Related papers: Planar semilattices and nearlattices with eighty-t…

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 $127\cdot 2^{n-8}-1$…

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…

A (Hasse) diagram of a finite partially ordered set (poset) P will be called quasiplanar if for any two incomparable elements u and v, either v is on the left of all maximal chains containing u, or v is on the right of all these chains.…

For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq…

In the current paper we study extremal semilattices with respect to their equational properties. In the class $\mathbf{S}_n$ of all semilattices of order $n$ we find semilattices which have maximal (minimal) number of consistent equations.…

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…

A planar semimodular lattice is slim if it does not contain $M_3$ as a sublattice. An SPS lattice is a slim, planar, semimodular lattice. A recent result of G\'abor Cz\'edli proves that there is an eight element (planar) distributive…

A planar semimodular lattice $L$ is \emph{slim} if $\mathbf{M}_3$ is not a sublattice of $L$. In a recent paper, G. Cz\'edli introduced a very powerful diagram type for slim, planar, semimodular lattices. This short note proves the…

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of these…

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)…

Let $(L,\vee)$ be a finite n-element semilattice where $n\geq 5$. We prove that the fourth largest number of subuniverses of an $n$-element semilattice is $25\cdot 2^{n-5}$, the fifth largest number is $ 24.5\cdot 2^{n-5}$, and the sixth…

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…

We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\' edli, we prove that the third, fourth and fifth largest numbers of congruences…

For every natural number $n\geq 5$, we prove that the number of subuniverses of an $n$-element lattice is $2^n$, $13\cdot 2^{n-4}$, $23\cdot 2^{n-5}$, or less than $23\cdot 2^{n-5}$. By a subuniverse, we mean a sublattice or the emptyset.…

We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are…

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

A lattice in Euclidean $d$-space is called well-rounded if it contains $d$ linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The…

A planar (upper) semimodular lattice $L$ is slim if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular lattices as particular slim planar…