The possible values of critical points between varieties of lattices

Journal of Algebra, 362 (2012), 30-55
http://dx.doi.org/10.1016/j.jalgebra.2012.04.006

We denote by View the MathML sourceConcL the (?,0)(?,0)-semilattice of all finitely generated congruences of a lattice L . For varieties (i.e., equational classes) VV and WW of lattices such that VV is contained neither in WW nor in its dual, and such that every simple member of WW contains a prime interval, we prove that there exists a bounded lattice A?VA?V with at most ?2?2 elements such that View the MathML sourceConcA is not isomorphic to View the MathML sourceConcB for any B?WB?W. The bound ?2?2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices.