Cover-preserving Embedding of Modular Lattices

In this note we prove: If a subdirect product of ̄nitely many ̄nite projective geometries has the cover-preserving embedding property, then so does each factor. In what follows all the lattices will be ̄nite modular ones. A ̄nite lattice K has the cover-preseving embedding property, abbreviated as CPEP with respect a variety V of lattices if whenever K can be embedded into a ̄nite lattice L in V , then K has a cover-preserving embedding into L, that is an embedding f with the property that if a covers b in K then f (a) covers f (b) in L. In a paper of E. Fried, G. GrÄatzer and H. Lakser, [1] it was proved that a ̄nite projective geometry has the cover-preserving embedding property with respect to the variety M of all modular lattices if and only if one of the following three conditions hold: (i) the length of P is 1; (ii) the length of P is 2 and P is isomorphic to M3; (iii) the length of P is greater then 2 and either P is non-arguesian or P is arguesian and for some prime p each interval of P of length 2 contains p + 1 atoms (i.e. P is a projective geometry over a prime ̄eld). Later in E. T. Schmidt, [2] the following theorem was formulated: Theorem 1. If a ̄nite modular lattice L has the CPEP with respect to M then L is the subdirect product of projective geometries of type (i)-(iii). Really in [2] the following was proved: If a ̄nite modular lattice L has the CPEP with respect to M then L is the subdirect product of projective geometries. The proof that the subdirect components are just the projective geometries (i)-(iii) was missing. This statment seems in the ̄rst moment quite trivial, but it is far not so. Date: April 2, 2000. 1991 Mathematics Subject Classi ̄cation. Primary 06B10; Secondary 06D05.