Congruence Lattice Problem - A First Application of Kuratowski's Free Set Theorem

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.

Theorem (Wehrung 1998). 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. 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 ℵ₂.

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 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 (∨,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 ≥ 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 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 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 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.

Definition (Ploščica, Tůma, and Wehrung 1998). 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 and of elements in S such that a_i ∨ b_i=e for all i in I, there exists a family of elements of S such that the relations

• c_i,j ≤ a_i,b_j,

• c_i,j ∨ a_j ∨ b_i=e,

• 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.

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.

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.

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

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 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.