13 research outputs found
Distributive Lattices have the Intersection Property
Distributive lattices form an important, well-behaved class of lattices. They
are instances of two larger classes of lattices: congruence-uniform and
semidistributive lattices. Congruence-uniform lattices allow for a remarkable
second order of their elements: the core label order; semidistributive lattices
naturally possess an associated flag simplicial complex: the canonical join
complex. In this article we present a characterization of finite distributive
lattices in terms of the core label order and the canonical join complex, and
we show that the core label order of a finite distributive lattice is always a
meet-semilattice.Comment: 9 pages, 3 figures. Final version. Comments are very welcom
Generating all finite modular lattices of a given size
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. Here we adapt and improve this algorithm to construct and count
modular lattices up to size 24, semimodular lattices up to size 22, and
lattices of size 19. We also show that is a lower bound for the
number of nonisomorphic modular lattices of size .Comment: Preprint, 12 pages, 2 figures, 1 tabl
Simplifying modular lattices by removing doubly irreducible elements
Lattices are simplified by removing some of their doubly irreducible
elements, resulting in smaller lattices called racks. All vertically
indecomposable modular racks of elements are listed, and the numbers
of all modular lattices of elements are obtained by P\'olya
counting. SageMath code is provided that allows easy access both to the listed
racks, and to the modular lattices that were not listed. More than 3000-fold
savings in storage space are demonstrated.Comment: 13 page
Obtainable Sizes of Topologies on Finite Sets
We study the smallest possible number of points in a topological space having
k open sets. Equivalently, this is the smallest possible number of elements in
a poset having k order ideals. Using efficient algorithms for constructing a
topology with a prescribed size, we show that this number has a logarithmic
upper bound. We deduce that there exists a topology on n points having k open
sets, for all k in an interval which is exponentially large in n. The
construction algorithms can be modified to produce topologies where the
smallest neighborhood of each point has a minimal size, and we give a range of
obtainable sizes for such topologies.Comment: Final version, to appear in Journal of Combinatorial Theory, Series
The number of slim rectangular lattices
Slim rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. They are finite semimodular lattices L such that the ordered set Ji L of join-irreducible elements of L is the cardinal sum of two nontrivial chains. After describing these lattices of a given length n by permutations, we determine their number, |SRectL(n)|. Besides giving recursive formulas, which are effective up to about n = 1000, we also prove that |SRectL(n)| is asymptotically (n - 2)! · (Formula presented.). Similar results for patch lattices, which are special rectangular lattices introduced by G. Czédli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given. © 2015 Springer International Publishin
Lie Algebras with a finite number of ideals
In this paper we focus on the structure of the variety of Lie algebras with a
finite number of ideals and their graph representations using Hasse diagrams.
The large number of necessary conditions on the algebraic structure of this
type of algebras leads to the explicit description of those algebras in the
variety with trivial Frattini subalgebra. To illustrate our results, we have
included and discussed lots of examples throughout the paper.Comment: 19 pages, 7 figure