Tight embedding of modular lattices into partition lattices: progress and program
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.
Keywords
Cite
@article{arxiv.1704.03715,
title = {Tight embedding of modular lattices into partition lattices: progress and program},
author = {Marcel Wild},
journal= {arXiv preprint arXiv:1704.03715},
year = {2018}
}
Comments
36 pages, 22 figures, to appear (up to stylistic changes) in Algebra Universalis