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 continuously many congruence classes of locally finite varieties of (bounded) modular lattices. A large part of our work involves a categorical theory of partial algebras endowed with a partial subalgebra together with a semilattice-valued distance, that we call gamps. This part of the theory is formulated in any variety of (universal) algebras.

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. 2010. ⟨hal-00468048v1⟩