17 research outputs found
CD-independent subsets in meet-distributive lattices
A subset of a finite lattice is CD-independent if the meet of any two
incomparable elements of equals 0. In 2009, Cz\'edli, Hartmann and Schmidt
proved that any two maximal CD-independent subsets of a finite distributive
lattice have the same number of elements. In this paper, we prove that if
is a finite meet-distributive lattice, then the size of every CD-independent
subset of is at most the number of atoms of plus the length of . If,
in addition, there is no three-element antichain of meet-irreducible elements,
then we give a recursive description of maximal CD-independent subsets.
Finally, to give an application of CD-independent subsets, we give a new
approach to count islands on a rectangular board.Comment: 14 pages, 4 figure
Finite convex geometries of circles
Let F be a finite set of circles in the plane. We point out that the usual
convex closure restricted to F yields a convex geometry, that is, a
combinatorial structure introduced by P. H Edelman in 1980 under the name
"anti-exchange closure system". We prove that if the circles are collinear and
they are arranged in a "concave way", then they determine a convex geometry of
convex dimension at most 2, and each finite convex geometry of convex dimension
at most 2 can be represented this way. The proof uses some recent results from
Lattice Theory, and some of the auxiliary statements on lattices or convex
geometries could be of separate interest. The paper is concluded with some open
problems.Comment: 22 pages, 7 figure
Quasiplanar diagrams and slim semimodular lattices
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. Every planar diagram is quasiplanar, and P has a quasiplanar diagram
iff its order dimension is at most 2. A finite lattice is slim if it is
join-generated by the union of two chains. We are interested in diagrams only
up to similarity. The main result gives a bijection between the set of the
(similarity classes of) finite quasiplanar diagrams and that of the (similarity
classes of) planar diagrams of finite, slim, semimodular lattices. This
bijection allows one to describe finite posets of order dimension at most 2 by
finite, slim, semimodular lattices, and conversely. As a corollary, we obtain
that there are exactly (n-2)! quasiplanar diagrams of size n.Comment: 19 pages, 3 figure
Hálóelmélet = Lattice theory
A pályázat rĂ©sztvevĹ‘i egyĂĽtt is Ă©s kĂĽlön-kĂĽlön is Ă©rtek el eredmĂ©nyeket; tĂşlnyomĂłrĂ©szt a hálĂłelmĂ©let, Ă©s nyomokban (a hálĂłelmĂ©lethez szorosan kapcsolĂłdĂł) univerzális algebra terĂĽletĂ©n. Az elĂ©rt eredmĂ©nyekbĹ‘l 32 tudományos cikk kĂ©szĂĽlt. Ezen cikkek közĂĽl 20 már megjelent (16 papĂron, 4 pedig a folyĂłiratok honlapján „on-line”), további kettĹ‘t közlĂ©sre elfogadtak, a maradĂ©k 10 pedig közlĂ©sre benyĂşjtott stádiumban van. A megjelent cikkek közĂĽl 14 a hálĂłelmĂ©let kĂ©t vezetĹ‘ folyĂłiratában jelent meg: 9 az Algebra Universalis, 5 pedig az Order hasábjain. KiemelĂ©st Ă©rdemel, hogy a 32 cikkbĹ‘l 5 a pályázatban rĂ©sztvevĹ‘k közös munkája. Az elĂ©rt eredmĂ©nyek Ă©s az azokbĂłl Ărt cikkek mennyisĂ©ge messze meghaladja a munkatervbeli cĂ©lkitűzĂ©st, amely nĂ©gy Ă©vre 7 cikket Ărt elĹ‘. | The participants of the project achieved results, both individually and together. The majority of these results belong to Lattice Theory, and a few of them to Universal Algebra, which is closely connected to Lattice Theory. Based on the results achieved, 32 scientific papers have been written. 20 of these papers have already appeared (16 in print and 4 on-line on the web sites of journals). Two additional papers are accepted for publication, and the remaining 10 papers are submitted. Fourteen of the twenty papers appeared in the two leading journals of Lattice Theory; 9 in Algebra Universalis and 5 in Order. It is worth emphasizing that five of the papers represent joint work of the two participants of the project. The amount of the results and that of the papers essentially exceed the original goal of the work plan, which promised 7 papers for the four-year-long duration of the project