Congruence lattice problem: Difference between revisions

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The '''Congruence Lattice Problem''' asks whether every [[Distributivity_%28order_theory%29Distributivity (order theory)|distributive]] [[compact element|algebraic lattice]] is [[isomorphic]] to the congruence lattice of (some other) lattice. It was one of the most famous and long-standing open problems in [[Lattice (order)|lattice theory]], an area of [[mathematics]]. Besides being open for a long time, it was distinguished by various related problems in other areas of mathematics and by a profound impact it had on the development of lattice theory itself. The problem was recently solved by F. Wehrung, in a most surprising fashion, by a counterexample construction.

The surprising aspect of the construction is twofold. There exists a counterexample with ℵℵ₂ many compact elements, though all distributive algebraic lattices with at most ℵℵ₁ many compact elements satisfy the conjecture. Also, this was proved without any additional assumptions to the standard ZFC axiom system of [[set theory]]. The key element which distinguishes ℵℵ₂ from ℵℵ₁ is the almost-forgotten old infinite combinatorics result by [[Kazimierz Kuratowski|Kuratowski]] called the [[Kuratowski's Free Set Theorem|Free Set Theorem]].

== Preliminaries ==

==Preliminaries==

We denote by Con ''A'' the congruence lattice of an [[Universal algebra|algebra]] ''A'', that is, the [[Lattice (order)|lattice]] of all [[Congruence relation|congruences]] of ''A'' under inclusion.

The following is a [[Universal algebra|universal-algebraic]] triviality. It says that for a congruence, being finitely generated is a lattice-theoretical property.<br /><br />

'''Lemma.'''

A congruence of an [[Universal algebra|algebra]] ''A'' is finitely generated if and only if it is a [[compact element]] of Con ''A''.

<br /><br />

As every congruence of an algebra is the join of the finitely generated congruences below it (e.g., every [[Module (mathematics)|submodule]] of a [[module (mathematics)|module]] is the union of all its finitely generated submodules), we obtain the following result, first published by Birkhoff and Frink in 1948.<br />

<br />

'''Theorem (Birkhoff and Frink 1948).'''

The congruence lattice Con ''A'' of any algebra ''A'' is an [[compact element|algebraic lattice]].

<br /><br />

While congruences of lattices lose something in comparison to [[group (mathematics)|groups]], [[module (mathematics)|modules]], [[ring (mathematics)|rings]] (they cannot be identified with [[subset|subsets]] of the universe), they also have a property unique among all the other structures encountered yet.<br /><br />

'''Theorem (Funayama and Nakayama 1942).'''

The congruence lattice of any lattice is [[Distributive lattice|distributive]].

<br /><br />

This says that &alpha;α &and;∧ (&beta;β &or;∨ &gamma;γ) = (&alpha;α &and;∧ &beta;β) &or;∨ (&alpha;α &and;∧ &gamma;γ), for any congruences &alpha;α, &beta;β, and &gamma;γ of a given lattice. The analogue of this result fails, for instance, for modules, as <math>A\cap(B+C)\neq(A\cap B)+(A\cap C)</math>, as a rule, for [[Module (mathematics)|submodules]] ''A'', ''B'', ''C'' of a given [[Module (mathematics)|module]].<br />

Soon after this result, [[Robert P. Dilworth|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 Gr&auml;tzerGrätzer and Schmidt 1962.<br /><br />

'''Theorem (Dilworth ≈1940, Gr&auml;tzerGrätzer and Schmidt 1962).'''

Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice.<br /><br />

It is important to observe that the solution lattice found in Gr&auml;tzerGrätzer and Schmidt's proof is ''sectionally complemented'', that is, it has a least element (true for any finite lattice) and for all elements ''a'' &le;≤ ''b'' there exists an element ''x'' with ''a'' &or;∨ ''x'' = ''b'' and ''a'' &and;∧ ''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 Gr&auml;tzerGrätzer's 2005 monograph.<br /><br />

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'''The congruence lattice problem (CLP):'''

Is every distributive algebraic lattice isomorphic to the congruence lattice of some lattice?

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<br />

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).'''

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.<br /><br />

'''Theorem (Gr&auml;tzer and Schmidt 1963).'''

Every algebraic lattice is isomorphic to the congruence lattice of some algebra.

<br /><br />

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.<br /><br />

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 (mathematics)|field]] ''F''. Then Con ''A'' isomorphic to Sub ''V'' implies that ''A'' has at least card ''F'' operations, for any algebra ''A''.<br /><br />

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.<br /><br />

'''Theorem (Lampe 1982).'''

Every algebraic lattice with compact unit is isomorphic to the congruence lattice of some [[Magma (algebra)|groupoid]].

== Semilattice formulation of CLP==

The congruence lattice Con ''A'' of an [[Universal algebra|algebra]] ''A'' is an [[Compact element|algebraic lattice]]. The (∨,0)-[[semilattice]] of [[Compact element|compact elements]] of Con ''A'' is denoted by Con_c ''A'', and it is sometimes called the ''congruence semilattice'' of ''A''. Then Con ''A'' is isomorphic to the [[Ideal (order theory)|ideal lattice]] of Con_c ''A''. By using the classical [[Equivalence of categories|equivalence]] between the category of all (∨,0)-semilattices and the category of all algebraic lattices (with suitable definitions of [[morphisms]]), as it is outlined [[Semilattice|here]], we obtain the following semilattice-theoretical formulation of CLP.

The congruence lattice Con ''A'' of an [[Universal algebra|algebra]] ''A'' is an [[Compact element|algebraic lattice]]. The (&or;,0)-[[semilattice]] of [[Compact element|compact elements]] of Con ''A'' is denoted by Con_c ''A'', and it is sometimes called the ''congruence semilattice'' of ''A''. Then Con ''A'' is isomorphic to the [[Ideal (order theory)|ideal lattice]] of Con_c ''A''. By using the classical [[Equivalence of categories|equivalence]] between the category of all (&or;,0)-semilattices and the category of all algebraic lattices (with suitable definitions of [[morphisms]]), as it is outlined [[Semilattice|here]], we obtain the following semilattice-theoretical formulation of CLP.<br /><br />

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'''Semilattice-theoretical formulation of CLP:'''

Is every [[Distributivity (order theory)|distributive]] (&or;∨,0)-semilattice isomorphic to the congruence semilattice of some lattice?

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<br />

Say that a distributive (&or;∨,0)-semilattice is ''representable'', if it is isomorphic to Con_c ''L'', for some lattice ''L''. So CLP asks whether every distributive (&or;∨,0)-semilattice is representable.

Many investigations around this problem involve ''diagrams'' of semilattices or of algebras. A most useful folklore result about these is the following.

<br /><br />

'''Theorem.'''

The functor Con_c, defined on all algebras of a given [[signature (logic)|signature]], to all (&or;∨,0)-semilattices, [[Limit (category theory)|preserves direct limits]].<br /><br />

== Schmidt's approach via distributive join-homomorphisms ==

We say that a (∨,0)-semilattice satisfies ''Schmidt's Condition'', if it is isomorphic to the quotient of a [[generalized Boolean semilattice]] ''B'' under some [[Distributive homomorphism|distributive join-congruence]] of ''B''. One of the deepest results about representability of (∨,0)-semilattices is the following.

We say that a (&or;,0)-semilattice satisfies ''Schmidt's Condition'', if it is isomorphic to the quotient of a [[generalized Boolean semilattice]] ''B'' under some [[Distributive homomorphism|distributive join-congruence]] of ''B''. One of the deepest results about representability of (&or;,0)-semilattices is the following.<br /><br />

'''Theorem (Schmidt 1968).'''

Any (&or;∨,0)-semilattice satisfying Schmidt's Condition is representable.

<br /><br />

This raised the following problem, stated in the same paper.<br /><br />

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'''Problem 1 (Schmidt 1968).'''

Does any (&or;∨,0)-semilattice satisfiy Schmidt's Condition?

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<br />

Partial positive answers are the following.<br /><br />

'''Theorem (Schmidt 1981).'''

Every distributive ''lattice'' with zero satisfies Schmidt's Condition; thus it is representable.

<br /><br />

This result has been improved further as follows, ''via'' a very long and technical proof, using forcing and Boolean-valued models.<br /><br />

'''Theorem (Wehrung 2003).'''

Every [[direct limit]] of a countable sequence of distributive ''lattices'' with zero and (&or;∨,0)-homomorphisms is representable.<br /><br />

Other important representability results are related to the [[cardinality]] of the semilattice. The following result was prepared for publication by Dobbertin after Huhn's passing away in 1985. The two corresponding papers were published in 1989.<br /><br />

'''Theorem (Huhn 1985).''' Every distributive (&or;∨,0)-semilattice of cardinality at most &alefsym;ℵ₁ satisfies Schmidt's Condition. Thus it is representable.<br /><br />

By using different methods, Dobbertin got the following result.<br /><br />

'''Theorem (Dobbertin 1986).'''

Every distributive (&or;∨,0)-semilattice in which every principal [[Ideal (order theory)|ideal]] is at most [[countable]] is representable.<br /><br />

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== Pudl&aacute;kPudlák's approach; lifting diagrams of (&or;∨,0)-semilattices ==

The approach of CLP suggested by Pudlák in his 1985 paper is different. It is based on the following result, Fact 4, p. 100 in Pudlák's 1985 paper, obtained earlier by Ju.L. Ershov as the main theorem in Section 3 of the Introduction of his 1977 monograph.

'''Theorem (Ershov 1977, Pudlák 1985).'''

The approach of CLP suggested by Pudl&aacute;k in his 1985 paper is different. It is based on the following result, Fact 4, p. 100 in Pudl&aacute;k's 1985 paper, obtained earlier by Ju.L. Ershov as the main theorem in Section 3 of the Introduction of his 1977 monograph.<br /><br />

Every distributive (∨,0)-semilattice is the directed union of its finite distributive (∨,0)-subsemilattices.

'''Theorem (Ershov 1977, Pudl&aacute;k 1985).'''

Every distributive (&or;,0)-semilattice is the directed union of its finite distributive (&or;,0)-subsemilattices.

<br /><br />

This means that every finite subset in a distributive (&or;∨,0)-semilattice ''S'' is contained in some finite ''distributive'' (&or;∨,0)-subsemilattice of ''S''. Now we are trying to represent a given distributive (&or;∨,0)-semilattice ''S'' as Con_c ''L'', for some lattice ''L''. Writing ''S'' as a directed union <math>S=\bigcup(S_i\mid i\in I)</math> of finite distributive (&or;∨,0)-subsemilattices, we are ''hoping'' to represent each ''S_i'' as the congruence lattice of a lattice ''L_i'' with lattice homomorphisms ''f_i^j : L_i→ L_j'', for ''i &le;≤ j'' in ''I'', such that the diagram <math>\mathcal{S}</math> of all ''S_i'' with all inclusion maps ''S_i→S_j'', for ''i &le;≤ j'' in ''I'', is [[Equivalence of categories|naturally equivalent]] to <math>(\mathrm{Con_c}\,L_i,\mathrm{Con_c}\,f_i^j\mid i\leq j\text{ in }I)</math>, we say that the diagram <math>(L_i,f_i^j\mid i\leq j\text{ in }I)</math> lifts <math>\mathcal{S}</math> (with respect to the Con_c functor). If this can be done, then, as we have seen that the Con_c functor preserves direct limits, the direct limit <math>L=\varinjlim_{i\in I}L_i</math> satisfies <math>{\rm Con_c}\,L\cong S</math>.

While the problem whether this could be done in general remained open for about 20 years, Pudl&aacute;kPudlák could prove it for distributive ''lattices'' with zero, thus extending one of Schmidt's results by providing a ''functorial'' solution.<br /><br />

'''Theorem (Pudl&aacute;kPudlák 1985).'''

There exists a direct limits preserving functor Φ, from the category of all distributive lattices with zero and 0-lattice embeddings to the category of all lattices with zero and 0-lattice embeddings, such that Con_cΦ is [[Equivalence of categories|naturally equivalent]] to the identity. Furthermore, Φ(''S'') is a finite [[Lattice (order)|atomistic lattice]], for any finite distributive (&or;∨,0)-semilattice ''S''.

<br /><br />

This result is improved further, by an even far more complex construction, to ''locally finite, sectionally complemented modular lattices'' by R&#367;&#382;i&#269;ka in 2004 and 2006.

This result is improved further, by an even far more complex construction, to ''locally finite, sectionally complemented modular lattices'' by Růžička in 2004 and 2006.

Pudl&aacute;k asked in 1985 whether his result above could be extended to the whole category of distributive (&or;,0)-semilattices with (&or;,0)-embeddings. The problem remained open until it was recently solved in the negative by T&#367;ma and Wehrung.<br /><br />

Pudlák asked in 1985 whether his result above could be extended to the whole category of distributive (∨,0)-semilattices with (∨,0)-embeddings. The problem remained open until it was recently solved in the negative by Tůma and Wehrung.

'''Theorem (Tůma and Wehrung 2006).'''

There exists a [[Diagram (category theory)|diagram]] ''D'' of finite Boolean (∨,0)-semilattices and (∨,0,1)-embeddings, indexed by a finite partially ordered set, that cannot be lifted, with respect to the Con_c functor, by any diagram of lattices and lattice homomorphisms.

'''Theorem (T&#367;ma and Wehrung 2006).'''

There exists a [[Diagram (category theory)|diagram]] ''D'' of finite Boolean (&or;,0)-semilattices and (&or;,0,1)-embeddings, indexed by a finite partially ordered set, that cannot be lifted, with respect to the Con_c functor, by any diagram of lattices and lattice homomorphisms.

<br /><br />

In particular, this implies immediately that CLP has no ''functorial'' solution.

Furthermore, it follows from deep 1998 results of universal algebra by Kearnes and Szendrei in so-called ''commutator theory of varieties'' that the result above can be extended from the variety of all lattices to any variety <math>\mathcal{V}</math> such that all Con ''A'', for <math>A\in\mathcal{V}</math>, satisfy a fixed nontrivial identity in the signature (&or;∨,&and;∧) (in short, ''with a nontrivial congruence identity'').

We should also mention that many attempts at CLP were also based on the following result, first proved by Bulman-Fleming and McDowell in 1978 by using a categorical 1974 result of Shannon, see also Goodearl and Wehrung in 2001 for a direct argument.<br /><br />

'''Theorem (Bulman-Fleming and McDowell 1978).'''

Every distributive (&or;∨,0)-semilattice is a direct limit of finite [[Boolean algebra (structure)|Boolean]] (&or;∨,0)-semilattices and (&or;∨,0)-homomorphisms.

<br /><br />

It should be observed that while the transition homomorphisms used in the Ershov-Pudl&aacute;kPudlák Theorem are (&or;∨,0)-embeddings, the transition homomorphisms used in the result above are not necessarily one-to-one, for example when one tries to represent the three-element chain. Practically this does not cause much trouble, and makes it possible to prove the following results.<br /><br />

'''Theorem.'''

Every distributive (&or;∨,0)-semilattice of cardinality at most &alefsym;ℵ₁ is isomorphic to<br />

(1) Con_c ''L'', for some locally finite, relatively complemented modular lattice ''L'' (T&#367;maTůma 1998 and Gr&auml;tzerGrätzer, Lakser, and Wehrung 2000).<br />

(2) The semilattice of finitely generated two-sided ideals of some (not necessarily unital) von Neumann regular ring (Wehrung 2000).<br />

(3) Con_c ''L'', for some sectionally complemented modular lattice ''L'' (Wehrung 2000).<br />

(4) The semilattice of finitely generated [[normal subgroup|normal subgroups]] of some [[locally finite group]] (R&#367;&#382;i&#269;kaRůžička, T&#367;maTůma, and Wehrung 2006).<br />

(5) The submodule lattice of some right module over a (non-commutative) ring (R&#367;&#382;i&#269;kaRůžička, T&#367;maTůma, and Wehrung 2006).

== Congruence lattices of lattices and nonstable K-theory of von Neumann regular rings ==

We recall that for a (unital, associative) [[ring (mathematics)|ring]] ''R'', we denote by ''V(R)'' the (conical, commutative) monoid of isomorphism classes of finitely generated projective right ''R''-modules, see [[Refinement monoid|here]] for more details. Recall that if ''R'' is von [[Von Neumann regular ring|Neumann regular]], then ''V(R)'' is a [[refinement monoid]]. Denote by Id_c ''R'' the (∨,0)-semilattice of finitely generated [[Ideal (ring theory)|two-sided ideals]] of ''R''. We denote by ''L(R)'' the lattice of all principal right ideals of a von Neumann regular ring ''R''. It is well-known that ''L(R)'' is a [[Complemented lattice|complemented]] [[modular lattice]].

The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.

We recall that for a (unital, associative) [[ring (mathematics)|ring]] ''R'', we denote by ''V(R)'' the (conical, commutative) monoid of isomorphism classes of finitely generated projective right ''R''-modules, see [[Refinement monoid|here]] for more details. Recall that if ''R'' is von [[Von Neumann regular ring|Neumann regular]], then ''V(R)'' is a [[refinement monoid]]. Denote by Id_c ''R'' the (&or;,0)-semilattice of finitely generated [[Ideal (ring theory)|two-sided ideals]] of ''R''. We denote by ''L(R)'' the lattice of all principal right ideals of a von Neumann regular ring ''R''. It is well-known that ''L(R)'' is a [[Complemented lattice|complemented]] [[modular lattice]].

The following result was observed by Wehrung, building on earlier works mainly by J&oacute;nsson and Goodearl.<br /><br />

'''Theorem (Wehrung 1999).'''

Let ''R'' be a von Neumann regular ring. Then the (&or;∨,0)-semilattices Id_c ''R'' and Con_c ''L(R)'' are both isomorphic to the [[maximal semilattice quotient]] of ''V(R)''.

<br /><br />

Bergman proves in a well-known unpublished note from 1986 that any at most countable distributive (&or;,0)-semilattice is isomorphic to Id_c ''R'', for some [[Refinement monoid|locally matricial]] ring ''R'' (over any given field). This result is extended to semilattices of cardinality at most &alefsym;₁ in 2000 by Wehrung, by keeping only the regularity of ''R'' (the ring constructed by the proof is not locally matricial). The question whether ''R'' could be taken locally matricial in the &alefsym;₁ case remained open for a while, until it was disproved by Wehrung in 2004. Translating back to the lattice world by using the theorem above and using a lattice-theoretical analogue of the ''V(R)'' construction, called the ''dimension monoid'', introduced by Wehrung in 1998, yields the following result.<br /><br />

Bergman proves in a well-known unpublished note from 1986 that any at most countable distributive (∨,0)-semilattice is isomorphic to Id_c ''R'', for some [[Refinement monoid|locally matricial]] ring ''R'' (over any given field). This result is extended to semilattices of cardinality at most ℵ₁ in 2000 by Wehrung, by keeping only the regularity of ''R'' (the ring constructed by the proof is not locally matricial). The question whether ''R'' could be taken locally matricial in the ℵ₁ case remained open for a while, until it was disproved by Wehrung in 2004. Translating back to the lattice world by using the theorem above and using a lattice-theoretical analogue of the ''V(R)'' construction, called the ''dimension monoid'', introduced by Wehrung in 1998, yields the following result.

'''Theorem (Wehrung 2004).'''

There exists a distributive (&or;∨,0,1)-semilattice of cardinality &alefsym;ℵ₁ that is not isomorphic to Con_c ''L'', for any modular lattice ''L'' every finitely generated sublattice of which has finite length.<br /><br />

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== A first application of [[Kuratowski's Free Set Theorem]] ==

The abovementioned Problem 1 (Schmidt), Problem 2 (Dobbertin), and Problem 3 (Goodearl) were solved simultaneously in the negative in 1998.

<br /><br />

'''Theorem (Wehrung 1998).'''

There exists a [[Refinement monoid|dimension vector space]] ''G'' over the rationals with [[Monoid|order-unit]] whose positive cone ''G''⁺ is not isomorphic to ''V(R)'', for any von Neumann regular ring ''R'', and is not [[Refinement monoid|measurable]] in Dobbertin's sense. Furthermore, the [[maximal semilattice quotient]] of ''G''⁺ does not satisfy Schmidt's Condition. Furthermore, ''G'' can be taken of any given cardinality greater than or equal to &alefsym;ℵ₂.

<br /><br />

It follows from the previously mentioned works of Schmidt, Huhn, Dobbertin, Goodearl, and Handelman that the &alefsym;₂ bound is optimal in all three negative results above.

It follows from the previously mentioned works of Schmidt, Huhn, Dobbertin, Goodearl, and Handelman that the ℵ₂ bound is optimal in all three negative results above.

As the &alefsym;₂ bound suggests, infinite combinatorics are involved. The principle used is [[Kuratowski's Free Set Theorem]], first published in 1951. Only the case ''n=2'' is used here.

As the ℵ₂ bound suggests, infinite combinatorics are involved. The principle used is [[Kuratowski's Free Set Theorem]], first published in 1951. Only the case ''n=2'' is used here.

The semilattice part of the result above is achieved ''via'' an infinitary semilattice-theoretical statement URP (''Uniform Refinement Property''). If we want to disprove Schmidt's problem, the idea is (1) to prove that any generalized Boolean semilattice satisfies URP (which is easy), (2) that URP is preserved under homomorphic image under a weakly distributive homomorphism (which is also easy), and (3) that there exists a distributive (&or;,0)-semilattice of cardinality &alefsym;₂ that does not satisfy URP (which is difficult, and uses Kuratowski's Free Set Theorem).

The semilattice part of the result above is achieved ''via'' an infinitary semilattice-theoretical statement URP (''Uniform Refinement Property''). If we want to disprove Schmidt's problem, the idea is (1) to prove that any generalized Boolean semilattice satisfies URP (which is easy), (2) that URP is preserved under homomorphic image under a weakly distributive homomorphism (which is also easy), and (3) that there exists a distributive (∨,0)-semilattice of cardinality ℵ₂ that does not satisfy URP (which is difficult, and uses Kuratowski's Free Set Theorem).

Schematically, the construction in the theorem above can be described as follows. For a set Ω, we consider the partially ordered vector space ''E(Ω)'' defined by generators 1 and ''a_i,x'', for ''i<2'' and ''x'' in Ω, and relations ''a_0,x+a_1,x=1'', ''a_0,x &ge; 0'', and ''a_1,x &ge; 0'', for any ''x'' in Ω. By using a Skolemization of the theory of dimension groups, we can embed ''E(Ω)'' functorially into a [[Refinement monoid|dimension vector space]] ''F(Ω)''. The vector space counterexample of the theorem above is ''G=F(Ω)'', for any set Ω with at least &alefsym;₂ elements.

Schematically, the construction in the theorem above can be described as follows. For a set Ω, we consider the partially ordered vector space ''E(Ω)'' defined by generators 1 and ''a_i,x'', for ''i<2'' and ''x'' in Ω, and relations ''a_0,x+a_1,x=1'', ''a_0,x ≥ 0'', and ''a_1,x ≥ 0'', for any ''x'' in Ω. By using a Skolemization of the theory of dimension groups, we can embed ''E(Ω)'' functorially into a [[Refinement monoid|dimension vector space]] ''F(Ω)''. The vector space counterexample of the theorem above is ''G=F(Ω)'', for any set Ω with at least ℵ₂ elements.

This counterexample has been modified subsequently by Plo&#353;&#269;ica and T&#367;ma to a direct semilattice construction. For a (&or;,0)-semilattice, the larger semilattice ''R(S)'' is the (&or;,0)-semilattice freely generated by new elements ''t(a,b,c)'', for ''a, b, c'' in ''S'' such that ''c &le; a &or; b'', subjected to the only relations ''c=t(a,b,c) &or; t(b,a,c)'' and ''t(a,b,c) &le; a''. Iterating this construction gives the ''free distributive extension'' <math>D(S)=\bigcup(R^n(S)\mid n<\omega)</math> of ''S''. Now, for a set Ω, let ''L(Ω)'' be the (&or;,0)-semilattice defined by generators 1 and ''a_i,x'', for ''i<2'' and ''x'' in Ω, and relations ''a_0,x &or; a_1,x=1'', for any ''x'' in Ω. Finally, put ''G(Ω)=D(L(Ω))''.

This counterexample has been modified subsequently by Ploščica and Tůma to a direct semilattice construction. For a (∨,0)-semilattice, the larger semilattice ''R(S)'' is the (∨,0)-semilattice freely generated by new elements ''t(a,b,c)'', for ''a, b, c'' in ''S'' such that ''c ≤ a ∨ b'', subjected to the only relations ''c=t(a,b,c) ∨ t(b,a,c)'' and ''t(a,b,c) ≤ a''. Iterating this construction gives the ''free distributive extension'' <math>D(S)=\bigcup(R^n(S)\mid n<\omega)</math> of ''S''. Now, for a set Ω, let ''L(Ω)'' be the (∨,0)-semilattice defined by generators 1 and ''a_i,x'', for ''i<2'' and ''x'' in Ω, and relations ''a_0,x ∨ a_1,x=1'', for any ''x'' in Ω. Finally, put ''G(Ω)=D(L(Ω))''.

In most related works, the following ''uniform refinement property'' is used. It is a modification of the one introduced by Wehrung in 1998 and 1999.<br /><br />

In most related works, the following ''uniform refinement property'' is used. It is a modification of the one introduced by Wehrung in 1998 and 1999.

'''Definition (Plo&#353;&#269;ica, T&#367;ma, and Wehrung 1998).'''

Let ''e'' be an element in a (&or;,0)-semilattice ''S''. We say that the ''weak uniform refinement property'' WURP holds at ''e'', if for all families <math>(a_i)_{i\in I}</math> and <math>(b_i)_{i\in I}</math> of elements in ''S'' such that ''a_i &or; b_i=e'' for all ''i'' in ''I'', there exists a family <math>(c_{i,j}\mid (i,j)\in I\times I)</math> of elements of ''S'' such that the relations

'''Definition (Ploščica, Tůma, and Wehrung 1998).'''

• ''c_i,j &le; a_i,b_j'',<br />

Let ''e'' be an element in a (∨,0)-semilattice ''S''. We say that the ''weak uniform refinement property'' WURP holds at ''e'', if for all families <math>(a_i)_{i\in I}</math> and <math>(b_i)_{i\in I}</math> of elements in ''S'' such that ''a_i ∨ b_i=e'' for all ''i'' in ''I'', there exists a family <math>(c_{i,j}\mid (i,j)\in I\times I)</math> of elements of ''S'' such that the relations

• ''c_i,j &or;≤ a_ji &or; ,b_ij=e'',<br />

• ''c_i,kj &le;∨ ca_i,j&or; c∨ b_j,ki=e''<br />,

• ''c_i,k ≤ c_i,j∨ c_j,k''

hold for all ''i, j, k'' in ''I''. We say that ''S'' satisfies WURP, if WURP holds at every element of ''S''.

<br /><br />

hold for all ''i, j, k'' in ''I''. We say that ''S'' satisfies WURP, if WURP holds at every element of ''S''.

By building on Wehrung's abovementioned work on dimension vector spaces, Plo&#353;&#269;ica and T&#367;ma proved that WURP does not hold in ''G(Ω)'', for any set Ω of cardinality at least &alefsym;₂. Hence ''G(Ω)'' does not satisfy Schmidt's Condition. It is to be noted that all negative representation results mentioned here always make use of some ''uniform refinement property'', including the first one about dimension vector spaces.

However, the semilattices used in these negative results are relatively complicated. The following result, proved by Plo&#353;&#269;ica, T&#367;ma, and Wehrung in 1998, is more striking, because it shows examples of ''representable'' semilattices that do not satisfy Schmidt's Condition. We denote by F_'''V'''(Ω) the free lattice on Ω in '''V''', for any variety '''V''' of lattices.<br /><br />

By building on Wehrung's abovementioned work on dimension vector spaces, Ploščica and Tůma proved that WURP does not hold in ''G(Ω)'', for any set Ω of cardinality at least ℵ₂. Hence ''G(Ω)'' does not satisfy Schmidt's Condition. It is to be noted that all negative representation results mentioned here always make use of some ''uniform refinement property'', including the first one about dimension vector spaces.

'''Theorem (Plo&#353;&#269;ica, T&#367;ma, and Wehrung 1998).'''

The semilattice Con_c F_'''V'''(Ω) does not satisfy WURP, for any set Ω of cardinality at least &alefsym;₂ and any non-distributive variety '''V''' of lattices. Consequently, Con_c F_'''V'''(Ω) does not satisfy Schmidt's Condition.

<br /><br />

However, the semilattices used in these negative results are relatively complicated. The following result, proved by Ploščica, Tůma, and Wehrung in 1998, is more striking, because it shows examples of ''representable'' semilattices that do not satisfy Schmidt's Condition. We denote by F_'''V'''(Ω) the free lattice on Ω in '''V''', for any variety '''V''' of lattices.

It is proved by T&#367;ma and Wehrung in 2001 that Con_c F_'''V'''(Ω) is not isomorphic to Con_c ''L'', for any lattice ''L'' with permutable congruences. By using a slight weakening of WURP, this result is extended to arbitrary [[Universal algebra|algebras]] with permutable congruences by R&#367;&#382;i&#269;ka, T&#367;ma, and Wehrung in 2006. Hence, for example, if Ω has at least &alefsym;₂ elements, then Con_c F_'''V'''(Ω) is not isomorphic to the normal subgroup lattice of any group, or the submodule lattice of any module.

'''Theorem (Ploščica, Tůma, and Wehrung 1998).'''

== Solving CLP: the Erosion Lemma ==

The semilattice Con_c F_'''V'''(Ω) does not satisfy WURP, for any set Ω of cardinality at least ℵ₂ and any non-distributive variety '''V''' of lattices. Consequently, Con_c F_'''V'''(Ω) does not satisfy Schmidt's Condition.

The following recent theorem solves CLP.<br /><br />

It is proved by Tůma and Wehrung in 2001 that Con_c F_'''V'''(Ω) is not isomorphic to Con_c ''L'', for any lattice ''L'' with permutable congruences. By using a slight weakening of WURP, this result is extended to arbitrary [[Universal algebra|algebras]] with permutable congruences by Růžička, Tůma, and Wehrung in 2006. Hence, for example, if Ω has at least ℵ₂ elements, then Con_c F_'''V'''(Ω) is not isomorphic to the normal subgroup lattice of any group, or the submodule lattice of any module.

==Solving CLP: the Erosion Lemma==

The following recent theorem solves CLP.

'''Theorem (Wehrung 2007).'''

The semilattice ''G(Ω)'' is not isomorphic to Con_c ''L'' for any lattice ''L'', whenever the set Ω has at least &alefsym;ℵ_&omega;ω+1 elements.<br /><br />

Hence, the counterexample to CLP had been known for nearly ten years, it is just that nobody knew why it worked! All the results prior to the theorem above made use of some form of permutability of congruences. The difficulty was to find enough structure in congruence lattices of non-congruence-permutable lattices.

We shall denote by &epsilon;ε the `parity function' on the natural numbers, that is, &epsilon;ε(''n'')=''n'' mod 2, for any natural number ''n''.

We let ''L'' be an [[Universal algebra|algebra]] possessing a structure of semilattice (''L'',&or;∨) such that every congruence of ''L'' is also a congruence for the operation &or;∨ . We put

:<math> U\vee V=\{u\vee v \mid (u,v)\in U\times V\},\quad

\text{for all }U,V\subseteq L,

</math>

and we denote by Con_c^''U'' ''L'' the (&or;∨,0)-subsemilattice of Con_c ''L'' generated by all principal congruences Θ(''u'',''v'') ( = least congruence of ''L'' that identifies ''u'' and ''v''), where (''u'',''v'') belongs to ''U'' ×''U''. We put Θ⁺(''u'',''v'')=Θ(''u &or;∨ v'',''v''), for all ''u, v'' in ''L''.<br /><br />

'''The Erosion Lemma (Wehrung 2007).'''

Line 250 ⟶ 238:

\pmod{\theta_0\vee\theta_1}\quad\text{and}\quad

\theta_j\subseteq\alpha_j\cap\Theta_L^+(z_n,x_j),\text{ for all }j<2.

</math>

<br /><br />

(Observe the faint formal similarity with [[Resolution (logic)|first-order resolution]] in mathematical logic. Could this analogy be pushed further?)

The proof of the theorem above runs by setting a ''structure'' theorem for congruence lattices of semilattices---namelysemilattices—namely, the Erosion Lemma, against ''non-structure'' theorems for free distributive extensions ''G(Ω)'', the main one being called the ''Evaporation Lemma''. While the latter are technically difficult, they are, in some sense, predictable. Quite to the contrary, the proof of the Erosion Lemma is elementary and easy, so it is probably the strangeness of its statement that explains that it has been hidden for so long.

More is, in fact, proved in the theorem above: ''For any algebra L with a congruence-compatible structure of join-semilattice with unit and for any set Ω with at least &alefsym;ℵ_&omega;ω+1 elements, there is no weakly distributive homomorphism &mu;μ: Con_c L → G(Ω) containing 1 in its range''. In particular, CLP was, after all, not a problem of lattice theory, but rather of [[universal algebra]]---even—even more specifically, [[Semilattice|''semilattice theory'']]! These results can also be translated in terms of a ''uniform refinement property'', denoted by CLR in Wehrung's paper presenting the solution of CLP, which is noticeably more complicated than WURP.

Finally, the cardinality bound &alefsym;ℵ_&omega;ω+1 has been improved to the optimal bound &alefsym;ℵ₂ by R&#367;&#382;i&#269;kaRůžička.<br /><br />

'''Theorem (R&#367;&#382;i&#269;kaRůžička 2008).'''

The semilattice ''G(Ω)'' is not isomorphic to Con_c ''L'' for any lattice ''L'', whenever the set Ω has at least &alefsym;ℵ₂ elements.

<br /><br />

R&#367;&#382;i&#269;ka's proof follows the main lines of Wehrung's proof, except that it introduces an enhancement of [[Kuratowski's Free Set Theorem]], called there ''existence of free trees'', which it uses in the final argument involving the Erosion Lemma.

Růžička's proof follows the main lines of Wehrung's proof, except that it introduces an enhancement of [[Kuratowski's Free Set Theorem]], called there ''existence of free trees'', which it uses in the final argument involving the Erosion Lemma.

== A positive representation result for distributive semilattices ==

==A positive representation result for distributive semilattices==

The proof of the negative solution for CLP shows that the problem of representing distributive semilattices by compact congruences of lattices already appears for congruence lattices of ''semilattices''. The question whether the structure of [[partially ordered set]] would cause similar problems is answered by the following result.

<br /><br />

'''Theorem (Wehrung 2008).''' For any distributive (&or;,0)-semilattice ''S'', there are a (&and;,0)-semilattice ''P'' and a map &mu; : ''P'' × ''P'' → ''S'' such that the following conditions hold:

(1) ''x'' &le; ''y'' implies that &mu;(''x'',''y'')=0, for all ''x'', ''y'' in ''P''.

(2) &mu;(''x'',''z'') &le; &mu;(''x'',''y'') &or; &mu;(''y'',''z''), for all ''x'', ''y'', ''z'' in ''P''.

(3) For all ''x'' &ge; ''y'' in ''P'' and all &alpha;, &beta; in ''S'' such that &mu;(''x'',''y'') &le; &alpha; &or; &beta;, there are a positive integer ''n'' and elements ''x''=''z''₀ &ge; ''z''₁ &ge; ... &ge; ''z''_2''n''=''y'' such that &mu;(''z''_i'',''z''_i+1'') &le; &alpha; (resp., &mu;(''z''_i'',''z''_i+1'') &le; &beta;) whenever ''i'' < 2''n'' is even (resp., odd).

(4) ''S'' is generated, as a join-semilattice, by all the elements of the form &mu;(''x'',0), for ''x'' in ''P''.

Furthermore, if ''S'' has a largest element, then ''P'' can be assumed to be a lattice with a largest element.<br /><br />

It is not hard to verify that conditions (1)--(4) above imply the distributivity of ''S'', so the result above gives a ''characterization'' of distributivity for (&or;,0)-semilattices.

== References ==

* G.M. Bergman, ''Von Neumann regular rings with tailor-made ideal lattices'', Unpublished note (26 October 1986).

* [[Garrett Birkhoff|G. Birkhoff]], ''Lattice Theory'', rev. ed. Amer. Math. Soc. New York, 1948.

* [[Garrett Birkhoff|G. Birkhoff]] and O. Frink, ''Representations of lattices by sets'', Trans. Amer. Math. Soc. '''64''', no. 2 (1948), 299--316.

* S. Bulman-Fleming and K. McDowell, ''Flat semilattices'', Proc. Amer. Math. Soc. '''72''', no. 2 (1978), 228--232.

* K.P. Bogart, R. Freese, and J.P.S. Kung (editors), ''The Dilworth Theorems. Selected papers of Robert P. Dilworth'', Birkh&auml;user Verlag, Basel - Boston - Berlin, 1990. xxvi+465 p. ISBN 0-8176-3434-7

* [[Hans Dobbertin|H. Dobbertin]], ''Refinement monoids, Vaught monoids, and Boolean algebras'', Math. Ann. '''265''', no. 4 (1983), 473--487.

* [[Hans Dobbertin|H. Dobbertin]], ''Vaught measures and their applications in lattice theory'', J. Pure Appl. Algebra '''43''', no. 1 (1986), 27--51.

* E.G. Effros, D.E. Handelman and C.-L. Shen, ''Dimension groups and their affine representations'', Amer. J. Math. '''102''', no. 2 (1980), 385--407.

* G.A. Elliott, ''On the classification of inductive limits of sequences of semisimple finite-dimensional algebras'', J. Algebra '''38''', no. 1 (1976), 29--44.

* Ershov, Ju.L., Theory of Numerations (Russian), Monographs in Mathematical Logic and Foundations of Mathematics, Nauka, Moscow, 1977. 416 p.

* R. Freese, W.A. Lampe, and W. Taylor, ''Congruence lattices of algebras of fixed similarity type. I'', Pacific J. Math. '''82''' (1979), 59--68.

* N. Funayama and T. Nakayama, ''On the distributivity of a lattice of lattice congruences'', Proc. Imp. Acad. Tokyo '''18''' (1942), 553--554.

* K.R. Goodearl, von Neumann regular rings. Second edition. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. xviii+412 p. ISBN 0-89464-632-X

* K.R. Goodearl and D. Handelman, ''Simple self-injective rings'', Comm. Algebra '''3''', no. 9 (1975), 797--834.

* K.R. Goodearl and D. Handelman, ''Tensor products of dimension groups and K₀ of unit-regular rings'', Canad. J. Math. '''38''', no. 3 (1986), 633--658.

* K.R. Goodearl and F. Wehrung, ''Representations of distributive semilattices in ideal lattices of various algebraic structures'', Algebra Universalis '''45''', no. 1 (2001), 71--102.

* G. Gr&auml;tzer, General Lattice Theory. Second edition, new appendices by the author with B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E.T. Schmidt, S.E. Schmidt, F. Wehrung, and R. Wille. Birkh&auml;user Verlag, Basel, 1998. xx+663 p. ISBN 3-7643-5239-6

* G. Gr&auml;tzer, The Congruences of a Finite Lattice: a ''Proof-by-Picture'' Approach, Birkh&auml;user Boston, 2005. xxiii+281 p. ISBN 978-0-8176-3224-3; 0-8176-3224-7

* G. Gr&auml;tzer, H. Lakser, and F. Wehrung, ''Congruence amalgamation of lattices'', Acta Sci. Math. (Szeged) '''66''' (2000), 339--358.

* G. Gr&auml;tzer and E.T. Schmidt, ''On congruence lattices of lattices'', Acta Math. Sci. Hungar. '''13''' (1962), 179--185.

* G. Gr&auml;tzer and E.T. Schmidt, ''Characterizations of congruence lattices of abstract algebras'', Acta Sci. Math. (Szeged) '''24''' (1963), 34--59.

* G. Gr&auml;tzer and E.T. Schmidt, ''Finite lattices and congruences. A survey'', Algebra Universalis '''52''', no. 2-3 (2004), 241--278.

* P.A. Grillet, ''Directed colimits of free commutative semigroups'', J. Pure Appl. Algebra '''9''', no. 1 (1976), 73--87.

* A.P. Huhn, ''On the representation of algebraic distributive lattices II'', Acta Sci. Math. (Szeged) '''53''' (1989), 3--10.

* A.P. Huhn, ''On the representation of algebraic distributive lattices III'', Acta Sci. Math. (Szeged) '''53''' (1989), 11--18.

* K.A. Kearnes and A. Szendrei, ''The relationship between two commutators'', Internat. J. Algebra Comput. '''8''', no. 4 (1998), 497--531.

* [[Kazimierz Kuratowski|C. Kuratowski]], ''Sur une caract&eacute;risation des alephs'', Fund. Math. '''38''' (1951), 14--17.

* W.A. Lampe, ''Congruence lattices of algebras of fixed similarity type. II'', Pacific J. Math. '''103''' (1982), 475--508.

* [[John von Neumann|J. von Neumann]], ''On regular rings'', Proc. Nat. Acad. Sci. U.S.A. '''22'''(12) (December 1936), 707–713.

* M. Plo&#353;&#269;ica and J. T&#367;ma, ''Uniform refinements in distributive semilattices'', Contributions to General Algebra '''10''', Proceedings of the Klagenfurt Conference, May 29 -- June 1, 1997. Verlag Johannes Heyn, Klagenfurt 1998.

* M. Plo&#353;&#269;ica, J. T&#367;ma, and F. Wehrung, ''Congruence lattices of free lattices in nondistributive varieties'', Colloq. Math. '''76''', no. 2 (1998), 269--278.

* P. Pudl&aacute;k, ''On congruence lattices of lattices'', Algebra Universalis '''20''' (1985), 96--114.

* P. R&#367;&#382;i&#269;ka, ''Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem'', Israel J. Math. '''142''' (2004), 1--28.

* P. R&#367;&#382;i&#269;ka, ''Liftings of distributive lattices by locally matricial algebras with respect to the Id_c functor'', Algebra Universalis '''55''', no. 2-3 (August 2006), 239--257.

* P. R&#367;&#382;i&#269;ka, ''Free trees and the optimal bound in Wehrung's theorem'', Fund. Math. <B>198</B> (2008), 217--228.

* P. R&#367;&#382;i&#269;ka, J. T&#367;ma, and F. Wehrung, ''Distributive congruence lattices of congruence-permutable algebras'', J. Algebra '''311''' (2007), 96--116.

* E.T. Schmidt, ''Zur Charakterisierung der Kongruenzverb&auml;nde der Verb&auml;nde'', Mat. Casopis Sloven. Akad. Vied '''18''' (1968), 3--20.

* E.T. Schmidt, ''The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice'', Acta Sci. Math. (Szeged) '''43''' (1981), 153--168.

* E.T. Schmidt, A Survey on Congruence Lattice Representations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], '''42'''. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1982. 115 p.

* R.T. Shannon, ''Lazard's theorem in algebraic categories'', Algebra Universalis '''4''' (1974), 226--228.

* [[Alfred Tarski|A. Tarski]], Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni J&oacute;nsson and Alfred Tarski. Oxford University Press, New York, N. Y., 1949. xii+326 p.

* J. T&#367;ma, ''On the existence of simultaneous representations'', Acta Sci. Math. (Szeged) '''64''' (1998), 357--371.

* J. T&#367;ma and F. Wehrung, ''Simultaneous representations of semilattices by lattices with permutable congruences'', Internat. J. Algebra Comput. '''11''', no. 2 (2001), 217--246.

* J. T&#367;ma and F. Wehrung, ''A survey of recent results on congruence lattices of lattices'', Algebra Universalis '''48''', no. 4 (2002), 439--471.

* J. T&#367;ma and F. Wehrung, ''Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties'', Internat. J. Algebra Comput. '''16''', no. 3 (2006), 541--550.

'''Theorem (Wehrung 2008).''' For any distributive (∨,0)-semilattice ''S'', there are a (∧,0)-semilattice ''P'' and a map μ : ''P'' × ''P'' → ''S'' such that the following conditions hold:

* F. Wehrung, ''Non-measurability properties of interpolation vector spaces'', Israel J. Math. '''103''' (1998), 177--206.

(1) ''x'' ≤ ''y'' implies that μ(''x'',''y'')=0, for all ''x'', ''y'' in ''P''.

* F. Wehrung, ''The dimension monoid of a lattice'', Algebra Universalis '''40''', no. 3 (1998), 247--411.

(2) μ(''x'',''z'') ≤ μ(''x'',''y'') ∨ μ(''y'',''z''), for all ''x'', ''y'', ''z'' in ''P''.

* F. Wehrung, ''A uniform refinement property for congruence lattices'', Proc. Amer. Math. Soc. '''127''', no. 2 (1999), 363--370.

(3) For all ''x'' ≥ ''y'' in ''P'' and all α, β in ''S'' such that μ(''x'',''y'') ≤ α ∨ β, there are a positive integer ''n'' and elements ''x''=''z''₀ ≥ ''z''₁ ≥ ... ≥ ''z''_2''n''=''y'' such that μ(''z''_i'',''z''_i+1'') ≤ α (resp., μ(''z''_i'',''z''_i+1'') ≤ β) whenever ''i'' < 2''n'' is even (resp., odd).

* F. Wehrung, ''Representation of algebraic distributive lattices with &alefsym;₁ compact elements as ideal lattices of regular rings'', Publ. Mat. (Barcelona) '''44''' (2000), 419--435.

(4) ''S'' is generated, as a join-semilattice, by all the elements of the form μ(''x'',0), for ''x'' in ''P''.

* F. Wehrung, ''Forcing extensions of partial lattices'', J. Algebra '''262''', no. 1 (2003), 127--193.

Furthermore, if ''S'' has a largest element, then ''P'' can be assumed to be a lattice with a largest element.

* F. Wehrung, ''Semilattices of finitely generated ideals of exchange rings with finite stable rank'', Trans. Amer. Math. Soc. '''356''', no. 5 (2004), 1957--1970.

It is not hard to verify that conditions (1)–(4) above imply the distributivity of ''S'', so the result above gives a ''characterization'' of distributivity for (∨,0)-semilattices.

* F. Wehrung, ''Poset representations of distributive semilattices'', Internat. J. Algebra Comput. <B>18</B>, no. 2 (March 2008), 321--356.

==References==

* F. Wehrung, ''A solution to Dilworth's congruence lattice problem'', Adv. Math. <B>216</B>, no. 2 (2007), 610--625.

*G.M. Bergman, ''Von Neumann regular rings with tailor-made ideal lattices'', Unpublished note (26 October 1986).

*[[Garrett Birkhoff|G. Birkhoff]], ''Lattice Theory'', rev. ed. Amer. Math. Soc. New York, 1948.

*[[Garrett Birkhoff|G. Birkhoff]] and O. Frink, ''Representations of lattices by sets'', Trans. Amer. Math. Soc. '''64''', no. 2 (1948), 299–316.

*S. Bulman-Fleming and K. McDowell, ''Flat semilattices'', Proc. Amer. Math. Soc. '''72''', no. 2 (1978), 228–232.

*K.P. Bogart, R. Freese, and J.P.S. Kung (editors), ''The Dilworth Theorems. Selected papers of Robert P. Dilworth'', Birkhäuser Verlag, Basel - Boston - Berlin, 1990. xxvi+465 p. ISBN 0-8176-3434-7

*[[Hans Dobbertin|H. Dobbertin]], ''Refinement monoids, Vaught monoids, and Boolean algebras'', Math. Ann. '''265''', no. 4 (1983), 473–487.

*[[Hans Dobbertin|H. Dobbertin]], ''Vaught measures and their applications in lattice theory'', J. Pure Appl. Algebra '''43''', no. 1 (1986), 27–51.

*E.G. Effros, D.E. Handelman and C.-L. Shen, ''Dimension groups and their affine representations'', Amer. J. Math. '''102''', no. 2 (1980), 385–407.

*G.A. Elliott, ''On the classification of inductive limits of sequences of semisimple finite-dimensional algebras'', J. Algebra '''38''', no. 1 (1976), 29–44.

*Ershov, Ju.L., Theory of Numerations (Russian), Monographs in Mathematical Logic and Foundations of Mathematics, Nauka, Moscow, 1977. 416 p.

*R. Freese, W.A. Lampe, and W. Taylor, ''Congruence lattices of algebras of fixed similarity type. I'', Pacific J. Math. '''82''' (1979), 59–68.

*N. Funayama and T. Nakayama, ''On the distributivity of a lattice of lattice congruences'', Proc. Imp. Acad. Tokyo '''18''' (1942), 553–554.

*K.R. Goodearl, von Neumann regular rings. Second edition. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. xviii+412 p. ISBN 0-89464-632-X

*K.R. Goodearl and D. Handelman, ''Simple self-injective rings'', Comm. Algebra '''3''', no. 9 (1975), 797–834.

*K.R. Goodearl and D. Handelman, ''Tensor products of dimension groups and K₀ of unit-regular rings'', Canad. J. Math. '''38''', no. 3 (1986), 633–658.

*K.R. Goodearl and F. Wehrung, ''Representations of distributive semilattices in ideal lattices of various algebraic structures'', Algebra Universalis '''45''', no. 1 (2001), 71–102.

*G. Grätzer, General Lattice Theory. Second edition, new appendices by the author with B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E.T. Schmidt, S.E. Schmidt, F. Wehrung, and R. Wille. Birkhäuser Verlag, Basel, 1998. xx+663 p. ISBN 3-7643-5239-6

*G. Grätzer, The Congruences of a Finite Lattice: a ''Proof-by-Picture'' Approach, Birkhäuser Boston, 2005. xxiii+281 p. ISBN 978-0-8176-3224-3; 0-8176-3224-7

*G. Grätzer, H. Lakser, and F. Wehrung, ''Congruence amalgamation of lattices'', Acta Sci. Math. (Szeged) '''66''' (2000), 339–358.

*G. Grätzer and E.T. Schmidt, ''On congruence lattices of lattices'', Acta Math. Sci. Hungar. '''13''' (1962), 179–185.

*G. Grätzer and E.T. Schmidt, ''Characterizations of congruence lattices of abstract algebras'', Acta Sci. Math. (Szeged) '''24''' (1963), 34–59.

*G. Grätzer and E.T. Schmidt, ''Finite lattices and congruences. A survey'', Algebra Universalis '''52''', no. 2-3 (2004), 241–278.

*P.A. Grillet, ''Directed colimits of free commutative semigroups'', J. Pure Appl. Algebra '''9''', no. 1 (1976), 73–87.

*A.P. Huhn, ''On the representation of algebraic distributive lattices II'', Acta Sci. Math. (Szeged) '''53''' (1989), 3–10.

*A.P. Huhn, ''On the representation of algebraic distributive lattices III'', Acta Sci. Math. (Szeged) '''53''' (1989), 11–18.

*K.A. Kearnes and A. Szendrei, ''The relationship between two commutators'', Internat. J. Algebra Comput. '''8''', no. 4 (1998), 497–531.

*[[Kazimierz Kuratowski|C. Kuratowski]], ''Sur une caractérisation des alephs'', Fund. Math. '''38''' (1951), 14–17.

*W.A. Lampe, ''Congruence lattices of algebras of fixed similarity type. II'', Pacific J. Math. '''103''' (1982), 475–508.

*[[John von Neumann|J. von Neumann]], ''On regular rings'', Proc. Nat. Acad. Sci. U.S.A. '''22'''(12) (December 1936), 707–713.

*M. Ploščica and J. Tůma, ''Uniform refinements in distributive semilattices'', Contributions to General Algebra '''10''', Proceedings of the Klagenfurt Conference, May 29 – June 1, 1997. Verlag Johannes Heyn, Klagenfurt 1998.

*M. Ploščica, J. Tůma, and F. Wehrung, ''Congruence lattices of free lattices in nondistributive varieties'', Colloq. Math. '''76''', no. 2 (1998), 269–278.

*P. Pudlák, ''On congruence lattices of lattices'', Algebra Universalis '''20''' (1985), 96–114.

*P. Růžička, ''Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem'', Israel J. Math. '''142''' (2004), 1–28.

*P. Růžička, ''Liftings of distributive lattices by locally matricial algebras with respect to the Id_c functor'', Algebra Universalis '''55''', no. 2-3 (August 2006), 239–257.

*P. Růžička, ''Free trees and the optimal bound in Wehrung's theorem'', Fund. Math. '''198''' (2008), 217–228.

*P. Růžička, J. Tůma, and F. Wehrung, ''Distributive congruence lattices of congruence-permutable algebras'', J. Algebra '''311''' (2007), 96–116.

*E.T. Schmidt, ''Zur Charakterisierung der Kongruenzverbände der Verbände'', Mat. Casopis Sloven. Akad. Vied '''18''' (1968), 3–20.

*E.T. Schmidt, ''The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice'', Acta Sci. Math. (Szeged) '''43''' (1981), 153–168.

*E.T. Schmidt, A Survey on Congruence Lattice Representations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], '''42'''. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1982. 115 p.

*R.T. Shannon, ''Lazard's theorem in algebraic categories'', Algebra Universalis '''4''' (1974), 226–228.

*[[Alfred Tarski|A. Tarski]], Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni Jónsson and Alfred Tarski. Oxford University Press, New York, N. Y., 1949. xii+326 p.

*J. Tůma, ''On the existence of simultaneous representations'', Acta Sci. Math. (Szeged) '''64''' (1998), 357–371.

*J. Tůma and F. Wehrung, ''Simultaneous representations of semilattices by lattices with permutable congruences'', Internat. J. Algebra Comput. '''11''', no. 2 (2001), 217–246.

*J. Tůma and F. Wehrung, ''A survey of recent results on congruence lattices of lattices'', Algebra Universalis '''48''', no. 4 (2002), 439–471.

*J. Tůma and F. Wehrung, ''Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties'', Internat. J. Algebra Comput. '''16''', no. 3 (2006), 541–550.

*F. Wehrung, ''Non-measurability properties of interpolation vector spaces'', Israel J. Math. '''103''' (1998), 177–206.

*F. Wehrung, ''The dimension monoid of a lattice'', Algebra Universalis '''40''', no. 3 (1998), 247–411.

*F. Wehrung, ''A uniform refinement property for congruence lattices'', Proc. Amer. Math. Soc. '''127''', no. 2 (1999), 363–370.

*F. Wehrung, ''Representation of algebraic distributive lattices with ℵ₁ compact elements as ideal lattices of regular rings'', Publ. Mat. (Barcelona) '''44''' (2000), 419–435.

*F. Wehrung, ''Forcing extensions of partial lattices'', J. Algebra '''262''', no. 1 (2003), 127–193.

*F. Wehrung, ''Semilattices of finitely generated ideals of exchange rings with finite stable rank'', Trans. Amer. Math. Soc. '''356''', no. 5 (2004), 1957–1970.

*F. Wehrung, ''Poset representations of distributive semilattices'', Internat. J. Algebra Comput. '''18''', no. 2 (March 2008), 321–356.

*F. Wehrung, ''A solution to Dilworth's congruence lattice problem'', Adv. Math. '''216''', no. 2 (2007), 610–625.

[[category:Lattice theory]]