Free trees and the optimal bound in Wehrung's theorem

Pavel Růžička¹

¹ Department of Algebra Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic

Fundamenta Mathematicae, Tome 198 (2008) no. 3, pp. 217-228

Résumé

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.

Détail

DOI : 10.4064/fm198-3-2

Mots-clés : prove there distributive vee semilattice mathcal size aleph there weakly distributive vee homomorphism mathop con mathcal its range algebra either congruence compatible structure vee semi lattice congruence compatible structure lattice particular mathcal isomorphic vee semilattice compact congruences lattice improves wehrungs solution dilworths congruence lattice problem giving best cardinality bound possible main ingredient proof modification kuratowskis set theorem which involves what call trees