Distributive congruence lattices of congruence-permutable algebras
| dc.creator | Ruzicka, Pavel | |
| dc.creator | Tuma, Jiri | |
| dc.creator | Wehrung, Friedrich | |
| dc.date | 2005-05-18 | |
| dc.date | 2006-11-02 | |
| dc.date.accessioned | 2026-07-07T06:39:59Z | |
| dc.date.available | 2026-07-07T06:39:59Z | |
| dc.description | We prove that every distributive algebraic lattice with at most $\aleph\_1$ compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The $\aleph\_1$ bound is optimal, as we find a distributive algebraic lattice $D$ with $\aleph\_2$ compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E. T. Schmidt from 1969. Furthermore, $D$ may be taken as the congruence lattice of the free bounded lattice on $\aleph\_2$ generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued "distances" used by B. Jonsson in his proof of Whitman's embedding Theorem. In particular, the semilattice of compact elements of $D$ is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive join-semilattice with zero is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction. | |
| dc.description | to appear in Journal of Algebra | |
| dc.identifier | https://arxiv.org/abs/math/0505381 | |
| dc.identifier | http://arxiv.org/abs/math/0505381 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/101284 | |
| dc.subject | General Mathematics | |
| dc.subject | Primary 08A30; Secondary 06A12, 08B15 | |
| dc.title | Distributive congruence lattices of congruence-permutable algebras | |
| dc.type | text |