2-distributivity and lattices of sublattices closed under taking relative complements

TL;DR

This paper investigates the relationship between 2-distributivity and the structure of sublattices in finite-length modular lattices, establishing conditions for sublattices closed under relative complements to form ranked lattices.

Contribution

It proves that 2-distributivity is necessary for sublattices closed under relative complements to form ranked lattices in finite-length modular lattices.

Findings

2-distributivity is necessary for such sublattices

Sublattices closed under relative complements form ranked lattices

Distributivity ensures the property in the lattice

Abstract

For a modular lattice $L$ of finite length, we prove that the distributivity of $L$ is a sufficient condition while its 2-distributivity is a necessary condition that those sublattices of $L$ that are closed under taking relative complements form a ranked lattice of finite length.