Related papers: Atom-generated planar lattices

In addition to the unique cover $M^+$ of the variety of modular lattices, we also deal with those twenty-three known covers of $M^+$ that can be extracted from the literature. For $M^+$ and for each of these twenty-three known varieties…

Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove…

As the main achievement of the paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, the paper contains the following three observations. First, every coatomless three-generated…

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly…

Optical lattices can be loaded with atoms which can have strong interactions, such that the interaction of atoms at different lattice sites cannot be neglected. Moreover, the intersite interactions can be so strong that it can force the…

A lattice is $(1+1+2)$-generated if it has a four-element generating set such that exactly two of the four generators are comparable. We prove that the lattice Quo$(n)$ of all quasiorders (also known as preorders) of an $n$-element set is…

The possibility is considered for the formation in optical lattices of a heterogeneous state characterized by a spontaneous mesoscopic separation of the system into the spatial regions with different atomic densities. It is shown that such…

In optical lattices where each site is occupied in its lowest energy state by a superposition of zero, one and two atoms, one can in a controllable manner convert the atomic pair into a molecule while retaining the vacuum and one-atom…

A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is obtained by removing the top of $L$ and the bottom of $U$, and identifying the coatoms of $L$ with the atoms of $U$. This operation creates one or two nonisomorphic…

By median we mean a scheme that inputs three element of a lattice, and outputs an element that is an average of the three inputs in a certain sense. The medians of a given finite lattice form a new lattice that is usually larger than the…

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$…

We introduce the free Banach lattice generated by a lattice L. We give an explicit description of it and we study some of its properties for the case when $\mathbb{L}$ is a linear order, like the countable chain condition.

It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic…

Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Z\'adori gave a new proof of this fact with a particularly elegant…

We answer a question raised by Lanier about the possibility of generating $A_n$ and $S_n$ with two elements of order $k$, where $n \geqslant k \geqslant 3$. We show that this can always be done apart from some clear exceptions.

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…

Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices.

Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$…

We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and…

We construct all planar semimodular lattices in three simple steps from the direct product of two chains.