Related papers: Enumeration of lattices of nullity $k$ and contain…
In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. According to Bhavale and Waphare, if a dismantlable lattice of nullity k contains r reducible elements then 2 $\leq$ r $\leq$ 2k. In…
In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. In 2003 Pawar and Waphare counted all non-isomorphic lattices on n elements and n edges, which are precisely lattices of nullity one.…
In this paper, we count all non-isomorphic lattices on $n$ elements, containing four reducible elements and having nullity three. This work is in respect of Birkhoff's open problem (which is NP-complete) of counting all finite lattices on…
In $1941$ Dushnik and Miller introduced the concept of dimension of a poset. In $2020$ Bhavale and Waphare introduced the concept of an RC-lattice as a lattice in which all the reducible elements are lying on a chain. In this paper, we…
Given k sets such that no one is contained in another, there is an associated lattice on the power set P([k]) corresponding to inclusion relations among unions of the sets. Two lattices on P([k]) are equivalent if there is a permutation of…
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…
Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18.…
For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$…
Unitally nondistributive quantales are unital quantales such that the unit is approximable by the totally below relation and does not meet-distribute over arbitrary joins. It is shown that the underlying nondistributive complete lattice…
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…
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…
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…
A series of integral lattices parametrised by integers $k,m,n$ are introduced and investigated, where $n$ is the rank of the lattice, including the root lattices described in a uniform way and unimodular lattices such as the Niemeier…
Lattices are simplified by removing some of their doubly irreducible elements, resulting in smaller lattices called racks. All vertically indecomposable modular racks of $n \le 40$ elements are listed, and the numbers of all modular…
A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Cz\'edli and E.T. Schmidt as the duals of the lattices…
Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in…
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 provide the first examples of lattices on irreducible buildings that are not residually finite. Assuming that the normal subgroup property holds for them (which is expected) five of the lattices are simple.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct…
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…