Projective geometries as cover-preserving sublattices

Abstract

It is a well known result in the folklore of lattice theory that whenever M 3 (the five element modular nondistributive lattice) can be embedded into a finite modular lattice L, then M 3 also has a cover-preserving embedding into L, that is, an embedding ep with the property that if a covers b in M 3 , then ep(a) covers ep(b) in L. Therefore, if L is a finite nondistributive modular lattice, then L contains M3 as a cover-preserving sublattice. We formalize this concept: a finite lattice K has the cover-preserving embedding property, abbreviated as CPEP, with respect to a variety V of lattices, if whenever K can be embedded into a finite lattice L in V, then K has a cover-preserving embedding into L. In this note, we determine which finite projective geometries P satisfy the CPEP with respect to the variety M of modular lattices; from our point of view, a finite projective geometry is a finite complemented simple modular lattice. Parts of our main result are in the folklore; certainly, specialists in the field knew that M 3 has CPEP while M n , n > 3, does not. Those familiar with the deep results of A. Huhn on diamonds and of R. Freese on n-frames could easily deduce our results for projective geometries of dimension at least three. It appears to us that those techniques do not apply to projective planes. General references on projective geometries are [1, Chapter 13] and [6, Section IV.5]. We make particular use of two facts: a projective geometry of length ~ 4 (i.e., of geometric dimension ~ 3) is arguesian; any finite arguesian projective geometry of length m ~ 3 is isomorphic to the lattice Sub (V) of all subspaces of an