Free trees and the optimal bound in Wehrung's theorem

Volume 198 / 2008

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


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.


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

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