Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties
| dc.creator | Wehrung, Friedrich | |
| dc.creator | Tuma, Jiri | |
| dc.date | 2004-10-27 | |
| dc.date.accessioned | 2026-07-07T05:13:44Z | |
| dc.date.available | 2026-07-07T05:13:44Z | |
| dc.description | We construct a diagram D, indexed by a finite partially ordered set, of finite Boolean semilattices and (v,0,1)-embeddings, with top semilattice $2^4$, such that for any variety V of algebras, if D has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra $U$ in V such that the congruence lattice of $U$ contains, as a 0,1-sublattice, the five-element modular nondistributive lattice $M_3$. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive. Using earlier work of K.A. Kearnes and A.Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor F from finite Boolean semilattices and (v,0,1)-embeddings to lattices and lattice embeddings such that the composition Con F is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlak in 1985 about the existence of a functorial solution of the Congruence Lattice Problem. | |
| dc.identifier | https://arxiv.org/abs/math/0410576 | |
| dc.identifier | http://arxiv.org/abs/math/0410576 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73021 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Primary 08A30; Secondary 06A12, 08B15 | |
| dc.title | Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties | |
| dc.type | text |