Critical points between varieties generated by subspace lattices of vector spaces, preprint 2008, available online at http://hal.archives-ouvertes.fr/hal-00324397/fr

Abstract. 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 gen-erated 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 3)) = ℵ2, for any finite field F and any integer n such that 2+cardF ≤ n ≤ ω. Similarly crit(Var(SubF 3);Var(SubK3)) = ℵ2, for all finite fields F and K such that cardF> cardK. 1