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arxiv: math/0501370 · v1 · pith:76HOAHVCnew · submitted 2005-01-22 · 🧮 math.GM

Congruence amalgamation of lattices

classification 🧮 math.GM
keywords latticelatticescongruenceamalgamationcomplementedeveryfinitelocally
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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.

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