Basic Algebraic Applications of Stationary Sets?

Question

Background: I've been working my way through Thomas Jech's "Set Theory" because I'm working on some problems that have the potential to be logically independent of the usual axioms, or at least involve some hard set-theory about infinite sets, and I want to (eventually) better understand some of the set-theoretical techniques and methods that can be used to show such independence.

The first seven chapters of Jech's book have clear applications in my field of expertise, ring theory. I was already quite familiar with much of the material because of this fact.

Chapter 1: The basic axioms of set theory are used all the time in the language of ring theory, allowing for the formation of simple objects like unions, products, sequences, etc.

Chapter 2: The ordinal numbers have applications in infinitary constructions, and anywhere induction can be pushed further. For instance, one can define the "higher Wedderburn radicals" as a transfinite sequence of ideals which eventually stabilize at the prime radical.

Chapter 3: Everyone is familiar with how cardinal numbers give a very easy way to determine whether two objects are the same size, which helps prevent the existence of isomorphisms. Thus, this is a sort of first check one does to prove non-isomorphisms. There are other uses.

Chapter 4: Real numbers are a ring, with lots of interesting properties. What more needs to be said?

Chapter 5: The axiom of choice turns into Zorn's lemma. Using it you can prove all sorts of neat facts about existence of maximal ideals, algebraic closures, etc.

Chapter 6: Well-foundedness of relations comes up when trying to force processes to stop. One application I'm familiar with is in generalizing power series rings to allow for more products, but still requiring a "bottom" so that multiplication is well-defined.

Chapter 7: Boolean algebras, not much needs to be said here.

However, I'm just not familiar with any immediate uses of stationary sets in ring theory. This bothers me because (1) they seem quite useful in set theory, and (2) Jensen's Diamond, which settled the Whitehead problem (and was one reason I jumped into this current project) is stated in terms of stationary sets.

So, here is my formal question.

Question: What basic applications do stationary sets (in and of themselves, without appealing to axioms independent of ZFC) have in algebra? If applications to ring theory specifically could be provided, that would be even better.

Answer 1

Any group $G$ of size $\aleph_1$ is the union of an increasing smooth chain of countable groups $\langle G_\alpha\mid\alpha<\aleph_1\rangle$. Often times, $G$ has property $P$ iff $\{\alpha<\aleph_1\mid G_\alpha\text{ has property }Q\}$ is not stationary, or iff $\{\alpha<\aleph_1\mid G_{\alpha+1}/G_\alpha\text{ has property }Q\}$ is stationary, etc'. Some such examples may be found in Sections 5 and 6 of http://www.ams.org/mathscinet-getitem?mr=476511.

Answer 2

Stationary sets are exactly the positive-measure sets with respect to the club filter, which is a very natural measure on the subsets of $\omega_1$ or on higher cardinals with uncountable cofinality. Thus, they provide a notion of largeness, which could be used to provide a notion of large subalgebras.

For example, the non-stationary sets in $\omega_1$ form an ideal NS, whose quotient $P(\omega_1)/NS$ is a highly studied Boolean algebra in set theory. The properties of this algebra, such as whether it is precipitous or saturated, are connected with deep concepts in set theory, including large cardinals.

But you emphasized basic applications, so let me tell you one way of thinking about stationarity. This amounts to a fundamentally algebraic characterization of stationarity.

Theorem. A set $S\subseteq\omega_1$ is stationary if and only if for every algebra $\langle\omega_1,f_1,f_2,\ldots\rangle$ on $\omega_1$, allowing countably many functions of finite arity, there is an element $\gamma\in S$ such that $\gamma$ is a subalgebra.

Proof. The point is that every algebra has a closed unbounded collection of $\gamma$ that are closed under the functions of the algebra. And conversely, for every closed unbounded set, there is a function whose closure points are in the club: the function mapping every element to the next element of the club. Since a set is stationary just in case it has elements of every club set, one can also say that $S$ is stationary just in case it provides subalgebras for every algebra. $\Box$

The theorem generalizes to higher cardinals. This theorem is often used in set theory not in the context of rings, but rather by expanding a given structure by Skolem functions, so that the subalgebras of the expansion correspond to elementary substructures of the original structure. In this way, the stationary sets are those that provide elementary substructures of any given algebra.

The theorem also generalizes in a very attractive and natural way to the concept of generalized stationary, in the structure $P_{\omega_1}(X)$, consisting of the countable subsets of $X$. Here, a set $S\subseteq P_{\omega_1}(X)$ is defined to be stationary just in case for every algebra on $X$, there is a subalgeba in $S$.

Answer 3

One more application is in the automorphism tower problem.

If $G$ is a centreless group, there is a natural embedding of $G$ into $Aut(G).$ This can be iterated to give a transfinite continuous sequence $(G_α)$ (the "automorphism tower''), where $G_{α+1}≅Aut(G_α)$. Let $\tau(G)$ denote the least $\alpha$ such that $G_α=G_β$ for all $β>α$.

By a theorem of Simon Thomas, for any $G$ as above, $\tau(G) \leq (2^{|G|})^+$.

Then using Fodor's lemma (where stationarity is essential in it), one can in fact show that $\tau(G) < (2^{|G|})^+$. See The automorphism tower problem. II.

Answer 4

In his old paper on the Whitehead problem, Shelah proves that if $p$ is a prime number, $\kappa$ is uncountable regular, then there are $2^\kappa$ Abelian $p$-groups of cardinality $\kappa$, with no element of infinite height, none of them isomorphic to a subgroup to the other. This was a problem of Fuchs.