Free trees and the optimal bound in Wehrung's theorem

Pavel Růžička. "Free trees and the optimal bound in Wehrung's theorem." Fundamenta Mathematicae 198.3 (2008): 217-228. <http://eudml.org/doc/282838>.

@article{PavelRůžička2008,
abstract = {We prove that there is a distributive (∨,0,1)-semilattice of size ℵ₂ such that there is no weakly distributive (∨,0)-homomorphism from $Con_\{c\}A$ to 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, 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.},
author = {Pavel Růžička},
journal = {Fundamenta Mathematicae},
keywords = {algebraic lattice; congruence lattice; distributive lattice},
language = {eng},
number = {3},
pages = {217-228},
title = {Free trees and the optimal bound in Wehrung's theorem},
url = {http://eudml.org/doc/282838},
volume = {198},
year = {2008},
}

TY - JOUR
AU - Pavel Růžička
TI - Free trees and the optimal bound in Wehrung's theorem
JO - Fundamenta Mathematicae
PY - 2008
VL - 198
IS - 3
SP - 217
EP - 228
AB - We prove that there is a distributive (∨,0,1)-semilattice of size ℵ₂ such that there is no weakly distributive (∨,0)-homomorphism from $Con_{c}A$ to 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, 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.
LA - eng
KW - algebraic lattice; congruence lattice; distributive lattice
UR - http://eudml.org/doc/282838
ER -