![]() ![]() This says that α ∧ (β ∨ γ) = (α ∧ β) ∨ (α ∧ γ), for any congruences α, β, and γ of a given lattice. The congruence lattice of any lattice is distributive. While congruences of lattices lose something in comparison to groups, modules, rings (they cannot be identified with subsets of the universe), they also have a property unique among all the other structures encountered yet. ![]() The congruence lattice Con A of any algebra A is an algebraic lattice. It says that for a congruence, being finitely generated is a lattice-theoretical property.Ī congruence of an algebra A is finitely generated if and only if it is a compact element of Con A.Īs every congruence of an algebra is the join of the finitely generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain the following result, first published by Birkhoff and Frink in 1948. The following is a universal-algebraic triviality. We denote by Con A the congruence lattice of an algebra A, that is, the lattice of all congruences of A under inclusion. Wehrung provided a counterexample for distributive lattices with ℵ 2 compact elements using a construction based on Kuratowski's free set theorem. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ 1 compact elements, but F. ![]() Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory it had a deep impact on the development of lattice theory itself. In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. ![]()
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