CHAINS OF THEORIES AND COMPANIONABILITY

ÖZCAN KASAL AND DAVID PIERCE

Abstract. The theory of fields with a countably infinite family of commuting derivations is not companionable; but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable. Each of these two theories is the union of a chain of companionable theories. In a signature with predicates, in all finite numbers of arguments, for linear dependence of vectors, the two-sorted theory of vector-spaces with their scalar-fields is companionable, and it is the union of a chain of companionable theories, but its model-companion is not the union of the model-companions of the theories in the chain. Finally, the union of a chain of non-companionable theories may be companionable.

A theory in a given signature is a set of sentences, in the first-order logic of that signature, that is closed under logical implication. We shall consider chains (T_{m}: m ∈ ω) of theories: this means

T_{0} ⊆ T_{1} ⊆ T_{2} ⊆ ⋯  (∗)

The signature of T_{m} will be 𝒮_{m}, so automatically 𝒮_{0} ⊆ 𝒮_{1} ⊆ 𝒮_{2} ⊆ ⋯

In one motivating example, 𝒮_{m} is {0, 1, −, +, · , ∂_{0}, …, ∂_{m−1}}, the signature of fields with m additional singulary function-symbols; and T_{m} is m-DF, the theory of fields (of any characteristic) with m commuting derivations. In this example, each T_{m+1} is a conservative extension of T_{m}, that is, T_{m+1} ⊇ T_{m} and every sentence in T_{m+1} of signature 𝒮_{m} is already in T_{m}. We establish this by showing that every model of T_{m} expands to a model of T_{m+1}. (This condition is sufficient, but not necessary [3, §2.6, exer. 8, p. 66].) If (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, then (K, ∂_{0}, …, ∂_{m}) ⊧ (m + 1)-DF, where ∂_{m} is the 0-derivation.

The union of the theories m-DF can be denoted by ω-DF : it is the theory of fields with ω-many commuting derivations. Each of the theories m-DF has a model-companion, called m-DCF [11]; but we shall show (as Theorem 3 below) that ω-DF has no model-companion. Let us recall that a model-companion of a theory T is a theory T^{∗} in the same signature such that (1) T_{∀} = T^{∗}_{∀}, that is, every model of one of the theories embeds in a model of the other, and (2) T^{∗} is model-complete, that is, T^{∗} ∪ diag(𝔐) axiomatizes a complete theory for all models 𝔐 of T^{∗}. Here diag(𝔐) is the quantifier-free theory of 𝔐 with parameters: equivalently, diag(𝔐) is the theory of all structures in which 𝔐_{M} embeds. (These notions, with historical references, are reviewed further in [11].) A theory has at most one model-companion, by an argument with interwoven elementary chains.

Let m-DF_{0} be m-DF with the additional requirement that the field have characteristic 0. Then m-DF_{0} has a model-companion, called m-DCF_{0} [6]. We shall show (as Theorem 6 below) that m-DCF_{0} ⊆ (m + 1)-DCF_{0}. It will follow then that the union ω-DF _{0} of the m-DF_{0} has a model-companion, which is the union of the m-DCF_{0}. This is by the following general result, which has been observed also by Alice Medvedev [7, 8]. Again, the theories T_{k} are as in (∗) above.

Theorem 1. Suppose each theory T_{k} has a model-companion T_{k}^{∗}, and

T_{0}^{∗} ⊆ T_{1}^{∗} ⊆ T_{2}^{∗} ⊆ ⋯  (†)

Date: January 8, 2013.

Then the theory ⋃_{k∈ω} T_{k} has a model-companion, namely ⋃_{k∈ω} T_{k}^{∗}.

Proof. Write U for ⋃_{k∈ω} T_{k}, and U^{∗} for ⋃_{k∈ω} T_{k}^{∗}. Suppose 𝔄 ⊧ U, and Γ is a finite subset of U^{∗} ∪ diag(𝔄). Then Γ is a subset of T_{k}^{∗} ∪ diag(𝔄 ↾ 𝒮_{k}) for some k in ω, and also 𝔄 ↾ 𝒮_{k} ⊧ T_{k}. Since (T_{k}^{∗})_{∀} ⊆ T_{k}, we conclude that Γ is consistent. Therefore U^{∗} ∪ diag(𝔄) is consistent. Thus U^{∗}_{∀} ⊆ U. By symmetry U_{∀} ⊆ U^{∗}.

Similarly, if 𝔅 ⊧ U^{∗}, then T_{k}^{∗} ∪ diag(𝔅 ↾ 𝒮_{k}) axiomatizes a complete theory in each case, and therefore U^{∗} ∪ diag(𝔅) is complete. □

The foregoing proof does not require that the signatures 𝒮_{k} form a chain, but needs only that every finite subset of ⋃_{k∈ω} 𝒮_{k} be included in some 𝒮_{k}. This is the setting for Medvedev’s [8, Prop. 2.4, p. 6], which then has the same proof as the foregoing. Also in Medvedev’s setting, each T_{k+1}^{∗} is a conservative extension of T_{k}^{∗}; but only the weaker assumption T_{k}^{∗} ⊆ T_{k+1}^{∗} is needed in the proof.

Medvedev notes that many properties that the theories T_{k} might have are ‘local’ and are therefore preserved in ⋃_{k∈ω} T_{k}: examples are completeness, elimination of quantifiers, stability, and simplicity. In her main application, 𝒮_{n} is the signature of fields with singulary operation-symbols σ_{m/n!}, where m ∈ ℤ; and T_{n} is the theory of fields on which the σ_{m/n!} are automorphisms such that

σ_{k/n!} ∘ σ_{m/n!} = σ_{(k+m)/n!}.

Then T_{n} includes the theory S_{n} of fields with the single automorphism σ_{1/n!}. Using [12, §1] (which is based on [3, ch. 5]), we may observe at this point that reduction of models of T_{n} to models of S_{n} is actually an equivalence of the categories Mod^{⊆}(T_{n}) and Mod^{⊆}(S_{n}), whose objects are models of the indicated theories, and whose morphisms are embeddings. We thus have at hand a (rather simple) instance of the hypothesis of the following theorem.

Theorem 2. Suppose (I, J) is a bi-interpretation of theories S and T such that I is an equivalence of the categories Mod^{⊆}(S) and Mod^{⊆}(T). If S has the modelcompanion S^{∗}, and S ⊆ S^{∗}, then T also has a model-companion, which is the theory of those models 𝔅 of T such that J(𝔅) ⊧ S^{∗}.

Proof. The class of models 𝔅 of T such that J(𝔅) ⊧ S^{∗} is elementary. Let T^{∗} be its theory. Then T ⊆ T^{∗}. Suppose 𝔅 ⊧ T. Then J(𝔅) ⊧ S, so J(𝔅) embeds in a model 𝔄 of S^{∗}. Consequently I(J(𝔅)) embeds in I(𝔄). Also I(𝔄) ⊧ T^{∗}, since 𝔄 ≅ J(I(𝔄)). Since also 𝔅 ≅ I(J(𝔅)), we conclude that 𝔅 embeds in a model of T^{∗}. Finally, T^{∗} is model-complete. Indeed, suppose now 𝔅 and ℭ are models of T^{∗} such that 𝔅 ⊆ ℭ. Then J(𝔅) embeds in J(ℭ), and these structures are models of S^{∗}, so the embedding is elementary. Therefore an elementary embedding of I(J(𝔅)) in I(J(ℭ)) is induced. By the equivalence of the categories, 𝔅 ≼ ℭ. □

In the present situation, the theory S_{n} has a model-companion [5, 1]; let us denote this by ACFA_{n}. By the theorem then, T_{n} has a model-companion T_{n}^{∗}, which is axiomatized by T_{n} ∪ACFA_{n}. We have ACFA_{n} ⊆ T_{n+1}^{∗} by [1, 1.12, Cor. 1, p. 3013]. By Theorem 1 then, ⋃_{n∈ω} T_{n} has a model-companion, which is the union of the T_{n}^{∗}. Medvedev calls this union ℚACFA; she shows for example that it preserves the simplicity of the ACFA_{n}, as noted above, though it does not preserve their supersimplicity.

The following is similar to the result that the theory of fields with a derivation and an automorphism (of the field-structure only) has no model-companion [10]. The obstruction lies in positive characteristics p, where all derivatives of elements with p-th roots must be 0.

Theorem 3. The theory ω-DF has no model-companion.

Proof. We use that an ∀∃ theory T has a model-companion if and only if the class of its existentially closed models is elementary, and in this case the model-companion is the theory of this class [2]. (A model 𝔄 of T is an existentially closed model, provided that if 𝔅 ⊧ T and 𝔄 ⊆ 𝔅, then 𝔄 ≼_{1} 𝔅, that is, all quantifier-free formulas over A that are soluble in 𝔅 are soluble in 𝔄.) For each n in ω, the theory ω-DF has an existentially closed model 𝔄_{n}, whose underlying field includes 𝔽_{p}(α), where α is transcendental; and in this model,

∂_{k}α = \begin{cases}1, & if k = n, \\ 0, & otherwise. \end{cases}

Then α has no p-th root in 𝔄_{n}. Therefore, in a non-principal ultraproduct of the 𝔄_{n}, α has no p-th root, although ∂_{n}α = 0 for all n in ω, so that α does have a p-th root in some extension. Thus the ultraproduct is not an existentially closed model of ω-DF . Therefore the class of existentially closed models of ω-DF is not elementary. □

It follows then by Theorem 1 that m-DCF ⊈ (m + 1)-DCF for at least one m. We could contrive examples to show this independently; but this by itself would not be enough to establish the last theorem. For, by the results of [12], it is possible for each T_{k} to have a model-companion T_{k}^{∗}, while ⋃_{k∈ω} T_{k} has a model-companion that is not ⋃_{k∈ω} T_{k}^{∗}. We may even require T_{k+1} to be a conservative extension of T_{k}.

Indeed, if k > 0, then in the notation of [12], VS_{k} is the theory of vectorspaces with their scalar-fields in the signature {+, −, 0, ∘, 0, 1, ∗, P^{k}}, where ∘ is multiplication of scalars, and ∗ is the action of the scalar-field on the vector-space, and P^{k} is k-ary linear dependence. In particular, P^{2} may written also as ∥. Then VS_{k} has a model-companion, VS_{k}^{∗}, which is the theory of k-dimensional vectorspaces over algebraically closed fields [12, Thm 2.3]. Let VS_{ω} = ⋃_{1⩽k<ω} VS_{k}. (This was called VS_{∞} in [12].) This theory has the model-companion VS_{ω} ^{∗}, which is the theory of infinite-dimensional vector-spaces over algebraically closed fields [12, Thm 2.4].

Theorem 4. If 1 ⩽ n < ω, let

T_{n} = ⋃_{1⩽k⩽n} VS_{k} .

Then T_{n} has a model-companion T_{n}^{∗}, which is axiomatized by VS_{n}^{∗} ∪ T_{n}. Also T_{n+1} is a conservative extension of T_{n}. However, the model-companion VS_{ω} ^{∗} of the union VS_{ω} of the chain (T_{n}: 1 ⩽ n < ω) is not the union of the T_{n}^{∗}.

Proof. Every vector-space can be considered as a model of any VS_{k} and hence of any T_{k}. In particular, T_{n+1} is a conservative extension of T_{n}. The relation P^{1} is defined by x = 0. Let VS_{n}^{m} be axiomatized by VS_{n} and the requirement that the space have dimension at least n. If n > 2, then there are existential formulas that, in each model of VS_{n}^{m}, define the relation ∥ and its complement [12, §2, p. 431]. Similarly, if 2 ⩽ k < n − 1, then, using existential formulas, we can define P^{k+1} and its complement in models of VS_{k} ∪ VS_{n}^{m}: indeed, ¬P^{k+1}x_{0} ⋯ x_{k} is equivalent to ∃(x_{k+1}, …, x_{n−1}) P^{n}x_{0} ⋯ x_{n−1}, and P^{k+1}x_{0} ⋯ x_{k} is equivalent to

∃(x_{k+1}, …, x_{n}) (P^{k}x_{1} ⋯ x_{k} ∨ (¬P^{n}x_{1} ⋯ x_{n} ∧ ⋀_{j=k}^{n} P^{n}x_{0} ⋯ x_{j−1}x_{j+1} ⋯ x_{n})).

Therefore reduction from models of T_{n} to models of VS_{n}^{m} is an equivalence of the categories Mod^{⊆}(T_{n}) and Mod^{⊆}(VS_{n}^{m}) [12, Lem 1.1]. Since VS_{n} ⊆ VS_{n}^{m} and every model of VS_{n} embeds in a model of VS_{n}^{m}, the two theories have the same modelcompanion, namely VS_{n}^{∗}. Therefore VS_{n}^{∗} ∪ T_{n} axiomatizes T_{n}^{∗}, by Theorem 2. The rest has already been noted. □

Let VS_{n}^{r} comprise the sentences of VS_{n} having one-sorted signature {0, −, +, P^{n}}. If n > 1, then reduction of VS_{n} to VS_{n}^{r} is an equivalence of categories Mod^{⊆}(VS_{n}) and Mod^{⊆}(VS_{n}^{r}), and therefore VS_{n}^{r} is companionable. (This was mentioned at [12, p. 431], but the details, and in particular Theorem 2, were not spelled out.) So there is an alternative, one-sorted version of the last theorem, where T_{n} is ⋃_{1⩽k⩽n} VS_{k}^{r}.

The implication A ⇒ B in the following is used implicitly at [1, 1.12, p. 3013] to establish the result used above, that if (K, σ) is a model of ACFA, then so is (K, σ^{m}), assuming m ⩾ 1.

Theorem 5. Assuming as usual T_{0} ⊆ T_{1}, where each T_{k} has signature 𝒮_{k}, we consider the following conditions.

A. For every model 𝔄 of T_{1} and model 𝔅 of T_{0} such that

𝔄 ↾ 𝒮_{0} ⊆ 𝔅,  (‡)

there is a model ℭ of T_{1} such that

𝔄 ⊆ ℭ,  𝔅 ⊆ ℭ ↾ 𝒮_{0}.  (§)

B. The reduct to 𝒮_{0} of every existentially closed model of T_{1} is an existentially

closed model of T_{0}. C. T_{0} has the Amalgamation Property: if one model embeds in two others, then those two in turn embed in a fourth model, compatibly with the original embeddings.

D. T_{1} is ∀∃ (so that every model embeds in an existentially closed model). We have the two implications

A ⟹ B,  B & C & D ⟹ A,

but there is no implication among the four conditions that does not follow from these. This is true, even if T_{1} is required to be a conservative extension of T_{0}.

Proof. Suppose A holds. Let 𝔄 be an existentially closed model of T_{1}, and let 𝔅 be an arbitrary model of T_{0} such that (‡) holds. By hypothesis, there is a model ℭ of T_{1} such that (§) holds. Then 𝔄 ≼_{1} ℭ, and therefore 𝔄 ↾ 𝒮_{0} ≼_{1} ℭ ↾ 𝒮_{0}, and a fortiori 𝔄 ↾ 𝒮_{0} ≼_{1} 𝔅. Therefore 𝔄 ↾ 𝒮_{0} must be an existentially closed model of T_{0}. Thus B holds.

Suppose conversely B holds, along with C and D. Let 𝔄 ⊧ T_{1} and 𝔅 ⊧ T_{0} such that (‡) holds. We establish the consistency of T_{1} ∪ diag(𝔄) ∪ diag(𝔅). It is enough to show the consistency of

T_{1} ∪ diag(𝔄) ∪ {∃x φ(x)},  (¶)

where φ is an arbitrary quantifier-free formula of 𝒮_{0}(A) that is soluble in 𝔅. By D, there is an existentially closed model ℭ of T_{1} that extends 𝔄. By B then, ℭ ↾ 𝒮_{0} is an existentially closed model of T_{0} that extends 𝔄↾𝒮_{0}. By C, both 𝔅 and ℭ↾𝒮_{0} embed over 𝔄 ↾ 𝒮_{0} in a model of T_{0}. In particular, φ will be soluble in this model. Therefore φ is already soluble in ℭ ↾ 𝒮_{0} itself. Thus ℭ is a model of (¶). Therefore A holds.

The foregoing arguments eliminate the five possibilities marked X on the table below, where 0 means false, and 1, true. We give examples of each of the remaining cases, numbered according to the table. In each example, T_{0} will be the reduct of T_{1} to 𝒮_{0}. We shall denote by 𝒮_{f} the signature {+, · , −, 0, 1} of fields; and by 𝒮_{vs}, the signature {+, −, 0, ∘, 0, 1, ∗} of vector-spaces as two-sorted structures.

1 X 2 3 4 X 5 6 7 X 8 9 10 X X 11
A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
B 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
C 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
D 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

1. We first give an example in which none of the four lettered conditions hold. Let 𝒮_{0} = 𝒮_{f} ∪ {a, b} and 𝒮_{1} = 𝒮_{0} ∪ {c}. Let T_{1} be the theory of fields of characteristic p with distinguished elements a, b, and c such that {a, c} or {b, c} is p-independent, and if {b, c} is p-independent, then so is {b, c, d} for some d. Then T_{0} is the theory of fields of characteristic p in which, for some c, {a, c} or {b, c} is p-independent, and if {b, c} is p-independent, then so is {b, c, d} for some d. The negations of the four lettered conditions are established as follows. Throughout, a, b, c, and d will be algebraically independent over 𝔽_{p}. ¬A. We have

(𝔽_{p}(a, b^{1/p}, c), a, b, c) ⊧ T_{1},  (𝔽_{p}(a, b^{1/p}, c^{1/p}), a, b) ⊧ T_{0},

but if (𝔽_{p}(a, b^{1/p}, c), a, b, c) is a substructure of a model (K, a, b, c) of T_{1}, then

K cannot contain c^{1/p}.

¬B. T_{0} has no existentially closed models, since an element of a model that is p-independent from a or b will always have a p-th root in some extension. Similarly, no model of T_{1} in which {a, c} is not p-independent is existentially closed. But T_{1} does have existentially closed models, which are just the separably closed fields of characteristic p with p-basis {a, c} and with an additional element b.

¬C. T_{0} does not have the Amalgamation Property, since (𝔽_{p}(a, b^{1/p}, c), a, b) and (𝔽_{p}(a^{1/p}, b, c, d), a, b) are models that do not embed in the same model over

the common substructure (𝔽_{p}(a, b, c), a, b), which is a model of T_{0}.

¬D. T_{1} is not ∀∃, since, as we have already noted, models in which {a, c} is not

p-independent do not embed in existentially closed models.

2. For an example of the column headed by 2 in the table, we let 𝒮_{0} and 𝒮_{1} be as in 1; but now T_{1} is the theory of fields of characteristic p with distinguished elements a, b, and c such that {a, c, d} or {b, c, d} is p-independent {b, c, d} for some d. This ensures that T_{1} has no existentially closed models, so B holds vacuously; but the other three conditions still fail.

3. T_{0} and T_{1} are the same theory, so A and B hold trivially; and this theory is the theory of vector-spaces of dimension at least 2, in the signature of vector-spaces, so it neither has the Amalgamation Property, nor is ∀∃.

4. T_{1} is DF_{p} with the additional requirement that the field have p-dimension at least 2; and 𝒮_{0} = 𝒮_{f}, so T_{0} is the theory of fields of characteristic p with pdimension at least 2. The latter theory has the Amalgamation Property; but the other conditions fail. Indeed, let (𝔽_{p}(a, b), D) be the model of T_{1} in which Da = 1 and Db = 0: then the field 𝔽_{p}(a, b) embeds in 𝔽_{p}(a^{1/p}, b), which is a model of T_{0}, but D does not extend to this field. Also, T_{0} has no existentially closed models; but T_{1} does, and indeed it has a model-companion, namely DCF_{p}. Also T_{1} is not ∀∃, since T_{0} is not: there is a chain of models of the latter, whose union is not a model, and we can make the structures in the chain into models of T_{1} by adding the zero derivation.

5. 𝒮_{0} = 𝒮_{f}, and 𝒮_{1} = 𝒮_{0} ∪ {a}. T_{1} is the theory of fields of characteristic p with distinguished element a, which is p-independent from another element; so T_{0} is (as in 4) the theory of fields of characteristic p with p-dimension at least 2. Here T_{1} has no existentially closed models, so B holds trivially.

6. T_{0} and T_{1} are the same, namely the theory of fields of characteristic p of positive p-dimension, in the signature of fields, so this theory has the Amalgamation Property, but is not ∀∃.

7. 𝒮_{0} = 𝒮_{vs}, 𝒮_{1} = 𝒮_{0} ∪ {∥, a, b}, and T_{1} is axiomatized by VS_{2} ∪{a ∦ b}, so it is ∀∃. Then T_{0} is the theory of vector-spaces of dimension at least 2. As in Theorem 4 above, T_{1} has a model-companion, namely the theory of vector-spaces over algebraically closed fields with basis {a, b}. But T_{0} has no existentially closed models, since for all independent vectors a and b in some model, the equation x ∗ a + y ∗ b = 0 is always soluble in some extension. Thus B fails. Then T_{0} also does not have the Amalgamation Property, since the solutions of the given equation may satisfy 2x^{2} = y^{2} in one extension, but 3x^{2} = y^{2} in another. Similarly, A fails, since the reduct to 𝒮_{0} of a model of T_{1} may embed in a model of T_{0} in which a and b are parallel.

8. 𝒮_{0} = 𝒮_{vs} ∪ {∥}, 𝒮_{1} = 𝒮_{0} ∪ {a, b}, and T_{1} is axiomatized by VS_{2} together with

∀x ∀y (x ∗ a + y ∗ b = 0 → 2x^{2} = y^{2}).  (∥)

Then T_{0} is the theory of vector-spaces such that either the dimension is at least 2, or the scalar field contains √2. As in 7, T_{0} does not have the Amalgamation Property. The theory T_{1} is ∀∃. It also has the model (ℚ ∗ a ⊕ ℚ ∗ b, a, b), and ℚ ∗ a ⊕ ℚ ∗ b embeds in the model ℚ(√2, √3) ∗ a of T_{0} when we let b = √3 ∗ a; but then the latter space embeds in no space in which a and b are as required by (∥). So A fails. Finally, T_{1} has a model-companion, axiomatized by VS_{2}^{∗} together with

∃x ∃y (x ∗ a + y ∗ b = 0 ∧ 2x^{2} = y^{2} ∧ x ≠ 0);

and T_{0} has a model-companion, which is just VS_{2}^{∗}; so B holds.

9. T_{0} and T_{1} are both VS_{1}.

10. T_{1} = DF_{p}, and T_{0} is the reduct to 𝒮_{f}, namely field-theory in characteristic p.

11. T_{0} and T_{1} are both field-theory. □

Now let ω-DCF _{0} = ⋃_{m∈ω} m-DCF_{0}.

Theorem 6. For all m in ω,

m-DCF_{0} ⊆ (m + 1)-DCF_{0}.

Therefore ω-DF _{0} has a model-companion, which is ω-DCF _{0}. This theory admits full elimination of quantifiers, is complete, and is properly stable.

Proof. Suppose (L, ∂_{0}, …, ∂_{m−1}) is a model of m-DF_{0}, and L has a subfield K that is closed under the ∂_{i} (where i < m), and there is also a derivation ∂_{m} on K such that (K, ∂_{0} ↾ K, …, ∂_{m−1} ↾ K, ∂_{m}) is a model of (m + 1)-DF_{0}. We shall include (L, ∂_{0}, …, ∂_{m−1}) in another model of m-DF_{0}, namely a model that expands to a model of (m + 1)-DF_{0} that extends (K, ∂_{0}, …, ∂_{m}). By the last theorem, it will follow that m-DCF_{0} ⊆ (m + 1)-DCF_{0}. Since m is arbitrary, it will follow by Theorem 1 that ω-DCF _{0} is the model-companion of ω-DF _{0}.

If K = L, we are done. So suppose a ∈ L ∖ K. We shall define an extension K(a^{ξ}: ξ ∈ ω^{m+1}) of K, and for each i in m + 1, we shall define a derivation ˜∂_{i} on this extension so that

˜∂_{i} ↾ K = ∂_{i} ↾ K.  (∗∗)

For each σ in ω^{m+1}, we shall require

σ(m) = 0 ⟹ a^{σ} = ∂_{0}^{σ(0)} ⋯ ∂_{m−1}^{σ(m−1)}a.  (††)

If i < m + 1, let i denote the element of ω^{m+1} that takes the value 1 at i and 0 elsewhere. Then for each σ in ω^{m+1}, we shall require

σ(i) > 0 ⟹ ˜∂_{i}a^{σ−i} = a^{σ},  (‡‡)
σ(m) > 0 ∧ a^{σ−m} ∉ K(a^{ξ}: ξ < σ − m)^{alg} ⟹ a^{σ} ∉ L(a^{ξ}: ξ < σ)^{alg}.  (§§)

These conditions ensure that the derivations ˜∂_{i}, if they do exist, are unique.

This uniqueness of the ˜∂_{i} would be ensured, if the field L in the conclusion of (§§) were replaced with K. But the condition (§§), as it is, along with (∗∗), (††), and (‡‡), will ensure that, if i < m, then ˜∂_{i} will agree with ∂_{i} wherever they are both defined, that is, on K(a^{ξ}: ξ ∈ ω^{m+1}) ∩ L. Indeed, the conditions (∗∗), (††), and (‡‡) ensure this agreement on K(a^{ξ}: ξ(m) = 0). Moreover, we have

K(a^{ξ}: ξ(m) = 0) = K(a^{ξ}: ξ ∈ ω^{m+1}) ∩ L.

For, suppose σ(m) > 0 and a^{σ} ∈ L. Then by (§§) we have a^{σ−m} ∈ K(a^{ξ}: ξ < σ − m)^{alg}, and therefore, by (∗∗) and (‡‡), applying ˜∂_{m} to a^{σ−m} yields a^{σ} ∈ K(a^{ξ}: ξ < σ). Repeating this result, we obtain that a^{σ} is a rational function over K(a^{ξ}: ξ(m) = 0) of certain a^{η}, where η < σ; and these a^{η} are algebraically independent over L. Since a^{σ} ∈ L, we conclude a^{σ} ∈ K(a^{ξ}: ξ(m) = 0).

The last observation means that, once we have constructed K(a^{ξ}: ξ ∈ ω^{m+1}) as desired, then, if L ∖ K(a^{ξ}: ξ ∈ ω^{m+1}) is nonempty, we can repeat the process, using an element of this difference in place of a. Thus, ultimately, we shall obtain the desired model of (m + 1)-DF_{0} whose universe includes L.

We shall build up K(a^{ξ}: ξ ∈ ω^{m+1}) recursively, and we shall (simultaneously) establish by induction that the desired conditions are satisfied. We shall use the ordering of the σ in ω^{m+1} determined by the left-lexicographic ordering of

(σ(m), σ(0) + ⋯ + σ(m − 1), σ(0), σ(1), …, σ(m − 2)).

Then ω^{m+1} has the order-type of ω itself, and we shall have

K(a^{ξ}: ξ ∈ ω^{m+1}) = ⋃_{τ∈ω^{m+1}} K(a^{ξ}: ξ < τ).

When τ = (0, …, 0, 1), then, using (††) as a definition, we have the field K(a^{ξ}: ξ < τ) as desired. Suppose we have this field as desired for some τ in ω^{m+1} such that τ(m) > 0. In particular, for all i in m + 1, we have ˜∂_{i} as a derivation from K(a^{ξ}: ξ + i < τ) to K(a^{ξ}: ξ < τ), and the conditions (‡‡) and (§§) hold for all σ such that σ < τ. For defining a^{τ}, there are two cases to consider:

1. If a^{τ−m} is not algebraic over K(a^{ξ}: ξ < τ −m), then we let a^{τ} be transcendental over L(a^{ξ}: ξ < τ), as required by (§§). We are then free to define ˜∂_{m}a^{τ−m} as a^{τ}.

2. If a^{τ−m} is algebraic over K(a^{ξ}: ξ < τ − m), then ˜∂_{m}a^{τ−m} is determined

as an element of K(a^{ξ}: ξ < τ), and we let a^{τ} be this element. We now must check that, when i < m and τ(i) > 0, we can define ˜∂_{i}a^{τ−i} as a^{τ}. Again we consider two cases.

1. Suppose a^{τ−i} is algebraic over K(a^{ξ}: ξ < τ −i). Then ˜∂_{i}a^{τ−i} is determined as an element of K(a^{ξ}: ξ < τ). Thus the value of the bracket [˜∂_{i}, ˜∂_{m}] at a^{τ−i−m} is determined. But also, by (§§), a^{τ−i−m} must be algebraic over K(a^{ξ}: ξ < τ − i − m). Since the bracket is 0 on this field, it is 0 at a^{τ−i−m} as well [11, Lem. 4.2].

2. If a^{τ−i} is transcendental over K(a^{ξ}: ξ < τ − i), then since we are given ˜∂_{i} as a derivation whose domain is this field, we are free to define ˜∂_{i}a^{τ−i} as a^{τ}.

Thus we have obtained K(a^{ξ}: ξ ⩽ τ) as desired. Therefore ω-DF _{0} has the modelcompanion ω-DCF _{0}.

As noted, ω-DCF _{0} inherits quantifier-elimination, completeness, and stability from the m-DCF_{0}, which have these properties [6]. Although each m-DCF_{0} is actually ω-stable, ω-DCF _{0} is not even superstable, since if A is a set of constants (in the sense that all of their derivatives are 0), then as σ ranges over A^{ω} , the sets {∂_{m}x = σ(m): m ∈ ω} belong to distinct complete types. □

We may note that, in the foregoing proof, we cannot use Condition A of Theorem 5 in the stronger form in which the structure ℭ is required to be a mere expansion to 𝒮_{1} of 𝔅:

Theorem 7. If m > 0, there is a model of m-DF_{0} that does not expand to a model of (m + 1)-DF_{0}.

Proof. We generalize the example of [4] repeated in [9, Ex. 1.2, p. 927]. Suppose K is a pure transcendental extension ℚ(a^{σ}: σ ∈ ω^{m+1}) of ℚ, and ∂_{i}a^{σ} = a^{σ+i}. Let L be the pure transcendental extension K(b^{τ}: τ ∈ ω^{m−1}) of K, and if i < m − 1, let ∂_{i}b^{τ} = b^{τ+i}, while ∂_{m−1}b^{τ} = a^{(τ,0,0)}. Suppose ∂_{m} extends to L as well. We have

∂_{m−1}∂_{m}^{k}b^{τ} = ∂_{m}^{k}∂_{m−1}b^{τ} = ∂_{m}^{k}a^{(τ,0,0)} = a^{(τ,0,k)},

and these are all algebraically independent over ℚ(a^{σ}: σ ∈ ω^{m+1} ∧ σ(m − 1) > 0). However, ∂_{m−1}x is algebraic over this field whenever x is algebraic over K. Thus all of the ∂_{m}^{k}b^{τ} are algebraically independent over K; in particular, when k > 0, they do not belong to L. □

Finally, the union of a chain of non-companionable theories may be companionable:

Theorem 8. In the signature {f} ∪ {c_{k}: k ∈ ω}, where f is a singulary operationsymbol and the c_{k} are constant-symbols, let T_{0} be axiomatized by the sentences

∀x ∀y (fx = fy → x = y)

and, for each k in ω,

∀x (f^{k+1}x ≠ x), ∀x (fx = c_{k} → x = c_{k+1}), fc_{k+2} = c_{k+1} → fc_{k+1} = c_{k}.

For each n in ω, let T_{n+1} be axiomatized by

T_{n} ∪ {fc_{n+1} = c_{n}}.

Then

(1) each T_{n} is universally axiomatized, and a fortiori ∀∃, so it does have existentially closed models;

(2) each T_{n} has the Amalgamation Property;

(3) every existentially closed model of T_{n+1} is an existentially closed model of T_{n};

(4) no T_{n} is companionable;

(5) ⋃_{n∈ω} T_{n} is companionable.

Proof. Let 𝔄_{m} be the model of T_{0} with universe ω × ω such that

f^{𝔄_{m}}(k, ℓ) = (k, ℓ + 1),  c_{k}^{𝔄_{m}} = \begin{cases}(k − m, 0), & if k > m, \\ (0, m − k), & if k ⩽ m. \end{cases}

Let 𝔄_{ω} be the model of T_{0} with universe ℤ such that

f^{𝔄_{ω}} k = k + 1,  c_{k}^{𝔄_{ω}} = −k.

Then 𝔄_{m} is a model of each T_{k} such that k ⩽ m; and 𝔄_{ω} is a model of each T_{k}. Moreover, each model of T_{k} consists of a copy of some 𝔄_{β} such that k ⩽ β ⩽ ω, along with some (or no) disjoint copies of ω and ℤ in which f is interpreted as x ↦ x+1. Conversely, every structure of this form is a model of T_{k}. The β such that 𝔄_{β} embeds in a given model of T_{k} is uniquely determined by that model. Consequently T_{k} has the Amalgamation Property. Also, a model of T_{k} is an existentially closed model if and only if includes no copies of ω (outside the embedded 𝔄_{β}): This establishes that every existentially closed model of T_{k+1} is an existentially closed model of T_{k}.

The existentially closed models of T_{k} are those models that omit the type {∀y fy ≠ x} ∪ {x ≠ c_{j}: j ∈ ω}. In particular, 𝔄_{m} is an existentially closed model of T_{k}, if k ⩽ m; but 𝔄_{m} is elementarily equivalent to a structure that realizes the given type. Thus T_{k} is not companionable.

Finally, the model-companion of ⋃_{k∈ω} T_{k} is axiomatized by this theory, together with ∀x ∃y fy = x. □

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Middle East Technical University, Northern Cyprus Campus

Mimar Sinan Fine Arts University, Istanbul

E-mail address: dpierce@msgsu.edu.tr