Finite Dismantlable Semidistributive Lattices are Planar

TL;DR

This paper proves that finite semidistributive lattices are dismantlable if and only if they are planar, extending known results for distributive lattices, and explores their breadth using canonical join representations.

Contribution

It establishes the equivalence between dismantlability and planarity for finite semidistributive lattices and introduces a method to compute their breadth.

Findings

Finite semidistributive lattices are dismantlable iff they are planar.

The breadth of such lattices can be computed via canonical join representations.

The breadth of finite semidistributive dismantlable lattices is at most 2.

Abstract

In this article, we prove that finite semidistributive lattices are dismantlable if and only if they are planar. This extends a well-known result by Kelly and Rival that states the same property for finite distributive lattices. Moreover, we show how the breadth of finite semidistributive lattices can be computed with the help of canonical join representations. We use this result to conclude that the breadth of a finite semidistributive dismantlable lattice cannot exceed $2$.