|
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. 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. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem. ==Preliminaries== We denote by Con ''A'' the congruence lattice of an algebra ''A'', that is, the lattice of all congruences of ''A'' under inclusion. The following is a universal-algebraic triviality. It says that for a congruence, being finitely generated is a lattice-theoretical property. Lemma. A congruence of an algebra ''A'' is finitely generated if and only if it is a compact element of Con ''A''. As 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. Theorem (Birkhoff and Frink 1948). The congruence lattice Con ''A'' of any algebra ''A'' is an algebraic lattice. 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. Theorem (Funayama and Nakayama 1942). The congruence lattice of any lattice is distributive. This says that α ∧ (β ∨ γ) = (α ∧ β) ∨ (α ∧ γ), for any congruences α, β, and γ of a given lattice. The analogue of this result fails, for instance, for modules, as , as a rule, for submodules ''A'', ''B'', ''C'' of a given module. Soon after this result, Dilworth proved the following result. He did not publish the result but it appears as an exercise credited to him in Birkhoff 1948. The first published proof is in Grätzer and Schmidt 1962. Theorem (Dilworth ≈1940, Grätzer and Schmidt 1962). Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. It is important to observe that the solution lattice found in Grätzer and Schmidt's proof is ''sectionally complemented'', that is, it has a least element (true for any finite lattice) and for all elements ''a'' ≤ ''b'' there exists an element ''x'' with ''a'' ∨ ''x'' = ''b'' and ''a'' ∧ ''x'' = ''0''. It is also in that paper that CLP is first stated in published form, although it seems that the earliest attempts at CLP were made by Dilworth himself. Congruence lattices of finite lattices have been given an enormous amount of attention, for which a reference is Grätzer's 2005 monograph. ---- The congruence lattice problem (CLP): Is every distributive algebraic lattice isomorphic to the congruence lattice of some lattice? ---- The problem CLP has been one of the most intriguing and longest-standing open problems of lattice theory. Some related results of universal algebra are the following. Theorem (Grätzer and Schmidt 1963). Every algebraic lattice is isomorphic to the congruence lattice of some algebra. The lattice Sub ''V'' of all subspaces of a vector space ''V'' is certainly an algebraic lattice. As the next result shows, these algebraic lattices are difficult to represent. Theorem (Freese, Lampe, and Taylor 1979). Let ''V'' be an infinite-dimensional vector space over an uncountable field ''F''. Then Con ''A'' isomorphic to Sub ''V'' implies that ''A'' has at least card ''F'' operations, for any algebra ''A''. As ''V'' is infinite-dimensional, the largest element (''unit'') of Sub ''V'' is not compact. However innocuous it sounds, the compact unit assumption is essential in the statement of the result above, as demonstrated by the following result. Theorem (Lampe 1982). Every algebraic lattice with compact unit is isomorphic to the congruence lattice of some groupoid. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「congruence lattice problem」の詳細全文を読む スポンサード リンク
|