NDLI: Quasiplanar Diagrams and Slim Semimodular Lattices

Content Provider	Springer Nature Link
Author	Czédli, Gábor
Copyright Year	2015
Abstract	For elements x and y in the (Hasse) diagram D of a finite bounded poset P, x is on the left of y, written as x λ y, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as x ϱ y, is defined analogously. The diagram D is quasiplanar if λ and ϱ are transitive and for any pair (x,y) of incomparable elements, if x is on the left of some maximal chain through y, then x λ y. A planar diagram is quasiplanar, and P has a quasiplanar diagram iff its order dimension is at most 2. We are interested in diagrams only up to similarity. A finite lattice is slim if it is join-generated by the union of two chains. 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.
Starting Page	239
Ending Page	262
Page Count	24
File Format	PDF
ISSN	01678094
Journal	Order
Volume Number	33
Issue Number	2
e-ISSN	15729273
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-08-27
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Semimodular lattice Planar lattice Slim lattice Quasiplanar diagram Antimatroid Join-distributive lattice Geometry Discrete Mathematics in Computer Science Theory of Computation
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Computational Theory and Mathematics Geometry and Topology

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The Asymptotic Number of Planar, Slim, Semimodular Lattice Diagrams

Slim Semimodular Lattices. I. A Visual Approach

The Matrix of a Slim Semimodular Lattice

Slim Semimodular Lattices. II. A Description by Patchwork Systems

Note on the description of join-distributive lattices by permutations

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Notes on Planar Semimodular Lattices. VII. Resections of Planar Semimodular Lattices

Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices

Isometrical Embeddings of Lattices into Geometric Lattices

Quasiplanar Diagrams and Slim Semimodular Lattices

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