The possible values of critical points between varieties of lattices

Pierre Gillibert (1, 2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 Department of Algebra (MFF-UK)

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

Mots clés: Lattice; variety; compact; congruence; congruence class; critical point; Congruence Lifting Lemma; partial algebra; gamp; chain diagram.

Mots clés: chain diagram

Domaines: Mathématiques générales [math.GM]; Logique [math.LO]; Catégories et ensembles [math.CT]

Pierre Gillibert. The possible values of critical points between varieties of lattices. Journal of Algebra, 2012, 362, pp.30-55. ⟨10.1016/j.jalgebra.2012.04.006⟩. ⟨hal-00468048v3⟩