Regular congruence-preserving extensions of lattices

Abstract.

In this paper, we prove that every lattice L has a congruence-preserving extension into a regular lattice $\tilde{L}$, moreover, every compact congruence of $\tilde{L}$ is principal. We construct $\tilde{L}$ by iterating a construction of the first author and F. Wehrung and taking direct limits.¶ We also discuss the case of a finite lattice L, in which case $\tilde{L}$ can be chosen to be finite, and of a lattice L with zero, in which case $\tilde{L}$ can be chosen to have zero and the extension can be chosen to preserve zero.