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.
[5] 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.
The (∨,0)-semilattice of compact elements of Con A is denoted by Conc A, and it is sometimes called the congruence semilattice of A.
Semilattice-theoretical formulation of CLP: Is every distributive (∨,0)-semilattice isomorphic to the congruence semilattice of some lattice?
The functor Conc, defined on all algebras of a given signature, to all (∨,0)-semilattices, preserves direct limits.
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 join-congruence of B.
It is based on the following result, Fact 4, p. 100 in Pudlák's 1985 paper,[13] obtained earlier by Yuri L. Ershov as the main theorem in Section 3 of the Introduction of his 1977 monograph.
While the problem whether this could be done in general remained open for about 20 years, Pudlák could prove it for distributive lattices with zero, thus extending one of Schmidt's results by providing a functorial solution.
Every distributive (∨,0)-semilattice of cardinality at most ℵ1 is isomorphic to (1) Conc L, for some locally finite, relatively complemented modular lattice L (Tůma 1998 and Grätzer, Lakser, and Wehrung 2000).
[20][21] (2) The semilattice of finitely generated two-sided ideals of some (not necessarily unital) von Neumann regular ring (Wehrung 2000).
[23] (5) The submodule lattice of some right module over a (non-commutative) ring (Růžička, Tůma, and Wehrung 2007).
[23] We recall that for a (unital, associative) ring R, we denote by V(R) the (conical, commutative) monoid of isomorphism classes of finitely generated projective right R-modules, see here for more details.
The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.
Bergman proves in a well-known unpublished note from 1986[25] that any at most countable distributive (∨,0)-semilattice is isomorphic to Idc R, for some locally matricial ring R (over any given field).
This result is extended to semilattices of cardinality at most ℵ1 in 2000 by Wehrung,[22] by keeping only the regularity of R (the ring constructed by the proof is not locally matricial).
[26] 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,[27] yields the following result.
[28] Is the positive cone of any dimension group with order-unit isomorphic to V(R), for some von Neumann regular ring R?
There exists a dimension vector space G over the rationals with order-unit whose positive cone G+ is not isomorphic to V(R), for any von Neumann regular ring R, and is not measurable in Dobbertin's sense.
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.
The semilattice part of the result above is achieved via an infinitary semilattice-theoretical statement URP (Uniform Refinement Property).
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
All negative representation results mentioned here always make use of some uniform refinement property, including the first one about dimension vector spaces.
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.
By using a slight weakening of WURP, this result is extended to arbitrary algebras with permutable congruences by Růžička, Tůma, and Wehrung in 2007.
, for j<2, such that (Observe the faint formal similarity with first-order resolution in mathematical logic.
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.
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.
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.
The question whether the structure of partially ordered set would cause similar problems is answered by the following result.