# Tight embedding of modular lattices into partition lattices: progress and program

## TL;DR

This paper discusses progress and future plans in embedding modular lattices into partition lattices with minimal set size, aiming for a complete characterization of such embeddings and posing open questions for further research.

## Contribution

The paper advances understanding of tight embeddings of modular lattices into partition lattices and outlines a program to fully characterize these lattices, including open problems.

## Key findings

- Progress after 24 years in embedding modular lattices into partition lattices.
- Proposes a program for complete characterization of lattices admitting tight embeddings.
- Poses eight open questions related to graph theory and matroid theory.

## Abstract

Representing lattices L by equivalence relations amounts to embed them into the lattice Part(V) of all partitions of a set V, and has a long history. Here we are concerned with MODULAR lattices L and aim for sets V as small as possible, i.e. |V| = d(L)+1 where d(L) is the length of L. In other words, we strive for a tight (=cover-preserving) lattice homomorphism from L into Part(V). After a 24 year break the author offers progress, and outlines a program to finally fully characterize the lattices L that admit a tight embedding. Not just 'modular latticians' but also combinatorists are encouraged to contribute. Specifically, eight open questions are posed, four of which purely graph- and matroid-theoretic in nature.