It belongs to the folklore of lattice theory that whenever \(M_ 3\) (the 5-element modular nondistributive lattice) can be embedded into a finite modular lattice L, then \(M_ 3\) also has a cover-preserving embedding into L. Formalizing this concept, the authors say that a finite lattice K has the cover-preserving embedding property (CPEP, for short) with respect to a variety \({\mathbb{V}}\) of lattices if whenever K can be embedded into a finite lattice in \({\mathbb{V}}\), then K has a cover-preserving embedding into L.
In the present paper the authors determine which finite projective geometries P satisfy the CPEP with respect to the variety \({\mathbb{M}}\) of modular lattices. (A finite projective geometry is treated here as a finite complemented simple modular lattice). The main result is the following:
Theorem. Let P be a finite projective geometry. Then P has the CPEP with respect to the variety \({\mathbb{M}}\) of all modular lattices if, and only if, one of the following conditions holds: (i) the length of P is 1; (ii) the length of P is 2 and P is isomorphic to \(M_ 3\); (iii) the length of P is greater than 2 and either p is nonarguesian or P is arguesian and, for some prime p, each interval in P of length 2 contains \(p+1\) atoms.