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

Abstract

A famous Theorem of Pudlak and Túma states that each finite lattice L occurs as sublattice of a finite partition lattice. Here we derive, for modular lattices L, necessary and sufficient conditions for cover-preserving embeddability. Aspects of our work relate to Bjarni Jónsson. arXiv:1704.03715v4 [math.CO]

Modular lattices of finite length (Part A)

“…Let M(E, cl) be a matroid and (J, Λ) a partial linear space. Following [W8,Def.4.1] we say that a bijection ψ : J → E line-preserving (line-pres) models (J, Λ) if cl(ψ(p), ψ(q)}) = ψ( ) for all 2-sets {p, q} ⊆ and all ∈ Λ. In particular ψ( ) is a dependent subset of E whenever | | ≥ 3.…”