NIP FOR SOME PAIR-LIKE THEORIES

GARETH BOXALL

Abstract. Generalising work from [2] and [5], we give sufficient conditions for a theory T_{P} to inherit NIP from T, where T_{P} is an expansion of the theory T by a unary predicate P. We apply our result to a theory, studied in [1], of the real field with a subgroup of the unit circle.

1. Introduction

We consider the situation where T is a complete one-sorted theory with infinite models, L is the language of T, P is a new unary predicate, L_{P} = L ∪ {P }, M ⊧ T and T_{P} is the complete theory of some expansion of M to L_{P}. Our main result provides sufficient conditions for T_{P} to inherit NIP from T. It is a common generalisation of two other recent results, one of Berenstein, Dolich and Onshuus in [2] and one of Günaydin and Hieronymi in [5]. With respect to the result from [2], our generalisation removes the assumption that P (M) be algebraically closed. With respect to the result from [5], it has the advantage that it works outside the setting where T is o-minimal. We apply our result to a theory of the real field with a subgroup of the unit circle which was studied by Belegradek and Zilber in [1].

I would like to thank Oleg Belegradek for pointing out errors in an earlier version and Hans Adler, John Baldwin and Alexander Berenstein for useful conversations about the content and presentation of this work. Above all I would like to thank my PhD supervisor Anand Pillay for much generous assistance with this material, including suggesting the main approach used.

2. A general result

First order logic is used throughout. The expansion of M to L_{P} is written as (M, P (M)). We work in L_{P} (or T_{P}) except where we specifically indicate L (or T). For example, acl denotes algebraic closure in the sense of T_{P} while acl_{L} denotes algebraic closure in the sense of T. Similarly tp(ā/B) is a complete type in the sense of T_{P} while tp_{L}(ā/B) is a complete type in the sense of T. Otherwise our notation is fairly standard. We abbreviate A ∪ B to AB and sometimes sets are treated as tuples or vice versa. For a tuple of variables x̄ = x_{1}...x_{n} we abbreviate P (x_{1}) ∧ ... ∧ P (x_{n}) to P (x̄). Our main result is the following.

Theorem 2.1. T_{P} has NIP if, for any (M, P (M)) ≺ (N, P (N)) ≺ (\overline{M}, P (\overline{M})) ⊧ T_{P} such that all three models are sufficiently saturated, the following conditions are satisfied:

(i) acl_{L} is a pregeometry on \overline{M}, (ii) if b ∈ \overline{M} \ acl_{L}(NP (\overline{M})) then tp(b/N) is implied by tp_{L}(b/N) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})), (iii) there is some sufficiently saturated (N′, P (N′)) such that (N, P (N)) ≺ (N′, P (N′)) ≺ (\overline{M}, P (\overline{M})) and, for any finite n ≥ 1 and f̄ ∈ P (\overline{M})^{n}, there is some κ ≤ 2^{|M|} and f̂ ∈ P (\overline{M})^{κ} such that f̄ ∈ acl(f̂) and, for any extension of tp(f̂/M) to a complete type q(ẑ) over N such that q(ẑ) is finitely realisable in M, there is an extension of tp_{L}(f̂/M) to a complete L-type q′(ẑ) over N′ such that q′(ẑ) is finitely realisable in M and q(ẑ) is implied by q′(ẑ) in conjunction with tp(f̂/M), (iv) T has NIP.

Date: March 16, 2009.

Some of the generality of Theorem 2.1 is obtained at the expense of elegance. The thinking behind condition (iii) should become clear in the light of Sections 3 and 4. However it would probably be helpful at this stage to mention a neater version of it which is sufficient for some interesting applications:

(iii)′ for any finite n ≥ 1 and f̄ ∈ P (\overline{M})^{n}, tp(f̄/N) is implied by tp_{L}(f̄/N) in conjunction with the information that f̄ ∈ P (\overline{M})^{n}.

Clearly when (iii)′ replaces (iii) in the assumptions of Theorem 2.1, these assumptions are if anything strengthened. We shall comment further on (iii)′ in Section 3.

The independence property (the negation of NIP) was introduced by Shelah in [8]. Our proof of Theorem 2.1 uses the following fact due to a combination of Shelah and Poizat. Details are given in chapter 12 of [7].

Fact 2.2. The following conditions are equivalent and T has NIP if and only if they are true:

(1) for any M ≺ N ⊧ T such that both models are sufficiently saturated, there are at most 2^{|M|} complete one-types q(x) over N such that q(x) is finitely realisable in M,

(2) for any finite n ≥ 1 and any M ≺ N ⊧ T such that both models are sufficiently saturated, there are at most 2^{|M|} complete n-types q(x̄) over N such that q(x̄) is finitely realisable in M.

We conclude this section with a proof of Theorem 2.1. Let (M, P (M)) ≺ (N, P (N)) ≺ (\overline{M}, P (\overline{M})) ⊧ T_{P} be such that all three models are sufficiently saturated. Let b ∈ \overline{M} and suppose tp(b/N) is finitely realisable in M. We show that there are at most 2^{|M|} choices for tp(b/N).

Case 1: Suppose b ∉ acl_{L}(NP (\overline{M})). By condition (ii), tp(b/N) is implied by tp_{L}(b/N) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})). Clearly tp_{L}(b/N) is finitely realisable in M. By condition (iv) and Fact 2.2(1), there are at most 2^{|M|} choices for tp_{L}(b/N). Therefore there are, in this case, at most 2^{|M|} choices for tp(b/N).

Case 2: Suppose b ∈ acl_{L}(NP (\overline{M})). Let āc̄f̄ be a tuple such that ā ∈ M^{k}, c̄ ∈ N^{l} and f̄ ∈ P (\overline{M})^{n}, for some k, l, n < ω, and b ∈ acl_{L}(āc̄f̄). We may assume c̄ is of minimal length. Suppose c̄ is not empty (that is to say l ≠ 0). It follows by condition (i) that c̄ is not acl_{L}- independent over ābf̄. Let φ(x̄, y, z̄, w̄) be an L-formula which is realised by ā, b, c̄, f̄ and which witnesses the fact that c̄ is not acl_{L}-independent over ābf̄. Since tp(b/āc̄) is finitely realisable in M, the formula ψ(ā, y, c̄) ≡ (∃w̄)[P (w̄) ∧ φ(ā, y, c̄, w̄)] is realisable in M and so c̄ is not acl_{L}-independent over MP (\overline{M}). This contradicts the minimality of the length of c̄. Therefore c̄ is empty. Therefore b ∈ acl_{L}(Mf̄).

Let (N′, P (N′)), κ and f̂ be as in condition (iii). Let p(yŵ) = r(y) ∧ s(yŵ) where r(y) = tp(b/N) and s(yŵ) = tp(bf̂/M). Then p(yŵ) is finitely realisable in M. By a well known argument (you extend a filter-base to an ultrafilter and extract what you want from that), p(yŵ) extends to a complete type p′(yŵ) over N which is finitely realisable in M. Let b′f̂′ ⊧ p′(yŵ). Then tp(b′/N) = tp(b/N), tp(f̂′/M) = tp(f̂/M) and b′ ∈ acl(Mf̂′).

By condition (iii), tp(f̂′/N) is implied by tp(f̂/M) in conjunction with some complete L-type q′(ŵ) over N′ which extends tp_{L}(f̂/M) and is finitely realisable in M. By condition (iv) and Fact 2.2(2), there are at most 2^{|M|}×κ choices for q′(ŵ). Therefore there are at most 2^{|M|}×κ×2^{|M|}×κ choices for tp(f̂′/N). Therefore there are, in this case, at most 2^{|M|}×κ×2^{|M|}×κ×|M|×κ = 2^{|M|} choices for tp(b/N).

Adding the two cases together, there are at most 2^{|M|} choices for tp(b/N). Therefore T_{P} has NIP, by Fact 2.2(1).

3. Comparison with results in [2] and [5]

Theorem 2.7 of [2] makes use of the notion of P -independence which is defined as follows and makes sense provided acl_{L} is a pregeometry.

Definition 3.1. A set A ⊆ M is said to be P -independent if A is acl_{L}-independent from P (M) over A ∩ P (M).

We shall also want to speak of P -tp(ā) by which we mean the information that tells us which members of the tuple ā belong to P (M). The following is Theorem 2.7 from [2].

Theorem 3.2. T_{P} has NIP if, for any sufficiently saturated (M, P (M)) ⊧ T_{P}, the following conditions are satisfied:

(a) acl_{L} is a pregeometry on M,

(b) for any finite n ≥ 1, if ā, b̄ ∈ M^{n} are such that both ā and b̄ are P -independent and both tp_{L}(ā) = tp_{L}(b̄) and P -tp(ā) = P -tp(b̄) then tp(ā) = tp(b̄),

(c) acl_{L}(P (M)) = P (M),

(d) T has NIP.

The following result establishes that Theorem 2.1 is a generalisation of Theorem 3.2. The proof is standard and trivial. Note that when we speak of P -independence we mean with respect to the larger model (\overline{M}, P (\overline{M})).

Proposition 3.3. The assumptions of Theorem 3.2 excluding (c) imply the assumptions of Theorem 2.1 even when (iii)′ replaces (iii).

Proof. Suppose the assumptions of Theorem 3.2 are satisfied with the possible exception of (c). Let (M, P (M)) ≺ (N, P (N)) ≺ (\overline{M}, P (\overline{M})) ⊧ T_{P} be such that all three models are sufficiently saturated. Conditions (i) and (iv) follow immediately. Since (N, P (N)) is a model it is clear that, for any finite tuple ā from N, there is a finite tuple ḡ from P (N) such that āḡ is P - independent. Let b ∈ \overline{M} \ acl_{L}(NP (\overline{M})). For any P -independent tuple ā from N, it is clear that bā is also P -independent. It follows from condition (b) that tp(b/N) is implied by tp_{L}(b/N) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})). So condition (ii) is satisfied. Let n ≥ 1 be finite and f̄ ∈ P (\overline{M})^{n}. For any P -independent tuple ā from N it is clear that f̄ā is also P -independent. It follows from condition (b) that tp(f̄/N) is implied by tp_{L}(f̄/N) in conjunction with the information that f̄ ∈ P (\overline{M})^{n}. So condition (iii)′ is satisfied. □

Theorem 3.2 is used in [2] to show that if T has NIP and is a geometric theory and T_{P} is the theory of lovely pairs of models of T, as defined in [3], then T_{P} has NIP. This provides an interesting class of examples to which Theorem 2.1 applies even when condition (iii) is replaced by (iii)′.

There is a typo in the version of [5] currently available from the MODNET Preprint server, but through private correspondence with one of the authors we understand that their Theorem 1.3 should be as follows.

Theorem 3.4. Suppose T is dense o-minimal. T_{P} has NIP if, for any (M, P (M)) ⊧ T_{P}, the following conditions are satisfied:

(e) for any finite n ≥ 1, any definable subset of P (M)^{n} is a boolean combination of sets of the form X ∩ Y where X is ∅-definable and Y is L-definable,

(f) any formula ψ(x̄) without parameters is equivalent modulo T_{P} to a boolean combination of formulas of the form (∃z̄)[P (z̄) ∧ φ(x̄, z̄)] where φ(x̄, z̄) is an L-formula which also has no parameters,

(g) (M, P (M)) has o-minimal open core.

The following result establishes that Theorem 2.1 is a generalisation of Theorem 3.4. The proof overlaps with the argument used in [5] to prove Theorem 3.4.

Proposition 3.5. Suppose T is dense o-minimal. The assumptions of Theorem 3.4 imply the assumptions of Theorem 2.1.

Proof. Suppose the assumptions of Theorem 3.4 are satisfied. Let (M, P (M)) ≺ (N, P (N)) ≺ (\overline{M}, P (\overline{M})) ⊧ T_{P} be such that all three models are sufficiently saturated. Conditions (i) and (iv) are well known consequences of T being a dense o-minimal theory. Let b ∈ \overline{M} \ acl_{L}(NP (\overline{M})). It follows from condition (f) that to know tp(b/N) it is enough to know which formulas of the form ψ(y, ā) ≡ (∃z̄)[P (z̄) ∧ φ(y, ā, z̄)] belong to tp(b/N), where φ(ȳ, x̄, z̄) is an L-formula with no parameters and ā ∈ N^{|x|}. Knowing that b ∉ acl_{L}(NP (\overline{M})), it is enough to consider the case where, for each f̄ ∈ P (\overline{M})^{|z|}, φ(y, ā, f̄) defines an open interval in \overline{M}. In this case the set defined by ψ(y, ā) is an open subset of \overline{M}. By condition (g) this set is L-definable and so, since (N, P (N)) is a model, it is L-definable over N. Therefore tp(b/N) is implied by tp_{L}(b/N) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})). So condition (ii) is satisfied.

Let (N′, P (N′)) be such that (N, P (N)) ≺ (N′, P (N′)) ≺ (\overline{M}, P (\overline{M})), (N′, P (N′)) is sufficiently saturated and N′ contains enough parameters so that, whenever Z is definable over N and Z is a boolean combination of sets of the form X ∩ Y where X is ∅-definable and Y is L-definable, it is possible to choose the sets Y to be L-definable over N′. Let n ≥ 1 be finite and f̄ ∈ P (\overline{M})^{n}. Let f̄′ ⊧ tp(f̄/M) be such that tp(f̄′/N) is finitely realisable in M. Let f̄′′ ⊧ tp(f̄′/N) be such that tp_{L}(f̄′′/N′) is finitely realisable in M. It follows from condition (e) that tp(f̄′/N) is implied by tp_{L}(f̄′′/N′) in conjunction with tp(f̄/M). So condition (iii) is satisfied with κ = n and f̂ = f̄. □

4. An example

We now consider a theory studied by Belegradek and Zilber in [1]. Let ℝ be the real field and 𝕊 the unit circle thought of as a subgroup of the multiplicative group of the complex field ℂ. Let Γ(ℝ) ≤ 𝕊 be a subgroup. Let Γ(ℝ)^{[n]} = {g^{n} : g ∈ Γ(ℝ)}. With reference to [6], Γ(ℝ) is said in [1] to have the Lang property if, for any finite n ≥ 1 and algebraic set X ⊆ ℂ^{n}, X ∩Γ(ℝ)^{n} is a finite union of cosets of subgroups of Γ(ℝ)^{n}. Assume Γ(ℝ) satisfies the following three conditions:

• |Γ(ℝ)| = ℵ_{0},

• |Γ(ℝ)/Γ(ℝ)^{[n]}| < ℵ_{0} for every finite n ≥ 1, • Γ(ℝ) has the Lang property.

Let L = {<, +, ·, 0, 1, Re(g), Im(g)}_{g∈Γ(ℝ)} where Re(g) and Im(g) are suggestively named constant symbols for the real part and the imaginary part of each member of Γ(ℝ). Let T be the resulting L-theory of ℝ. Let Re : 𝕊 → ℝ be the function which assigns to each member of 𝕊 its real part. Let P be a new unary predicate and L_{P} = L ∪ {P }. Interpret P such that P (ℝ) = Re(Γ(ℝ)). Let T_{P} be the resulting L_{P} -theory of ℝ. Let Γ be a new binary predicate and L_{Γ} = L ∪ {Γ}. Let the suggestively named Γ(ℝ) be the interpretation of Γ in ℝ. Let T_{Γ} be the resulting L_{Γ}-theory of ℝ. As is noted in [1], Γ(ℝ) = Re^{−1}(Re(Γ(ℝ))) and so T_{P} and T_{Γ} are definitionally equivalent.

T_{Γ} was introduced and studied by Belegradek and Zilber in [1]. They gave axioms for it and proved a near model completeness result. Expecting a positive answer, they asked if T_{Γ} has NIP. We use Theorem 2.1 to obtain a positive answer. A similar result is given in [5] for a theory of the real field with a multiplicative subgroup which has the Mann property, this theory having been studied by van den Dries and Günaydin in [4].

We check that T_{P} satisfies the assumptions of Theorem 2.1. Let (M, P (M)) ≺ (N, P (N)) ≺ (\overline{M}, P (\overline{M})) ⊧ T_{P} be such that all three models are sufficiently saturated. Since T is an expansion by constants of the theory of real closed ordered fields, it is clear that conditions (i) and (iv) are satisfied. We deduce conditions (ii) and (iii) from the results in [1]. The argument overlaps with the reasoning in [1]. Since the predicate Γ is ∅-definable in T_{P} we shall feel free to use it. Re will now denote the real part map from the unit circle in the big model (\overline{M}, P (\overline{M})). So Γ(\overline{M}) = Re^{−1}(P (\overline{M})). Since L contains constant symbols for all real parts and all imaginary parts of members of Γ(ℝ), we may assume Γ(ℝ) ≤ Γ(M). Let Γ(\overline{M})_{d} be the largest divisible subgroup of Γ(\overline{M}). As is observed in [1], Γ(\overline{M})_{d} has a direct complement H in Γ(\overline{M}) such that |H| ≤ 2^{ℵ_{0}}. Let Γ(\overline{M})_{d}^{T} be the torsion subgroup of Γ(\overline{M})_{d}. Clearly |Γ(\overline{M})_{d}^{T} | ≤ ℵ_{0}. Let K = \{h × g : h ∈ H and g ∈ Γ(\overline{M})_{d}^{T} \}. Given the sufficient saturation of M, we may assume K ≤ Γ(M).

The following is proved in [1] by means of a back-and-forth argument.

Theorem 4.1. Let F ≤ Γ(\overline{M}) be a subgroup with the following properties: (α) F = \{k × d : k ∈ K and d ∈ D\} for some divisible subgroup D ≤ Γ(\overline{M}), (β) |F | ≤ 2^{ℵ_{0}}, (γ) Γ(ℝ) ≤ F.

Let n ≥ 1 be finite and let ā ∈ \overline{M}^{n} be acl_{L}-independent over P (\overline{M}). Then tp(F ā) is implied by the information so far mentioned in conjunction with tp_{L}(F ā).

Let b ∈ \overline{M} \acl_{L}(NP (\overline{M})). Let ḡ be a finite tuple from P (N) and c̄ a finite tuple from N such that c̄ is acl_{L}-independent over P (N) and hence over P (\overline{M}). Let F̃ ≤ Γ(N) satisfy conditions (α), (β) and (γ) and be such that ḡ ∈ dcl_{L}(F̃). It follows from Theorem 4.1 that tp(b/ḡc̄) is implied by tp_{L}(b/F̃c̄) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})). Therefore tp(b/N) is implied by tp_{L}(b/N) in conjunction with the information that b ∉ acl_{L}(NP (\overline{M})). So condition (ii) is satisfied.

Let n ≥ 1 be finite and f̄ ∈ P (\overline{M})^{n}. Let F ≤ Γ(\overline{M}) satisfy conditions (α), (β) and (γ) and be such that f̄ ∈ acl(F). Let F′ ⊧ tp(F/M) be such that tp(F′/N) is finitely realisable in M. Let ḡ be a finite tuple from P (N) and c̄ a finite tuple from N such that c̄ is acl_{L}-independent over P (N) and hence over P (\overline{M}). Let F̃ ≤ Γ(N) satisfy conditions (α), (β) and (γ) and be such that ḡ ∈ dcl_{L}(F̃). It is clear that the subgroup {a × b : a ∈ F′ and b ∈ F̃} also satisfies conditions (α), (β) and (γ). It is also clear that this is a consequence of tp(F′/M) in conjuction with tp(F̃/M). It follows from Theorem 4.1 that tp(F′/ḡc̄) is implied by tp_{L}(F′/F̃c̄) in conjunction with tp(F′/M). Therefore tp(F′/N) is implied by tp_{L}(F′/N) in conjunction with tp(F′/M). Setting f̂ = Re(F), condition (iii) is satisfied with (N′, P (N′)) = (N, P (N)) and κ ≤ 2^{ℵ_{0}}.

References

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