# EL-Shelling on Comodernistic Lattices

## TL;DR

This paper establishes that comodernistic lattices are EL-shellable by proving their recursive atom orderings are independent of roots, simplifying the understanding of their combinatorial structure.

## Contribution

It proves the equivalence of EL-shellability and root-independent recursive atom ordering, and demonstrates this for comodernistic lattices, including order congruence lattices.

## Key findings

- Comodernistic lattices are EL-shellable.
- Recursive atom orderings can be independent of roots.
- Simplified EL-shelling for order congruence lattices.

## Abstract

We prove the equivalence of EL-shellability and the existence of recursive atom ordering independent of roots. We show that a comodernistic lattice, as defined by Schweig and Woodroofe, admits a recursive atom ordering independent of roots, therefore is EL-shellable. We also present and discuss a simpler EL-shelling on one of the most important classes of comodernistic lattice, the order congruence lattices.