CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ 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 $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. 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 résztvevői együtt is és külön-külön is értek el eredményeket; túlnyomórészt a hálóelmélet, és nyomokban (a hálóelmélethez szorosan kapcsolódó) univerzális algebra területén. Az elért eredményekből 32 tudományos cikk készült. Ezen cikkek közül 20 már megjelent (16 papíron, 4 pedig a folyóiratok honlapján „on-line”), további kettőt közlésre elfogadtak, a maradék 10 pedig közlésre benyújtott stádiumban van. A megjelent cikkek közül 14 a hálóelmélet két vezető folyóiratában jelent meg: 9 az Algebra Universalis, 5 pedig az Order hasábjain. Kiemelést érdemel, hogy a 32 cikkből 5 a pályázatban résztvevők közös munkája. Az elért eredmények és az azokból írt cikkek mennyisége messze meghaladja a munkatervbeli célkitűzést, amely négy évre 7 cikket írt elő. | 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