Critical Points between Varieties Generated by Subspace Lattices of Vector Spaces

We denote by Conc A the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc V the class of all semilattices isomorphic to Conc A for some A ∈ V. Given V1 and V2 varieties of algebras, the critical point of V1 under V2 is defined as crit(V1;V2) = min{cardD | D ∈ Conc V1 − Conc V2}. Given a finitely generated variety V of modular lattices, we obtain an integer l, depending of V, such that crit(V;Var(SubFn)) ≥ א2 for any n ≥ l and any field F . In a second part, using tools introduced in [4], we prove that: crit(Mn;Var(SubF )) = א2, for any finite field F and any integer n such that 2+cardF ≤ n ≤ ω. Similarly crit(Var(SubF );Var(SubK)) = א2, for all finite fields F and K such that cardF > cardK.