On a question of Herzog and Rothmaler

A.Pillay  M.Ziegler

1 Introduction

Herzog and Rothmaler gave the following purely topological characterization of stable theories. (See the exercises 11.3.4 – 11.3.7 in [2]).

A complete theory T is stable iff for any model M and any extension M ⊂ B the restriction map S(B) → S(M) has a continuous section.

In fact, if T is stable, taking the unique non-forking extension defines a continuous section of S(B) → S(A) for all subsets A of B, provided A is algebraically closed in T^{eq}. Herzog and Rothmaler asked, if, for stable T, there is a continuous section for any subset A of B. Or, equivalently, if for any A, S(acl^{eq}(A)) → S(A) has a continuous section .

This is an interesting problem, also for unstable T. Is it true that for any T and any set of parameters A the restriction map S(acl(A)) → S(A) has a continuous section? We answer the question by the following two theorems.

Theorem 1 Let A be a subset of a model of T. Assume that the Boolean algebra of acl(A)–definable formulas is generated by

• some countable set of formulas,

• all A–definable formulas,

• all formulas which are atomic over acl(A).

Then S(acl(A)) → S(A) has a continuous section.

The conditions of the theorems are satisfied if, for example, L and A are countable, or, if there are only countably many non–isolated types over acl(A).

Theorem 2 There is a theory of Morley rank 2 and Morley degree 1 such that S(acl(∅)) → S(∅) has no continuous section.

The second author thanks Sabine Koppelberg and Philipp Rothmaler for several discussions.

2 Proof of Theorem 1

Theorem 1 follows immediately from the next lemma. (Note that the map S(acl(A)) → S(A) is always open).

Lemma 3 Let A be a subalgebra of the Boolean algebra B such that the projection of Stone spaces S(B) → S(A) is open. Assume that B can be generated by

• some countable subalgebra C of B,

• the elements of A,

• all atoms of B.

Then the projection has a continuous section σ : S(A) → S(B)

Without atoms as generators, the lemma is well–known. See for example Proposition 2.9 in S. Koppelberg’s chapter on projective Boolean algebras in [1].

Proof: That the projection S(B) → S(A) is open means that for each b there is a smallest element π(a) of A which contains b.^{1}

We first define a homomorphism h : C → A which satisfies

h(c) ⊂ π(c)  (1)

for each c ∈ C. Since C is countable, it is enough to show that we can extend any h defined on a finite subalgebra D to any finite extension D′. Let d_{1}, …, d_{n} be the atoms of D. We can assume that D′ just splits d_{1} into two new atoms d′ and d′′. Since h(d_{1}) ⊂ π(d_{1}) = π(d′) ∪ π(d′′), we can extend h to D′ by setting h(d′) = π(d′) ∩ h(d_{1}) and h(d′′) = h(d_{1}) \ h(d′).

The condition (1) is equivalent to

c ⊂ a ⇒ h(c) ⊂ a  (2)

for all c ∈ C and a ∈ A. Let C′ be the subalgebra generated by C and A. (2) implies that we can extend h (uniquely) to a homomorphism h : C′ → A which is the identity on A.

We have constructed so far a continuous section τ : S(A) → S(C′). We claim that any p ∈ S(C′) which has two different extensions to S(B) is projected to an isolated point of S(A). Indeed, let q_{1} and q_{2} be two extension of p. Since B is generated over C′ by atoms, q_{1} and q_{2} can be separated by an atom of B, so one, say q_{1}, is isolated. Since the projection from S(B) → S(A) is open, q_{1} is projected to an isolated point of S(A).

Now we can apply the next lemma, which shows that any section σ′ : S(C′) → S(B) gives a continuous section σ = σ′ ∘ τ : S(A) → S(B). QED.

1A is then called a relatively complete subalgebra of B.

Lemma 4 Let τ : X → Y and π : Z → Y be continuous maps, π surjective and closed. Assume that for every non-isolated element x ∈ X, τ(x) has exactly one preimage in Z. Then for every section of σ′ of π the composition σ′ ∘τ : X → Z is continuous.

Proof: Let σ′ : Y → Z be any section of π and σ = σ′ ∘ τ. Consider a closed subset C of Z. Since π is closed, π(C) is closed and therefore also τ^{−1}(π(C)). σ^{−1}(C) is a subset of τ^{−1}(π(C)). The difference consists of isolated points, whence also σ^{−1}(C) is closed. QED.

3 Proof of Theorem 2

We consider the language

L = {p_{i} | i < ω} ∪ {P_{α} | α < 2^{ω}} ∪ {E_{α} | α < 2^{ω}}

with unary predicates p_{i} and P_{α} and binary relations E_{α}. Fix a family (X_{α})_{α<2^{ω}} of infinite subsets of ω which are pairwise almost disjoint.

Consider the (incomplete) theory T^{∗} which says that

a) The p_{i} are pairwise disjoint and each of them has exactly two elements

b) p_{i} is a subset of P_{α} if i ∈ X_{α} and otherwise disjoint from P_{α}.

c) The intersection of two different P_{α} and P_{β} is the union of the (finitely many) p_{i} with i ∈ X_{α} ∩ X_{β}.

d) E_{α} is an equivalence relation on P_{α}, where it has two classes and cuts every p_{i} contained in P_{α} in two pieces.

T^{∗} is incomplete since it does not tell us how the E_{α} interact on the intersections of the P_{α}. But it is already clear that each completion has Morley rank 2 and degree 1. This is because the (parametrically) definable sets are, up to finite number of elements, Boolean combinations of the E_{α}–classes, which are infinite and almost disjoint.

To describe a completion T we construct a model M of T^{∗}. We have not much choice for the p_{i} and the P_{α}: We let ω be the universe of M and define p_{i}(M) = {2i, 2i + 1} and

P_{α}(M) = ⋃_{i∈X_{α}} p_{i}(M).

Fix a list (f_{α})_{α<2^{ω}} of all choice functions which pick an element from each p_{i}:

f_{α}(i) ∈ p_{i}(M).

We complete the construction of M by choosing, for each α, the two classes of E_{α} in such a way that Axiom d) is satisfied and both classes contain infinitely many elements of

{f_{α}(i) | i ∈ X_{α}}.

It is clear from the construction that M has an automorphism which swaps the elements of each p_{i}. This shows that the only ∅–definable sets are Boolean combinations of the p_{i} and the P_{α}. Also it is clear that M = acl(∅).

Now we show that there is no continuous section σ : S(∅) → S(M). We need some notation.

Let p_{i}^{∗} ∈ S(∅) denote the complete type axiomatized by p_{i}(x), P_{α}^{∗} ∈ S(∅) the unique non–algebraic type containing P_{α}(x).

For each element a of M let q_{a} be the type of a over M. Let C_{α}^{1} and C_{α}^{2} be the two classes of E_{α} in P_{α}. These are two strongly minimal sets and we denote by e_{α}^{1} and e_{α}^{2} the corresponding strongly minimal types in S(M).

Now suppose that σ is a continuous section. Then each σ(p_{i}) is q_{a} for some a ∈ p_{i}(M). So there is an α < 2^{ω} such that

σ(p_{i}) = q_{f_{α}(i)}

for all i.

Let, for j = 1, 2, I^{j} be the set of all i ∈ X_{α} with f_{α}(i) in C_{α}^{j}. I^{1} and I^{2} form a partition of X_{α} and both sets are infinite by construction.

The sequence (p_{i}^{∗})_{i∈I^{1}} converges (in S(∅)) to P_{α}^{∗}, the sequence

(σ(p_{i}^{∗}))_{i∈I^{1}} = (q_{f_{α}(i)})_{i∈I^{1}}

converges in S(M) towards e_{α}^{1}. Whence σ(P_{α}^{∗}) = e_{α}^{1}. In the same way it follows that σ(P_{α}^{∗}) = e_{α}^{2}, which is a contradiction.

References

[1] J. Donald Monk and Robert Bonnet, editors. Handbook of Boolean Algebras, volume 3. North–Holland, 1989.

[2] Philipp Rothmaler. Introduction to Model Theory. Gordon & Breach Science Publishers, 2000.

Department of Mathematics University of Illinois at Urbana–Champaign

Mathematisches Institut Universität Freiburg, Germany