Abstract
Let Con L denote, up to isomorphism, the class of congruence lattices of lattices and let DA denote the class of all distributive algebraic lattices. For every lattice L, it it clear that the congruence lattice Con L is algebraic. By a 1942 result of N. Funayama and T. Nakayama, Con L is also distributive, so Con L ⊆ DA. Is the converse true: Is every distributive algebraic lattice isomorphic to the congruence lattice of a suitable lattice? This is one of the most famous open questions of lattice theory. We shall briefly review this topic here, together with its related results; for a more complete overview (up to 1998), see Appendix C in (Grätzer 1998); we shall only reference later papers here.
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© 2002 Springer Science+Business Media Dordrecht
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Grätzer, G. et al. (2002). Lattices. In: The Concise Handbook of Algebra. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3267-3_6
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DOI: https://doi.org/10.1007/978-94-017-3267-3_6
Publisher Name: Springer, Dordrecht
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