We prove that there is a distributive (∨,0,1)-semilattice of size ℵ₂ such that there is no weakly distributive (∨,0)-homomorphism from 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-2,
author = {Pavel R\r u\v zi\v cka},
title = {Free trees and the optimal bound in Wehrung's theorem},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {217-228},
zbl = {1147.06003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-2}
}
Pavel Růžička. Free trees and the optimal bound in Wehrung's theorem. Fundamenta Mathematicae, Tome 201 (2008) pp. 217-228. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-2/