Lattices

George Grätzer, E. Tamás Schmidt, Ralph Freese, Viacheslav N. Saliĭ, Gudrun Kalmbach, Carlton J. Maxson, Günter F. Pilz, Viacheslav N. Saliĭ and Rudolf Wille

Chapter Chapter F in The Concise Handbook of Algebra, 2002, pp 417-450 from Springer

Abstract: 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.

Keywords: Prime Ideal; Boolean Algebra; Distributive Lattice; Complete Lattice; Congruence Lattice (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-94-017-3267-3_6

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