saturated set of vertices and…
One of the longstanding problems in universal algebra is the question of which finite lattices are isomorphic to the congruence lattices of finite algebras. This question can be phrased as which finite lattices can be represented as…
This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be…