J. Tuma proved an interesting "congruence amalgamation" result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: ---A.P. Huhn proved that every distributive algebraic lattice $D$ with at most $\aleph_1$ compact elements can be represented as the congruence lattice of a lattice $L$. We show that $L$ can be constructed as a locally finite relatively complemented lattice with zero. ---We find a large class of lattices, the $\omega$-congruence-finite lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruence-preserving extension.
@article{hal-00004029,
author = {Gr\"atzer, George and Lakser, Harry and Wehrung, Friedrich},
title = {Congruence amalgamation of lattices},
journal = {HAL},
volume = {2000},
number = {0},
year = {2000},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00004029}
}
Grätzer, George; Lakser, Harry; Wehrung, Friedrich. Congruence amalgamation of lattices. HAL, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/hal-00004029/