Quasiplanar diagrams and slim semimodular lattices

TL;DR

This paper establishes a bijection between quasiplanar diagrams of finite posets with order dimension at most 2 and planar diagrams of slim, semimodular lattices, enabling new characterizations of these structures.

Contribution

It introduces a bijection linking quasiplanar diagrams of posets to slim, semimodular lattices, providing a new way to analyze order dimension 2 posets.

Findings

Exactly (n-2)! quasiplanar diagrams of size n exist.

Quasiplanar diagrams correspond to planar diagrams of slim, semimodular lattices.

The bijection facilitates characterizations of finite posets with order dimension at most 2.

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.