1 research outputs found
Tight embedding of modular lattices into partition lattices: progress and program
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.Comment: 36 pages, 22 figures, to appear (up to stylistic changes) in Algebra
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