Quasiplanar diagrams and slim semimodular lattices

Abstract

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.