Dimension of RC-lattices

TL;DR

This paper explores the properties of RC-lattices, establishing bounds on their dimension, and characterizing non-planar RC-lattices as those with dimension three.

Contribution

It introduces bounds on the dimension of dismantlable lattices and characterizes non-planar RC-lattices based on their dimension.

Findings

Dimension of the adjunct sum of two lattices is obtained.

Bound on the dimension of dismantlable lattices in terms of nullity.

RC-lattice on n elements has dimension at most three.

Abstract

In $1941$ Dushnik and Miller introduced the concept of dimension of a poset. In $2020$ Bhavale and Waphare introduced the concept of an RC-lattice as a lattice in which all the reducible elements are lying on a chain. In this paper, we obtain the dimension of adjunct sum of two lattices. We obtain a bound on the dimension of a dismantlable lattice in terms of its nullity. We also prove that the dimension of an RC-lattice on $n$ elements is at the most three. Consequently, we prove that an RC-lattice is non planar if and only if its dimension is three.