Cover-preserving embedding of modular lattices (Q1329223)

A finite lattice \(K\) has the cover-preserving property with respect to a variety \(V\) of lattices, if whenever \(K\) can be embedded into a finite lattice \(L\in V\), then \(K\) has an embedding \(f\) into \(L\) such that if \(a\) covers \(b\) in \(K\), then \(f(a)\) covers \(f(b)\) in \(L\). It is proved that if a subdirect product of finitely many finite projective geometries has the cover-preserving embedding property, then so does each factor.