Critical points between varieties generated by subspace lattices of vector spaces

Pierre Gillibert (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We denote by Conc(A) the semilattice of 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 in V. Given varieties V1 and V2 varieties of algebras, the critical point of V1 under V2, denote by crit(V1;V2) is the smalest cardinality of a semilattice in Conc(V1) but not in Conc(V2). Given a finitely generated variety V of modular lattices, we obtain an integer l, depending of V, such that crit(V;Var(Sub F^n)) is at least aleph_2 for any n > 1 and any field F. In a second part, we prove that crit(Var(Mn);Var(Sub F^3))=aleph_2, for any finite field F and any integer n such that 1+card F< n. Similarly crit(Var(Sub F^3);Var(Sub K^3))=aleph_2, for all finite fields F and K such that card F>card K.

Mots clés: Lattice; semilattice; poset; diagram; lifting; compact; congruence; homomorphism; condensate; critical point; norm-covering; supported; support; ideal; dimension monoid; regular ring; dimension monoid.

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

Pierre Gillibert. Critical points between varieties generated by subspace lattices of vector spaces. Journal of Pure and Applied Algebra, 2010, 214 (8), pp.1306-1318. ⟨10.1016/j.jpaa.2009.10.013⟩. ⟨hal-00324397v3⟩