Finite covers of disintegrated sets

Martin Ziegler

January 1992

1 Introduction

Let X be structure. A 2-sorted structure Ē = (E, X) is a finite cover of X if

1. there is 0-definable surjection π : E → X whose fibers E_{x} = π^{−1}(x) have a bounded finite cardinality

2. There is no new structure induced on X, i.e. every in Ē 0-definable relation on X is 0definable in X.

In this note we determine what the finite covers are in the simplest of all cases: where X is a countable set without structure, a disintegrated set.

Remark 1.1 Let Ē be a finite cover of X. Since Ē is algebraic over X every automorphism of X extends to an automorphism of Ē. This shows that a finite cover of an ω–categorical structure is also ω–categorical.

Since countable ω–categorical structures are - up to interdefinability - determined by their automorphism groups we will instead of giving the actual structures of the finite covers give their automorphism groups.

The following lemma is easy to prove.

Lemma 1.2 A permutation group G on E ∪ X is the automorphism group of a finite cover of X if the following conditions are satisfied:

1. G is a closed subgroup of Sym(E ∪ X)

2. Every element of G respects the fibration π.

3. Every permutation of X is induced by an element of G.

The kernel of a finite cover Ē is the set of all automorhisms σ which leave X pointwise fixed i.e. Aut(Ē/X). All reducts Ē′ of Ē which are also finite covers are determined solely by their kernel K′. For one computes easily that Aut(Ē′) = Aut(Ē)K′.

Let F be a set with f_{0} elements. We make (F × X, X) to a finite covering by adding the natural projection π : F × X → X and for every f ∈ F a predicate for the set {(f, x) | x ∈ X}. We denote this structure by F × X. The reducts of F × X (and the covers sets isomorphic to such reducts) are called splitting covers. Splitting covers are determined by their kernels. One sees easily that this are exactly the closed subgroups of ∏_{x∈X} Sym(F) which are invariant under conjugation with elements of id × Sym(X).

In the next section we will prove that all finite covers of the disintegrated set split. In the last section we determine the kernels of the splitting covers.

2 The covers split

Let Ē = (E, X) be a finite cover of X. For subsets A of X let E_{A} denote the union of the fibers above A i.e. π^{−1}(A).

Lemma 2.1 Let A be a subset of X. Then E_{A} and X are orthogonal over A in the sense that

tp(E_{A}/A) ⊢ tp(E_{A}/X)

Proof: We can assume that A is finite. Then by restriction the finite group Aut(E_{A}/A)/Aut(E_{A}/X) is a homomorphic image of Aut(E/A)/Aut(E/X), which is again by restriction isomorphic to Aut(X/A) ≅ Sym(X \ A). But an infinite symmetric group has no non-trivial finite quotients. Whence Aut(E_{A}/A) = Aut(E_{A}/X), which we had to show: two tuples from E_{A} which are conjugate over A are conjugate over X.

This is just a group-proof of a more general fact: Let Ē be a finite cover of a strongly minimal set X. If X weakly eliminates imaginairies and A is an algebraically closed subset of X then E_{A} and X are orthogonal over A.

Corollary 2.2 Let σ ∈ Sym(X) have support B. Then σ can be extended to σ̄ ∈ Aut(E) which is the identity on E_{X\B}.

Proof: Since both X and E_{A} are ⋀–definable over A

tp(E_{A}/A) ⊢ tp(E_{A}/X)

is equivalent to

tp(X/A) ⊢ tp(X/E_{A}).

This, applied to A = X \ B proves the assertion.

Lemma 2.3 For all a ≠ b ∈ X there is a σ_{ab} ∈ Aut(E) such that

1. σ_{ab} induces the transposition (ab) on X,

2. σ_{ab} fixes E_{X\{ab}} pointwise,

3. σ_{ab}^{2} = id

Proof: By the corollary choose τ_{ab} which satisfies 1 and 2. Then ρ = τ_{ab}^{2} leaves everything fixed except possibly E_{a} and E_{b}. If we conjugate ρ by τ_{bc} we obtain an automorphism ρ_{c} which leaves everything fixed except possibly E_{a} and E_{c}. Furthermore ρ_{c} agrees with ρ on E_{a}. If we fix a and b and let c tend to infinity the sequence (ρ_{c}) will converge to an automorphism σ which agrees with ρ on E_{a} and is the identity else. Now set σ_{ab} = σ^{−1}τ_{ab}

Theorem 2.4 Every finite cover of a disintegrated set splits

Proof: We have to prove that there is a bijection between Ē and F × X which respects the fibration and such that every 0-definable relation of Ē is mapped to a 0-definable relation of F × X i.e. every automorphism of F × X is the image of an automorphism of Ē. This amounts to finding a family (β_{x})_{x∈X} of bijections β_{x} : F → E_{x} such that every permutation σ of X can be lifted to an automorphism σ̄ of Ē such that every diagram

[figure: F; E_{x}; E_{σ(x)}; σ̄; β_{x}; β_{σ(x)}]

commutes. Fix an element a ∈ X and a bijection β_{a} : F → E_{a}. Now chose for every b ≠ a the bijection β_{b} : F → E_{b} in such a way that the diagram above commutes for σ = (ab), σ̄ = σ_{ab} and x = a. But then the diagram commutes for all x ∈ X. This is clear if x ≠ b and follows from σ_{ab}^{2} = id if x = b. If σ ∈ Sym(X) has finite support it can be written as a product of transpositions (ab_{i}). If we define σ̄ to be product of the σ_{ab_{i}} the diagram commutes. Finally if σ is the limit of permutations σ_{i} of finite support let σ̄ be an accumulation point of the σ̄_{i}.

3 The kernels

Finally we determine the possible kernels of finite covers of disintegrated sets. Since all covers split this is done by the following theorem:

Theorem 3.1 The closed subgroups K of ∏_{x∈X} Sym(F) which are invariant under conjugation with elements of id × Sym(X) are exactly the groups of the form

K_{H}^{G} = {α ∈ ∏_{x∈X} G | ∀x, y α_{x}H = α_{y}H},

where G is a subgroup of Sym(F) and H is a normal subgroup of G.

Proof: Clearly all groups K_{H}^{G} are closed and invariant under conjugation with elements of id × Sym(X). Let conversely K be a group with this property. Fix a ∈ X and set

G = {σ_{a} | σ ∈ K}

and

H = {α | ∃σ ∈ K σ_{a} = α ∧ ∀x ≠ a σ_{x} = id}.

We will prove that K = K_{H}^{G}. Let σ be an element of K. For each x ∈ X the a-component of σ^{(ax)} is σ_{x}. This shows that all σ_{x} lie in G. If x and y are given the components of the commutator κ = [σ, (xy)] are σ_{x}σ_{y}^{−1} at x, σ_{y}σ_{x}^{−1} at y and the identity everywhere else. The limit τ = lim_{z→∞} κ^{(yz)} is an element of K whose components are the identity except that τ_{x} = σ_{x}σ_{y}^{−1}. Then τ^{ax} ∈ K shows that σ_{x}σ_{y}^{−1} ∈ H. For the converse we remark first that by conjugation G and H do not depend on the choice of a. This implies immediately that ∏_{x∈X} H is contained in K. If now σ is an arbitrary element of K_{H}^{G} choose an element τ of K such that σ_{a} = τ_{a}. Since all σ_{x} and all τ_{x} (by the first part of the proof) are congruent mod H to σ_{a} = τ_{a} the quotient στ^{−1} belongs to ∏_{x∈X} H. This shows σ ∈ K.