A Remark on Morley Rank

M. Ziegler^{∗}

March 1997^{†}^{‡}^{§}^{¶}

1 Introduction

Let f : D → E be a definable map between definable classes. The following theorem is well known: Theorem 1 ([1], [2, V 6.8]) If E has Morley rank β and the Morley rank of all fibers f^{−1}(e) is bounded by α. Then

1. If α = 0 D has Morley rank at most β.

2. If α > 0 the Morley rank of D is bounded by α(β + 1).

It seems to be less well known that this theorem gives the optimal bound. We will prove: Lemma 2 For all α > 0 and all β there is a theory T and (in the monster model of T) a definable map f : D → E such that

a) E has Morley rank β

b) the Morley rank of all fibers of f is α

c) D has Morley rank α · (β + 1).

In section 4 we discuss a bound for the Morley rank of D if the Morley rank of all fibers of f is smaller than a limit ordinal α.

In the sequel let R(F) denote the (Morley) rank of the definable set F.

∗Revision : 1.9

†March 2011: Section 5 added.

‡May 2013: Correction of the first part of the proof of Theorem 1

§April 2014: Further simplification of the first part of the proof of Theorem 1. Typo in the formulation of the problem after Theorem 5.

¶March 2015: Remark 6 added.

2 Proof of Theorem 1

We include a proof of Theorem 1 for the convenience of the reader.

Let E have rank α and A be a set of parameters. We call an element e of E generic over A if it is not contained in any A–definable set of smaller rank than α. E has always generic elements (in the monster model). Note that all generics of E have the same type over A if E has degree one.

First we handle case 1, where α = 0. Let f : D → E have finite fibers. We prove

R(D) ≤ R(E)

by induction on β = R(E). We may assume that E has degree 1. Let D_{i} be an infinite family of disjoint definable subsets of D. We have to show that almost all of them have smaller rank than β. Let e ∈ E be generic over the parameters over which f, D, E and the D_{i} are defined. Almost all of the D_{i} do not intersect f^{−1}(e). So these f(D_{i}) do not contain e and have therefore smaller rank than β. So by induction almost all D_{i} have smaller rank than β.

For the case 2 we need a lemma. If E has Morley degree one and e ∈ E is generic over the relevant parameters we call the (possibly empty) set f^{−1}(e) a generic fiber of f.

Lemma 3 Let E have Morley degree one and α be the rank of the generic fiber of f. If

γ + α < R(D),

D has a definable subset D′ such that f ↾: D′ → E has finite generic fiber and

γ < R(D′).

Proof: We may assume α > 0. D contains then an infinite family D_{i} of definable disjoint sets having at least rank γ + α. Let e ∈ E be generic. Then for one index i the rank of D_{i} ∩ f^{−1}(e) is smaller than α. By induction on α D_{i} contains a D′ as required. QED.

We prove case 2 of the theorem by induction on β. We may assume that E has degree one. If

α · β + α < R(D),

by the last lemma, D contains a definable D′ of rank bigger than α · β such that the generic fiber of f ↾ D′ has finitely many, say k many, elements. For

E^{∗} = {e ∈ E | D′ ∩ f^{−1}(e) has cardinality k} ,

the complement E \E^{∗} has a rank β′ < β. Since (by case 1) D′ ∩f^{−1}(E^{∗}) has at most rank β, the rank of D′′ = D′ ∩f^{−1}(E \E^{∗}) is bigger than α·β ≥ α(β′ +1). This contradicts the induction hypothesis applied to f ↾ D′′ : D′′ → E \ E^{∗}.

3 Proof of Lemma 2

We deal only with countable α and β. (The proof in the uncountable case is essentially the same.) So in the sequel infinite means countably infinite.

For a fixed α > 0 and for all β we will construct models

𝔐_{β} = (D_{β}, E_{β}, f_{β}),

which consist of a two sorts D_{β} and E_{β}, a map f_{β} : D_{β} → E_{β} and unary predicates on D_{β} and E_{β} such that

a) E_{β} has Morley rank β

b) the Morley rank of all fibers of f_{β} is α

c) D_{β} has Morley rank α · (β + 1).

d) 𝔐_{β} is saturated and has quantifier elimination.

We start with a structure 𝔄 = (A, P_{i})_{i∈I}, where A is an infinite set and the P_{i} are unary predicates which ensure that 𝔄 has rank α (and is saturated). For the model 𝔐_{0} we take (A, E_{0}, f_{0}, P_{i})_{i∈I}, where E_{0} consists of one point and f_{0} is the constant map.

We give the following case a special treatment: Assume that α is finite and β is a limit cardinal. We take for E_{β} any set with unary predicates giving it rank β. Choose a surjection f_{β} : D_{β} → E_{β} with infinite fibers and sets X^{a} ⊂ D_{β} ,(a ∈ A), which intersect each fiber of f_{β} in exactly one point. From the predicates P_{i} we define the predicates Q_{i} = ⋃_{a∈P_{i}} X^{a}. This is our 𝔐_{β}. The sets X^{a} inherit rank β from E_{β}. Whence D_{β} has rank β + α, which in our case equals α(β + 1).

Now assume that α is infinite or β is a successor ordinal. Also assume that for all β′ < β the structures 𝔐_{β′} are constructed. Let α′ be such that 1+α′ = α and 𝔄′ = (A′, P′_{i})_{i∈I′} be the α′–version of 𝔄^{1}. To construct 𝔐_{β} we take infinite sets D_{β} and and E_{β} and a surjective map f_{β} : D_{β} → E_{β} with infinite fibers.

On D_{β} and E_{β} we choose two families (X^{a})_{a∈A′} and (E^{a,i})_{a∈A′,i∈ω} of disjoint subsets (and introduce predicates for them) such that

1. All intersections X^{a} ∩ f_{β}^{−1}(e) and the differences f_{β}^{−1}(e) \ ⋃_{a∈A′} X^{a} are infinite.

2. The difference E_{β} \ ⋃_{a∈A′,i∈ω} E^{a,i} is infinite. The cardinality of the E^{a,i} will be specified later.

1If α′ = 0 A′ is just any finite set

From the predicates P′_{i} we define also the predicates

Q_{i} = ⋃_{a∈P′_{i}} X^{a}.

Let (β_{i})_{i∈ω} be an enumeration of the ordinals β′ < β where all β′ occur infinitely often. In our last step, for each a ∈ A′ and i ∈ ω we introduce new predicates for subsets of E^{a,i} and for subsets of

D^{a,i} = f_{β}^{−1}(E^{a,i}) ∩ X^{a}

such that with the new predicates the structure

f_{β} ↾ D^{a,i} : D^{a,i} → E^{a,i}

looks like 𝔐_{β_{i}}. This also tells us the right cardinality of the E^{α,i}. This completes the construction of 𝔐_{β}.

It is easy to check that 𝔐_{β} has quantifier elimination and is saturated. Since the E_{a,i} have rank β_{i} E_{β} has rank β. Without the structure imprinted on the D_{a,i} the fibers look like 𝔄′ with each point blown up to an infinite set and have therefore rank 1 + α′ = α. The structure on D_{a,i} adds one set of rank α on the fiber. Whence the fibers have rank α.

Each D^{a,i} has rank α · (β_{i} + 1). We have to distinguish two cases:

1. β is a successor ordinal: Then X^{a} has at least rank α · β + 1 and D_{β} has at least rank α · β + 1 + α′ = α(β + 1).

2. β is a limit ordinal and α is infinite: Then X^{α} has at least rank α · β and D_{β} the rank α · β + α′ = α · β + α = α · (β + 1).

4 Fiber rank smaller than a limit ordinal

The following problem is left open by Theorem 1: Let α be a limit ordinal. If E has rank β and the ranks of all fibers of f : D → E are smaller than α can we say more about R(D) than just R(D) ≤ α(β + 1) ? The answer is yes:

Theorem 4 Let α be a limit ordinal and β be arbitrary

1. Let f : D → E be a definable map between definable classes: Assume E has Morley rank β and that the Morley rank of all fibers f^{−1}(e) is smaller than α. Then the Morley rank of D is smaller than α(β + 1).

2. For all γ < α and all β there is a theory T and (in the monster model of T) a definable map f : D → E such that a) E has Morley rank β

b) the Morley rank of all fibers of f is smaller than α

c) D has Morley rank α · β + γ.

Part 1 has the same proof as Theorem 1. But part 2 needs a modification of the construction in Lemma 2.

Again we deal only with countable α and β. By recursion on β we construct models

𝔐_{β}^{γ} = (D_{β}^{γ}, E_{β}^{γ}, f_{β}^{γ})

such that

a) E_{β}^{γ} has Morley rank β

b) the Morley rank of all fibers of f_{β}^{γ} is smaller that α

c) D_{β}^{γ} has Morley rank α · β + γ.

d) 𝔐_{β}^{γ} is saturated and has quantifier elimination.

We construct 𝔐_{0}^{γ} as in the proof of Lemma 2 from a structure 𝔄 = (A, P_{i})_{i∈I} of rank γ. If β > 0 assume that for all β′ < β (and all γ < α) the structures 𝔐_{β′}^{γ} are constructed. Take infinite sets D_{β}^{γ} and and E_{β}^{γ} and a surjective map f_{β}^{γ} : D_{β}^{γ} → E_{β}^{γ} with infinite fibers. Then choose two families (X^{a})_{a∈A} and (E^{a,i})_{a∈A,i∈ω} of disjoint subsets (and introduce predicates for them) such that

1. All intersections X^{a} ∩ (f_{β}^{γ})^{−1}(e) and the differences (f_{β}^{γ})^{−1}(e) \ ⋃_{a∈A} X^{a} are infinite.

2. The difference E_{β}^{γ} \ ⋃_{a∈A,i∈ω} E^{a,i} is infinite.

Define again the predicates Q_{i} = ⋃_{a∈P_{i}} X^{a}.

Finally we introduce new unary predicates on E^{a,i} and

D^{a,i} = (f_{β}^{γ})^{−1}(E^{a,i}) ∩ X^{a}

such that with the new predicates the structure

𝔑^{a,i} = (D^{a,i}, E^{a,i}, f_{β}^{γ} ↾ D^{a,i})

looks as follows:

Case 1: β = β′ + 1 is a successor.

Then choose an enumeration (γ_{i})_{i∈ω} of the ordinals below α and let 𝔑^{a,i} look like 𝔐_{β′}^{γ_{i}}.

Case 2: β is a limit ordinal.

Let (γ_{i})_{i∈ω} enumerate the ordinals below β and let 𝔑^{a,i} look like 𝔐_{β_{i}}^{0}.

In the successor case D^{a,i} has rank α · β′ + γ_{i}, X^{a} has rank α · β′ + α = α · β. In the limit case D^{a,i} has rank α · β_{i} and it follows again that X^{a} has rank α · β. This implies that D_{β}^{γ} has rank α · β + γ.

5 A better bound

The following theorem implies both Theorem 1 (2) and Theorem 4 (1):

Theorem 5 ([3, Exercise 6.4.4]) If E has Morley rank β, Morley rank of all fibers f^{−1}(e) is bounded by α > 0 and the Morley rank of the generic fibers is bounded by α^{gen}, then the Morley rank of D is bounded by

αβ + α^{gen}.

The proof is a slight variation of the proof of Theorem 1. One proves similarly:

Remark 6 If β is a limit ordinal, β < αβ, and α^{gen} is finite, then the Morley rank of D is smaller than αβ + α^{gen}.

Slight modifications of the constructions above show that this bounds are optimal: If β and 0 ≤ α^{gen} ≤ α are given, there are two cases:

1. If the conditions of Remark 6 are not satisfied, there is an example D whith Morley rank αβ + α^{gen}.

2. If the conditions of Remark 6 are satisfied, for every γ smaller than αβ + α^{gen} there is an example whith at least Morley rank γ.

References

[1] M. M. Erimbetov. Complete theories with 1-cardinal formulas. Algebra i Logika, 14:245–257, 1975.

[2] Saharon Shelah. Classification Theory. North Holland, Amsterdam, 1978.

[3] K. Tent and M. Ziegler. A Course in Model Theory. ASL Lecture Note Series. Cambridge University Press, 2012.