# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Free trees and the optimal bound in Wehrung's theorem

### Tom 198 / 2008

Fundamenta Mathematicae 198 (2008), 217-228 MSC: 06B15, 06B10, 06A12, 08A30, 08B10. DOI: 10.4064/fm198-3-2

#### Streszczenie

We prove that there is a distributive $(\vee,0,1)$-semilattice $\mathcal{G}$ of size $\aleph_2$ such that there is no weakly distributive $(\vee,0)$-homomorphism from $\mathop{\rm Con}_c A$ to $\mathcal{G}$ with $1$ in its range, for any algebra $A$ with either a congruence-compatible structure of a $(\vee,1)$-semi-lattice or a congruence-compatible structure of a lattice. In particular, $\mathcal{G}$ is not isomorphic to the $(\vee,0)$-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.

#### Autorzy

Faculty of Mathematics and Physics
Charles University
Sokolovská 83
186 75 Praha 8,
Czech Republic
e-mail

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