Critical points between varieties generated by subspace lattices of vector spaces

AbstractWe denote by ConcA the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by ConcV the class of all semilattices isomorphic to ConcA for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as crit(V;W)=min{cardD∣D∈ConcV−ConcW}. Given a finitely generated variety V of modular lattices, we obtain an integer ℓ, depending on V, such that crit(V;V ar(SubFn))≥ℵ2 for any n≥ℓ and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that: crit(Mn;V ar(SubF3))=ℵ2, for any finite field F and any ordinal n such that 2+cardF≤n≤ω. Similarly crit(V ar(SubF3);V ar(SubK3))=ℵ2, for all finite fields F and K such that cardF>cardK