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Congruence amalgamation of lattices

George Grätzer·Harry Lakser·Friedrich Wehrung
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Abstract

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 $ω$-congruence-finite lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruence-preserving extension.

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George Grätzer, Harry Lakser, Friedrich Wehrung. 2005-01-22. Congruence amalgamation of lattices. https://arxiv.org/abs/math/0501370

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