Quasiplanar Diagrams and Slim Semimodular Lattices

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 xλy, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as xϱy, 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 xλy. 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.