Cover-preserving embedding (Q1182527)

A finite lattice \(L\) has the cover-preserving embedding property (CPEP) if for every embedding \(L\to M\), \(M\) modular, there is a cover-preserving embedding. For \(L\) a projective geometry this amounts to being of dimension 0, dimension 1 and order 2, non-desarguesian or of dimension \(\geq 2\) and coordinatizable over a prime field (Fried et al.). In the present paper it is shown that any modular \(L\) with CPEP is a subdirect product of such lattices.