On many-sorted ω-categorical theories

Enrique Casanovas, Rodrigo Peláez, and Martin Ziegler^{∗}

March 17, 2011

Abstract

We prove that every many-sorted ω-categorical theory is completely interpretable in a one-sorted ω-categorical theory. As an application, we give a short proof of the existence of non G–compact ω-categorical theories.

1 Introduction

A many–sorted structure can be easily transformed into a one–sorted by adding new unary predicates for the different sorts. However ω-categoricity is not preserved. In this article we present a general method for producing ωcategorical one–sorted structures from ω-categorical many–sorted structures. This is stated in Corollary 3.2, the main theorem in this paper. Our initial motivation was to understand Alexandre Ivanov’s example (in [4]) of an ωcategorical non G-compact theory. In Corollary 3.3 we apply our results to offer a short proof of the existence of such theories.

Our method is based on the use of a particular theory T_{E} of equivalence relations E_{n} on n-tuples. The quotient by E_{n} is an imaginary sort containing a predicate P_{n} which can be used to copy the n-th sort of the given manysorted theory. Since the complexity of T_{E} is part of the complexity of the ω-categorical one–sorted theory obtained by our method, it is important to classify T_{E} from the point of view of stability, simplicity and related properties. It turns out that T_{E} is non-simple but it does not have SOP_{2}. A similar example of a theory with such properties has been presented by Shelah and Usvyatsov in [7]. Their proof, as ours, relies on Claim 2.11 of [3], which is known to have some gaps. A revised version of [3] will be posted in arxiv.org. In the meanwhile Kim and Kim have obtained a new proof of the same result: Proposition 2.3 from [5].

∗Work partially supported by MODNET, FP6 Marie Curie Training Network in Model Theory and its Applications with contract number MRTN-CT-2004-512234. The first author has been partially supported by grants MTM 2008-01545 and 2009SGR-187.

2010 Mathematics Subject Classification: Primary 03C45 ; Secondary 03C35.

Key words and phrases: many–sorted structures, ω-categoricity, G-compactness, strong order property, stable embedded predicates.

The one–sorted theory T_{E} is interdefinable with some many–sorted theory T^{∗} which is presented and discussed in Section 2. In order to describe T^{∗} we need a version of Fraïssé’s amalgamation method that can be applied to the many–sorted case (see Lemma 2.1). In Section 3 some results on stable embeddedness from the third author (in [8]) are extended and used to prove Corollary 3.2. Section 4 is devoted to classify T_{E} from the stability point of view.

A previous version of these results appeared in the second author’s Ph.D. Dissertation [6]. They have been corrected in some points and in general they have been elaborated and made more compact.

2 T^{∗} and Fraïssé’s amalgamation

Let L be a countable many–sorted language with sorts S_{i}, (i ∈ I), and let K be a class of finitely generated L–structures.^{1} We call an L–structure M a Fraïssé limit of K if the following holds:

1. K = Age(M), where Age(M) is the class of all finitely generated L– structures which are embeddable in M.

2. M is at most countable.

3. M is ultra–homogeneous i.e., any isomorphism between finitely generated substructures extends to an automorphism of M.

By a well–known argument K can only have one Fraïssé limit, up to isomorphism.

1We allow empty sorts if L has no constant symbols of that sort.

Lemma 2.1. Let K be as above. Then the following are equivalent:

a) The Fraïssé limit of K exists and is ω-categorical.

b) K has the amalgamation property AP, the joint embedding property JEP, the hereditary property HP (i.e., finitely generated L–structures which are embeddable in elements of K belong themselves to K) and satisfies

(∗) for all i_{1} … i_{n} ∈ I there are only finitely many quantifier-free types of tuples (a_{1}, …, a_{n}) where the a_{j} are elements of sort S_{i_{j}} in some structure A ∈ K.

If the Fraïssé limit of K exists, it has quantifier elimination.

Proof. a) ⇒ b). It is well known that the age of an ultra–homogeneous structure has AP, JEP and HP. All quantifier–free types which occur in elements of K are quantifier–free types of tuples of the Fraïssé limit. So property (∗) follows from the Ryll–Nardzewski theorem.

b) ⇒ a). The quantifier-free type qftp(ā) determines the isomorphism type of the structure generated by ā. Hence (∗) implies that K contains at most countably many isomorphism types. The existence of the Fraïssé limit M follows now from AP, JEP and HP.

If two sequences ā and b̄ have the same quantifier-free type in M, there is an automorphism of M which maps ā to b̄ and so it follows that ā and b̄ have the same type in M. Consider a formula φ(x̄) and the set P_{φ(x)} = {qftp(ā) : M ⊧ φ(ā)}. Then

M ⊧ φ(ā) ⇔ qftp(ā) ∈ P_{φ} ⇔ M ⊧ ⋁_{p∈P_{φ}} p(ā).

Now, (∗) implies that P_{φ} is finite and that in M all p = qftp(ā) are finitely axiomatisable, that is, p = ⟨χ_{p}⟩ for some quantifier-free χ_{p}(x). Then M ⊧ φ(ā) ⇔ ⋁_{p∈P_{φ}} χ_{p}(ā). So M has quantifier elimination and it is ω–categorical since there are only finitely many possibilities for the χ_{p}, depending only on the number and the sorts of the free variables of φ.

It is easy to see that the theory of the Fraïssé limit is the model–completion of the universal theory of K.

Definition 2.2. Let L^{∗} be the language with countably many sorts S, S_{1}, …, function symbols f_{i} : S^{i} → S_{i}, and constants c_{i} ∈ S_{i} and T^{0} the theory of all L^{∗}–structures A with

f_{i}(ā) = c_{i} ⇔ ā has some repetition

for all ā ∈ S^{i}(A). Furthermore let K^{∗} be the class of all finitely generated models of T^{0}.

Lemma 2.3. K^{∗} satisfies the conditions of Lemma 2.1.

Proof. The class of all models of T^{0} has AP and JEP and therefore also K^{∗}. (∗) follows easily from the fact that f_{i}(a_{m_{1}} …, a_{m_{i}}) = c_{i} for all i > k and {a_{m_{1}} …, a_{m_{i}}} ⊂ {a_{1}, …, a_{k}},

We define M^{∗} to be the Fraïssé limit of K^{∗} and T^{∗} to be the complete theory of M^{∗}. T^{∗} is the model–completion of T^{0}.

Recall the following definition from [2]:

Definition 2.4. Let T be a complete theory and P a 0–definable predicate. P is called stably embedded if every definable relation on P is definable with parameters from P.

Remarks

1. For many–sorted structures with sorts (S_{i})_{i∈I} this generalises to the notion of a sequence (P_{i})_{i∈I} of 0–definable P_{i} ⊂ S_{i} being stably embedded.

2. While the definition is meant in the monster model, an easy compactness argument shows that, if P (M) is stably embedded in M for some weakly saturated^{2} model M, then this is true for all models.

3. If M is saturated then P is stably embedded if and only if every automorphism (i.e. elementary permutation) of P (M) extends to an automorphism of M. This was claimed in [2] only for the case that |M| > |T |. But the proof can easily be modified to work for the general case. One has to use the fact that if A has smaller size than M, then any type over a subset of dcl^{eq}(A) can be realized in M.

2M is weakly saturated if every type over the empty set is realized in M.

4. If M is ω–categorical, it can be proved that for every finite tuple a ∈ M there is a finite tuple b ∈ P such that every relation on P which is definable over a can be defined using the parameter b.

Lemma 2.5. In T^{∗} the sequence of sorts (S_{1}, S_{2}, …) is stably embedded.

Proof. Clear since tp(ā/S_{1}, …) = tp(ā/f_{1}(ā), …). See also the discussion in [2].

For a complete theory T and a 0–definable predicate P the induced structure on P consists of all 0–definable relations on P. Note that the automorphisms of P with its induced structure are exactly the elementary permutations of P in the sense of T.

Lemma 2.6. In T^{∗} the induced structure on (S_{1}, S_{2}, …) equals its L_{>0}^{∗}– structure, where L_{>0}^{∗} is the sublanguage of L^{∗} which has only the sorts S_{1}, S_{2}, … and the constants c_{1}, c_{2}, . . ..

Proof. Quantifier elimination.

Let T_{>0}^{∗} denote the theory of all L_{>0}^{∗}–structures, where all sorts S_{i} are infinite. Clearly T_{>0}^{∗} is the restriction of T^{∗} to L_{>0}^{∗}.

Lemma 2.7. Every model of T_{>0}^{∗} can be expanded to a model of T^{∗}.

Proof. It is easy to see that the following amalgamation property is true:

Let N be a model of T^{0} with infinite sorts S_{i}(N). Let A be a finitely generated substructure of N and B ∈ K^{∗} an extension of A. Then B can be embedded over A in an extension N′ of N which is a model of T^{0} and such that S_{i}(N′) = S_{i}(N) for all i.

If a model of T_{>0}^{∗} is given, we expand it arbitrarily to a model N of T^{0} and apply the above amalgamation property repeatedly such that the union of the resulting chain is a model of T^{∗} which has the same sorts S_{i} as N.

Corollary 2.8. There is an ω–categorical one-sorted theory T_{E} with a series of 0–definable infinite predicates P_{1}, P_{2}, … in T_{E}^{eq} such that

1. (P_{1}, P_{2}, …) is stably embedded

2. The many–sorted structure induced on (P_{1}, P_{2}, …) is trivial.

3. For every sequence κ_{1}, κ_{2}, … of infinite cardinals there is a model N of T_{E} such that |P_{i}(N)| = κ_{i}.

Proof. The language L_{E} of T_{E} will contain for each i a symbol E_{i} for an equivalence relation between i–tuples. Let M = (S, S_{1}, S_{2}, …) be a model of T^{∗}. For a, b ∈ S^{i} define E_{i}(a, b) ⇔ f_{i}(a) = f_{i}(b). T_{E} is the theory of M_{E} = (S, E_{1}, E_{2}, …). The S_{i} live in M_{E}^{eq} as M_{E}^{i}/E_{i} and the c_{i} are 0–definable in M_{E}^{eq}. We set P_{i} = S_{i} \ {c_{i}}.

It is easy to see that T_{E} as constructed in the proof is the model– completion of the theory of all structures (M, E_{1}, E_{2}, …) where E_{n} is an equivalence relation on M^{n} where one equivalence class consists of all n– tuples which contain a repetition. That T_{E} has quantifier elimination can be proved as follows: Every formula φ(x̄) of L_{E} is equivalent to a quantifier– free L^{∗}–formula φ′(x̄). φ′(x̄) is a boolean combination of formulas of the form f_{i}(x̄′) ≐ f_{i}(x̄′′) and f_{i}(x̄′) ≐ c_{i}, which are equivalent to quantifier–free L_{E}–formulas: f_{i}(x̄′) ≐ f_{i}(x̄′′) is equivalent to E_{i}(x̄′, x̄′′), f_{i}(x′_{1}, …, x′_{i}) ≐ c_{i} is equivalent to ⋁_{1≤k<l≤i} x′_{k} ≐ x′_{l}.

3 Expansions of stably embedded predicates

Let T be complete theory with two sorts S_{0} and S_{1}. We consider S_{1} as a structure of its own carrying the structure induced from T and denote by T ↾ S_{1} the theory of S_{1}.

Lemma 3.1. Let T be complete theory with two sorts S_{0} and S_{1}. Let T̃_{1} be a complete expansion of T ↾ S_{1}. Assume that S_{1} is stably embedded. Then we have

1. T̃ = T ∪ T̃_{1} is complete.^{3} ([8, Lemma 3.1])

2. S_{1} is stably embedded in T̃ and T̃ ↾ S_{1} = T̃_{1}.

3. If T and T̃_{1} are ω–categorical, then T̃ is also ω–categorical.

Proof. 1. Let M̃ = (M_{0}, M̃_{1}) and M̃′ = (M′_{0}, M̃′_{1}) be saturated models of T̃ of the same cardinality and M = (M_{0}, M_{1}) and M′ = (M′_{0}, M′_{1}) their restrictions to the language of T. Since T and T̃_{1} are complete, there are isomorphisms f : M → M′ and g : M̃_{1} → M̃′_{1}. gf^{−1} is an automorphism of M′_{1}. Since M′_{1} is stably embedded in M′, gf^{−1} extends to an automorphism h of M′. hf is now an isomorphism from M to M′ which extends g.

3Actually we have: S_{1} is stably embedded if and only if T̃ is complete for all complete expansions T̃_{1}.

2. We use the same notation as in the proof of 1. Let M̃ be a saturated model of T̃. We have to show that every automorphism f of M̃_{1} extends to an automorphism of M̃. But f extends to an automorphism of M, which is automatically an automorphism of M̃.

3. Start with two countable models M̃ and M̃′ and proceed as in the proof of 1.

Corollary 3.2. Every many-sorted ω-categorical theory is completely (the induced structure is exactly this) interpretable in a one-sorted ω-categorical theory.

Proof. Let T be a complete theory with countably many sorts P_{1}, P_{2}, . . .. We consider T as an expansion of T_{E} ↾ (P_{1}, P_{2}, …) and set T̃ = T_{E} ∪ T. T̃ is a one-sorted complete theory. We have T̃ ↾ (P_{1}, P_{2}, …) = T. If T is ω–categorical, T̃ is also ω–categorical.

Corollary 3.3 (Ivanov). There is a one–sorted ω–categorical theory which is not G–compact.

Proof. By [1] there is a many–sorted ω–categorical theory T which is not G– compact. Interpret T in a one–sorted ω–categorical theory T̃ as in Corollary 3.2. Then T is also not G–compact. For this one has to check that if T̃ is G–compact, then every 0–definable subset with its induced structure is also G–compact. This follows from the following description of G–compactness: a, b of length ω are in the relation nc^{ω} if a and b are the first two elements of an infinite sequence of indiscernibles. A complete theory is G–compact, if the transitive closure of nc^{ω} is type–definable. (Note that (a, b) is in the transitive closure of nc^{ω} if and only if a and b have the same Lascar-strong type.)

4 Classification of T_{E}

Proposition 4.1. T_{E} has TP_{2}, the tree property of the second kind, and therefore it is not simple.

Proof. We show that φ(x; y, u, v) = E_{2}(xy, uv) has TP_{2}. Let (b_{i} : i < ω), (c_{i} : i < ω), and (d_{i} : i < ω) be pairwise disjoint sequences of different elements such that ¬E_{2}(c_{i}d_{i}, c_{j}d_{j}) for i ≠ j. For i, j ∈ ω, let ā_{j}^{i} = b_{i}c_{j}d_{j}. By compactness we can see that for any η ∈ ω^{ω}, the set {φ(x; ā_{η(i)}^{i}) : i < ω} is consistent, and since the c_{i}d_{i}’s are in different E_{2}-classes, for each i < ω, the set {φ(x; ā_{j}^{i}) : j < ω} is 2-inconsistent.

Lemma 4.2 (Independence lemma). Let a, b, c, d′, d′′ be tuples in the monster model of T_{E} and F a finite subset. Assume that a and c have only elements from F in common. If d′a ≡_{F} d′b ≡_{F} d′′b ≡_{F} d′′c, then there exists some d such that d′a ≡_{F} da ≡_{F} dc ≡_{F} d′′c.

[figure: a; d′; b; d′′; c; d]

Proof. Let A, B, C, D′ and D′′ denote the set of elements of the tuples a, b, c, d′ and d′′, respectively. We note first that we can assume that F is contained in A,B and C, since otherwise we can increase a, b and c by elements from F. Then we note that if A and D′ intersect in a subtuple f, this tuple also belongs to B and C and therefore to F. So we have that A ∩ D′ is contained in F and similarly that C ∩ D′′ is contained in F.

It suffices to find an L_{E}-structure M extending AC and containing a new tuple d with the same quantifier-free type as d′ over A and of d′′ over C. Take as d a new tuple of the right length which intersects A and B in the subtuple f. We have then d′a ≡_{F}^{eq} da ≡_{F}^{eq} dc ≡_{F}^{eq} d′′c, where g ≡_{F}^{eq} h means that g and h satisfy the same equality-formulas over F, i.e. g_{i} = g_{j} iff h_{i} = h_{j} and g_{i} = f_{j} iff h_{i} = f_{j}. If D denotes the elements of d, it follows that the intersection of any two of A, C and D belongs to F.

It remains to define the relations E_{n} on ACD. Let E_{n}^{0} denote the part of E_{n} which is already defined on AC. Let E′_{n} be the relation E_{n} transported from AD′ to AD via the identification d′ ↦ d and E′′_{n} the relation E_{n} transported from CD′′ to CD via the identification d′′ ↦ d. Note that d′ ≡_{F} d′′ implies that E′_{n} and E′′_{n} agree on DF. We define E_{n} on ACD as the transitive closure of

E_{n}^{0} ∪ E′_{n} ∪ E′′_{n} ∪ E_{n}^{rep} ∪ Δ,

where E_{n}^{rep} is the set of all pairs of n–tuples from ACD which contain repetitions and Δ is the identity on (ACD)^{n}.

We have to show that the new structure defined on AC agrees with the original structure. Also we must check that the structure on AD (and CD) agrees with the structure on AD′ (and CD′) via d ↦ d′ (and d ↦ d′′). Using the fact that an n-tuple which e.g. belongs to AC and AD belongs already to A, it is easy to see that we have to show the following:

For all n–tuples x ∈ A, y ∈ C and z ∈ DF
1. E′_{n}(x, z) ∧ E′′_{n}(z, y) ⇒ E_{n}^{0}(y, x)
2. E′′_{n}(z, y) ∧ E_{n}^{0}(y, x) ⇒ E′_{n}(x, z)
3. E_{n}^{0}(y, x) ∧ E′_{n}(x, z) ⇒ E′′_{n}(z, y)

Let z′ and z′′ be the subtuples of D′F and D′′F which correspond to z.

Proof of 1 : Assume E′_{n}(x, z) and E′′_{n}(z, y). We have then E_{n}(x, z′) and E_{n}(z′′, y). d′a ≡_{F} d′b implies z′a ≡_{F} z′b, which implies that there is a tuple x′ in B such that z′x ≡_{F} z′x′. So we have E_{n}(z′, x′). d′b ≡_{F} d′′b implies z′x′ ≡_{F} z′′x′ and whence E_{n}(z′′, x′). Now we can connect y and x as follows: y E_{n} z′′ E_{n} x′ E_{n} z′ E_{n} x.

Proof of 2 : Assume E′′_{n}(z, y) and E_{n}^{0}(y, x). We have then E_{n}(z′′, y). As above we find a tuple y′ ∈ B such that E_{n}(z′′, y′) and E_{n}(z′, y′). The chain x E_{n} y E_{n} z′′ E_{n} y′ E_{n} z′ shows that E′_{n}(x, z).

Proof of 3 : Symmetrical to the proof of 2.

In order to state [5, Proposition 2.3] we need the following terminology:

(1) A tuple η̄ = (η_{0}, …, η_{d−1}) of elements of 2^{<ω} is ∩-closed if the set {η_{0}, …, η_{d−1}} is closed under intersection.

(2) Two ∩-closed tuples η̄ and ν̄ are isomorphic if they have the same length and

(i) η_{i} ⊴ η_{j} iff ν_{i} ⊴ ν_{j}

(ii) η_{i}^{⌢}t ⊴ η_{j} iff ν_{i}^{⌢}t ⊴ ν_{j} for t = 0, 1.

(3) A tree (a_{η}: η ∈ 2^{<ω}) of tuples of the same length is modeled by (b_{η}: η ∈ 2^{<ω}) if for every formula ϕ(x̄) and every ∩-closed η̄ there is a ∩–closed ν̄ isomorphic to η̄ such that ⊧ ϕ(bη̄) ⇔ ⊧ ϕ(aν̄).

(4) (b_{η}: η ∈ 2^{<ω}) is indiscernible if ⊧ ϕ(bη̄) ⇔ ⊧ ϕ(bν̄) for all isomorphic ∩-closed η̄, ν̄.

Lemma 4.3 ([5, Proposition 2.3]. See also [3]). Let T be a complete theory. Then any tree of tuples can be modeled by an indiscernible tree.

Definition 4.4. The formula φ(x, y) has SOP_{2} in T if there is a binary tree (a_{η} : η ∈ 2^{<ω}) such that for every η ∈ 2^{ω}, {φ(x, a_{η↾n}) : n < ω} is consistent and for every incomparable η, ν ∈ 2^{<ω}, φ(x, a_{η}) ∧ φ(x, a_{ν}) is inconsistent. The theory T has SOP_{2} if some formula φ(x, y) ∈ L has SOP_{2} in T.

Remark 4.5 (H. Adler). The formula φ(x, y) has SOP_{2} in T if and only if φ(x, y) has the tree property of the first kind TP_{1}: there is a tree (a_{η} : η ∈ ω^{<ω}) such that for every η ∈ ω^{ω}, {φ(x, a_{η↾n}) : n < ω} is consistent and for every incomparable η, ν ∈ ω^{<ω}, φ(x, a_{η}) ∧ φ(x, a_{ν}) is inconsistent.

Proof. By compactness.

Proposition 4.6. T_{E} does not have SOP_{2}.

Proof. We follow ideas from a similar proof in [7]. Assume φ(x, y) has SOP_{2} in T_{E} and the tree (a_{η} : η ∈ 2^{<ω}) witnesses it. Choose for every η a tuple d_{η} such that ⊧ ϕ(d_{η}, a_{ν}) for all ν ⊊ η.

By Lemma 4.3 we can assume that the tree (d_{η}a_{η} : η ∈ 2^{<ω}) is indiscernible. Let us now look at the elements a_{00}, a_{⟨⟩}, a_{01}, d_{000}, d_{010}. We have by indiscernibility

d_{000}a_{00} ≡ d_{000}a_{⟨⟩} ≡ d_{010}a_{⟨⟩} ≡ d_{010}a_{01}.

If the tuples a_{00} and a_{01} are disjoint, we can apply the Independence Lemma to a = a_{00}, b = a_{⟨⟩}, c = a_{01}, d′ = d_{000}, d′′ = d_{010} to get a tuple d such that

d_{000}a_{00} ≡ da_{00} ≡ da_{01} ≡ d_{010}a_{01}.

It follows that ⊧ φ(d, a_{00})∧φ(d, a_{01}), which contradicts the SOP_{2} of the tree.

If a_{00} and a_{01} are not disjoint, we argue as follows: Assume that a_{00} and a_{01} have an element f in common, say f = a_{00,i} = a_{01,j}. Then a_{00}a_{01} ≡ a_{000}a_{01} implies a_{000,i} = a_{01,j}. So we have a_{000,i} = a_{00,i} and it follows from indiscernibility that f = a_{00,i} = a_{⟨⟩,i} = a_{01,i}. Let F be the set of elements which occur in both a_{00} and a_{01}. We have seen that the elements of F occur in a_{00}, a_{⟨⟩} and a_{01} at the same places. Therefore

d_{000}a_{00} ≡_{F} d_{000}a_{⟨⟩} ≡_{F} d_{010}a_{⟨⟩} ≡_{F} d_{010}a_{01}

and we can again apply the Independence Lemma.

References

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[2] Zoé Chatzidakis and Ehud Hrushovski. Model theory of difference fields. Trans. Amer. Math. Soc., 351(8):2997–3071, 1999.

[3] Mirna Dzamonja and Saharon Shelah. On <^{∗}-maximality. Annals of Pure and Applied Logic, 125:119–158, 2004. math.LO/0009087.

[4] A. A. Ivanov. A countably categorical theory which is not G-compact. Siberian Advances in Mathematics, 20:75 – 82, 2010. Based on ArXiv: math.LO/0602126.

[5] Byunghan Kim and Hyeung-Joon Kim. Notions around the tree property 1. Annals of Pure and Applied Logic, 2011. doi:10.1016/j.apal.2011.02.001.

[6] Rodrigo Peláez. About the Lascar group. Ph.D. Dissertation, University of Barcelona, April 2008.

[7] Saharon Shelah and Alex Usvyatsov. More on SOP_{1} and SOP_{2}. Annals of Pure and Applied Logic, 155:16–31, 2008. math.LO/0404178.

[8] Martin Ziegler. Fusion of structures of finite Morley rank. In Z. Chatzidakis, D. Macpherson, A. Pillay, and A. Wilkie, editors, Model Theory with Applications to Algebra and Analysis, volume 349 of London Mathematical Society Lecture Note Series, pages 225–248. Cambridge University Press, 2008.

Department of Logic, History and Philosopy of Science, University of Barcelona, Montalegre 5, 08001 Barcelona, Spain.

e.casanovas@ub.edu

Department of Logic, History and Philosopy of Science, University of Barcelona, Montalegre 5, 08001 Barcelona, Spain.

rpelaezpelaez@yahoo.com

Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany

ziegler@uni-freiburg.de