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

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

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

Patch extensions and trajectory colorings of slim rectangular lattices

With the help of our new tools in the title, we give an efficient representation of the congruence lattice of a slim rectangular lattice by an easy-to-visualize quasiordering on the set of its meet-irreducible elements or, equivalently, on the set of its trajectories

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

The Congruences of a Finite Lattice, A "Proof-by-Picture" Approach, third edition

The major topic of this book: Congruence lattices of finite lattices. It covers about 80 years of research and 250 papers.Comment: Contains Part I of the boo