Free trees and the optimal bound in Wehrung's theorem

• Autorzy: Pavel Růžička
• Języki publikacji: EN

Abstrakty

EN

We prove that there is a distributive (∨,0,1)-semilattice 𝒢 of size ℵ₂ such that there is no weakly distributive (∨,0)-homomorphism from $Con_{c}A$ to 𝒢 with 1 in its range, for any algebra A with either a congruence-compatible structure of a (∨,1)-semi-lattice or a congruence-compatible structure of a lattice. In particular, 𝒢 is not isomorphic to the (∨,0)-semilattice of compact congruences of any lattice. This improves Wehrung's solution of Dilworth's Congruence Lattice Problem, by giving the best cardinality bound possible. The main ingredient of our proof is the modification of Kuratowski's Free Set Theorem, which involves what we call free trees.

Kategorie tematyczne

• 06B15: Representation theory
• 06B10: Ideals, congruence relations
• 08B10: Congruence modularity, congruence distributivity
• 06A12: Semilattices
• 08A30: Subalgebras, congruence relations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2008
• Tom: 198
• Numer: 3
• Strony: 217-228

Daty

wydano

2008

Twórcy

autor

Pavel Růžička

• Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Identyfikatory

DOI

10.4064/fm198-3-2