Vector spaces with a dense-codense generic submodule

We study expansions of a vector space $V$ over a field $\mathbb F$, possibly with extra structure, with a generic submodule over a subring of $\mathbb F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP$_1$, NTP$_2$ and NSOP$_1$. We also study induced independence relations in the expansion.


Introduction
This paper brings together ideas of dense-codense expansions of geometric structures [6,3,8] with ideas about generic expansions by groups by D'Elbée [14]. Our base structures are vector spaces over a fixed field F, possibly with extra structure, such that the algebraic closure agrees with the F-linear span. We then fix a subring R of F and we study expansions by additive R-submodules satisfying some form of genericity (for technical details, see the definition of T U and T G in Definition 3.3), the main goal is to see how tame model-theoretic properties transfer from the original structure to the expansion.
There are many papers that deal with expansions by predicates and preservation of tame properties. There are general approaches [9,12] that study stable or NIP structures expanded by predicates, the main idea being that if the induced structure on the predicate is stable/NIP and if the formulas in the expansion are equivalent to bounded formulas (i.e. where the quantifiers range over the predicate), then the pair is again stable/NIP. On the other hand one can start with a geometric structure [6,3,8] and study preservation of properties like NTP 2 , strong dependence, supersimplicity or NSOP 1 under expansions by well-behaved dense codense predicates (for example, the predicate being an elementary substructure [3], a collection of algebraically independent elements [6] or a multiplicative substructure with the Mordell-Lang property [8,20,4]). This approach shares some ingredients with the previous one, formulas in the expanded language are equivalent to bounded formulas and the density property implies that the induced structure on the predicate is tame. One can also study different generic expansions, a classical example is the generic predicate, which preserves simplicity (see [10]). A more general construction is due to Winkler [26] in his thesis: given a model-complete L-theory T , and a language L ′ ⊃ L, the theory T can be considered as an L ′ -theory which has a model-companion, provided T has elimination of ∃ ∞ . Winkler [26] also considered the expansion of a theory by generic Skolem functions. Both expansions of Winkler were later shown to preserve the property NSOP 1 ( [21], [22]). One can also consider the expansion of a theory by a predicate for a reduct of this theory, for instance expanding a theory of fields by an additive or multiplicative generic subgroup (see [14,16,7]).
We start this paper (Section 2) by studying the expansion of a theory of F-vector space (in which the algebraic closure is the vector span) by a predicate for a generic R-submodule, where R stands for an integral domain such that F = Frac(R) = the fraction field of R. After adding predicates for pp-formulas in the language of R-modules, we characterize the expansions that are existentially closed, prove the existence of a model companion T G and in doing so show quantifier elimination for the expansion. As a Corollary of quantifier elimination, we show that T G is stable (resp. NIP) whenever T is stable (resp. NIP).
In Section 3, we study the general case where we drop the assumption F =R. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination and we prove a model-completeness result. In section 4 we use our description of definable sets to show that NIP and stability are preserved under these expansions. We would like to point out that these preservation results can be proved using the approach from [8].
In Section 5, we prove the main results of this paper: preservation of NTP 1 , NTP 2 and NSOP 1 in the expansion. The approach we follow is to check the property by doing a formula-by-formula analysis, separating the cases when the corresponding definable set is small (algebraic over the predicate) or large. To show the preservation of NSOP 1 we build on ideas presented in [23], the proofs for the preservation of NTP 1 , NTP 2 generalize ideas presented in [2,18]. In Section 6, we study independence notions in the expansion, assuming the original theory has a good notion of independence. As a Corollary, we give another proof for the preservation of simplicity and NSOP 1 .

A first example: the case F = Frac(R)
Let F be a field and let R be a subring of F such that F = Frac(R) (the fraction field of R). Let L 0 = {(λ·) λ∈F , +, 0} be the language of vector spaces over F and let L = +, 0, {λ·} λ∈F , . . . be an extension. Let T be a complete L-theory that expands the theory of vector spaces over F which has quantifier elimination in L and that satisfies the following properties: (i) Whenever M |= T and a ∈ M n , dcl( a) = acl( a) = span F ( a).
Let G be a unary predicate and for each formula φ( x) in the language L R−mod = {+, 0, (r·) r∈R } of R-modules, let P φ ( x) be a new predicate. Let L G be the expansion of L by G and P φ for all formulas φ in L R−mod . We will consider pairs (V, G) in the language L G that satisfy the following first order conditions: (A) G is an R-module and for all formulas φ( x) in the language of R-modules, ∀ x(P φ ( x) → G( x)) and ∀ x(G( x) → (P φ ( x) ↔φ( x)), whereφ( x) in the relativization of φ( x) to G and we interpret the relativization seeing G as an R-module. (B) (For all r ∈ R \ {0}, rG is dense in V ). For every L-formula φ(x, y), the axiom ∃ ∞ xφ(x, y) → ∃x(φ(x, y) ∧ rG(x)); (C) (Extension/co-density property) for any L-formulas φ(x, y) and ψ(x, y, z) and n ≥ 1, the axiom (∃ ∞ xφ(x, y) ∧ ∀ z∃ ≤n xψ(x, y, z)) → ∃x(φ(x, y) ∧ ∀ z(G( z) → ¬ψ(x, y, z))).
Let T G be the L G theory satisfying the schemes (A), (B) and (C). In all models (V, G) under consideration, the predicate G will interpret an R-module. Since modules have quantifier elimination up to boolean combinations of pp-formulas (see [27]), we can always assume that the predicates P φ ( x) are only defined for positive and negative instances of pp-formulas. Throughout this paper we will assume the reader is familiar with basic properties of pp-formulas inside R-modules. Notation 2. 1. In what follows, whenever (V, G) |= T G and A ⊂ V , we will write G(A) for A ∩ G(V ).
Our first goal is to show that the theory T G has quantifier elimination. We will start by proving properties of the divisible elements and the L-terms. Lemma 2.2 (Density of G div ). Let (V, G) be an |F| + -saturated model of T G . Let B ⊂ V such that |B| ≤ |F| and p(x) be a consistent non-algebraic L-type in a single variable over B. Let G div = r∈R\{0} rG. Then p(V ) ∩ G div (V ) is infinite.
Proof. By compactness, it is sufficient to show that for all r 1 , . . . , r n ∈ R, p(x) has infinitely realisations in n i=1 r i G. Let r = r 1 . . . r n . Now note that by axiom (B), the type rG(x) ∧ p(x) has infinitely many realisations. Lemma 2.3. Let t( x) be an L-term, then there exists L-formulas θ 1 ( x), . . . , θ n ( x) forming a partition of the universe, (i.e. such that V |= ∀ x( i θ i ( x))∧ i =j ¬∃ xθ i ( x)∧ θ j ( x)), and L 0 -terms t 1 ( x), . . . , t n ( x) such that Proof. Since T satisfies property (i), the algebraic closure agrees with the F-span, hence {t( x) = y}∪ λ · x = y | λ ∈ F n is inconsistent. Thus we can find λ 1 , . . . , λ n ∈ F n such that t( x) = y → i λ i · x = y. We may assume that λ i x = y define disjoint vector spaces. Now choose t i ( x) = λ i · x and let θ i ( x) be the formula t i ( x) = t( x). Theorem 2.4. The theory T G has quantifier elimination.
Proof. We show that the set of partial isomorphisms between two |L| + -saturated models (V, G), (V ′ , G ′ ) of T G has the back and forth property.
Let a ∈ V \ B. Since B = span F (B) and T satisfies property (i), we have that tp L (a/B) is non-algebraic. Every formula in tp QF LG (a/B), the quantifier free type in the extended language, is equivalent to a disjunction of formulas of the form is an L-term. By Lemma 2.3, up to a finite disjunction, we may assume that each term is an L 0 -term, i.e. of the form λx a tuple of linear combinations of x and b with coefficients in R) up to a finite disjunction of formulas of the form t j (x, b) / ∈ G.As a / ∈ B, conditions of the form t k (x, b) = 0 do not appear in tp QF LG (a/B). It turns out we only need to consider formulas of the form We will extend the map σ by cases.
Step 1. If a ∈ G. Then, as F =R, formulas of the form qx + b ∈ G, where q ∈ F, are equivalent to formulas of the form rx + g ∈ r ′ G for some r, r ′ ∈ R and g ∈ G(B) so conditions of the form rx + g ∈ r ′ G ∧ x ∈ G are equivalent to P ψ (x, g) for some L R−mod -formula ψ. By quantifier elimination in R-modules [27], the condition P ψ ( t(x, b)) is equivalent to a boolean combination of formulas of the form rx + r · b ∈ s 1 G + · · · + s n G. If x ∈ G, the latter is equivalent to rx + g ∈ s 1 G + · · · + s n G for some g ∈ G(B), so P ψ ( t(x, b)) is equivalent to some P ψ ′ (x, g) and tuple g ∈ G(B). Let q(x) = {P ψ (x, g) | G |= ψ(a, g), ψ ∈ L R−mod , g ∈ G(B)}, q σ (x) = σ(q)(x) and p(x) = σ(tp L (a/B)). To extend σ it is enough to show that there are infinitely many realisations of p(x) ∪ q σ (x) in V ′ . First using the fact that G(V ′ ) is |F| + -saturated and that |B ′ | < |F| + , there is a ′′ |= q σ (x). Let p sh (x) = p(x + a ′′ ), the type shifted by a ′′ . The type p sh is also non-algebraic, hence by density of R-divisible elements (Lemma 2.2), there exists d ∈ G div such that d |= p sh (x). Let a ′ = d + a ′′ .
Claim: a ′ |= q σ (x) ∪ p(x). First, as d |= p sh (x), a ′ = d + a ′′ |= p(x). By quantifier elimination in R-modules [27], every formula in q σ is a boolean combination of conditions of the form ra ′ + g ∈ r 1 G + · · · + r n G for r ∈ R and g ∈ G(B). As d ∈ G div , we have that rd ∈ r 1 G for all r ∈ R, hence ra ′ + g ∈ r 1 G + · · · + r n G if and only if ra + g ∈ r 1 G + · · · + r n G, so since a ′′ |= q σ (x), we also have a ′ |= q σ (x).
It follows that q σ (x) ∪ p(x) has infinitely many realisations.
Step 3. If a / ∈ span F (GB) then formulas of the form qx + b ∈ G do not appear in tp QF LG (a/B), and the formula P ψ (x, g) belong to tp QF LG (a/B) only when ψ is the negation of a pp-formula. Thus it is enough to show that σ(tp L (a/B)) have infinitely many realisations in V ′ \ span F (GB ′ ), which follows easily from condition (C), compactness and the fact that |B| ′ < |F| and the fact that Corollary 2.5. Let (V, G) and (V ′ , G ′ ) be two models of T G . Then whenever a ∈ V , a ′ ∈ V ′ are two tuples of the same length such that (1) tp R−mod (G(span R ( a))) = tp R−mod (G(span R ( a ′ ))) (the types agree in the sense of R-modules 1 ); (2) tp L ( a) = tp L ( a ′ ) (their types agree in the language L) ; Then tp G ( a) = tp G ( a ′ ).
Proof. From Theorem 2.4, T G has quantifier elimination, hence every formula in tp G ( a) is equivalent to a disjunction of formulas of the form for some quantifier-free L-formula φ( x), ψ an R-module formula and L-terms (t i ( x), t j ( x)) i,j , t( x). Using Lemma 2.3, we may assume that terms are F-linear combinations and that t( a) is a tuple of R-linear combinations (as in the proof of Theorem 2.4) in G. It follows that tp G ( a) is equivalent to a set of formulas of the form for q i , q j ∈ F. As F = Frac(R), each condition of the form q i · a ∈ G is equivalent to a condition r i · a ∈ s i G for r i , s i ∈ R. From condition (1), r i · a ∈ s i G if and only if r i · a ′ ∈ s i G. Also by (1), we have that P ψ ( t( a)) holds if and only if P ψ ( t( a ′ )) holds. Finally by condition (2), for φ( x) an L-formula, φ( a) holds if and only if φ( a ′ ) holds. This proves the desired result.
Lemma 2.6. Let T be a complete pregeometric theory with quantifier elimination. Then T has SAP.
Proof. We first prove the following claim. Claim. Let T be a complete theory with quantifier elimination. Then T has AP. Let M 0 , M 1 , M 2 |= T and assume there are embeddings f i : M 0 → M i for i = 1, 2. By quantifier elimination the maps are elementary embeddings. Let κ be the biggest cardinal among |M 1 |, |M 2 |. Let N |= T be κ + -saturated and κ + -strongly homogeneous. Then by κ + -saturation for each i = 1, 2 there is an elementary map 1 By which we mean that Since g 1 (f 1 (M 0 )) ≡ g 2 (f 2 (M 0 )) by strong homogeneity there is an automorphism h of N such that for each m 0 ∈ M 0 , h(g 2 (f 2 (m 0 ))) = g 1 (f 1 (m 0 )). Then h(g 2 (M 2 )) is an elementary copy of M 2 , g 1 (M 1 ) an elementary copy of M 1 and for each m 0 ∈ M 0 we have h(g 2 (f 2 (m 0 ))) = g 1 (f 1 (m 0 )) and the AP holds.
We now prove SAP. Let M 0 , M 1 , M 2 be three models of T and f i : M 0 → M i be embeddings. Since T has the AP there exists a model M 3 of T and g i : . The type of M ′ 1 over M ′ 0 has a free extension with respect to M ′ 2 (in the sense of the pregeometry), hence in a monster model Proof. We first check that every model of T U extends to a model of T G . This is done by a standard chain argument.
r the smallest R-module containing G and b1 r . One may apply this argument for all tuples a ∈ V with V |= ∃ ∞ xφ(x, a) and assume that φ(rG 1 , a) = ∅. Then we repeat the process for tuples in V 1 and build a chain {(V i , G i )} ∞ i=1 whose union has the desired properties. Second, one needs to check that T U has the amalgamation property. In fact, T U has the strong amalgamation property. Let (V 0 , U 0 ), (V 1 , U 1 ) and (V 2 , U 2 ) be three models of T U such that there exists embeddings f i : (V 0 , U 0 ) → (V i , U i ) for i = 1, 2. By Lemma 2.6, T has SAP hence there exists V 3 |= T and L-embeddings Without loss of generality, we may assume that i = 1. We have that g 1 (V 1 ) ∩ (g 1 (U 1 ) + g 2 (U 2 )) = g 1 (U 1 ) + g 1 (V 1 ) ∩ g 2 (U 2 ).
It follows from Theorem 2.4 that the theory T G has quantifier elimination in the extended language. We will use this fact below to show that several tameness properties are preserved in the expansion.
We start with a small lemma showing the stability of the new predicate.
Lemma 2.8. Let t( x, y) be a L 0 -term, then the formula G(t( x, y)) is stable.
Proof. Assume that t( x, y) = α · x + β · y and that there exist sequences ( a i ) i∈ω and ( b i ) i∈ω such that t( a i , b j ) ∈ G if and only if i < j. Now let c i = α · a i and d i = − β · b i . Then we have c i − d j ∈ G if and only if i < j. In particular we get c 1 ∈ d 2 + G, c 1 ∈ d 3 + G and c 2 ∈ d 3 + G. So the cosets d 3 + G, d 2 + G are equal and thus c 2 ∈ d 2 + G, which implies that c 2 − d 2 ∈ G, a contradiction. Corollary 2.9. If T is stable, then T G is stable. If T is NIP, then so is T G .
Proof. By Theorem 2.4 every formula in the extended language L G is equivalent to a disjunction of conjunctions of L-formulas, R-module formulas and formulas of the form t(x, y) ∈ G and their negation. Since the theory of modules over any ring is stable, any R-module formula is stable. Assume now that T is stable, then any L-formula is stable. By Lemma 2.3, any formula of the form t(x, y) ∈ G, where t( x, y) be a L-term, is equivalent to a formula of the form i t i (x, y) ∈ G ∧ θ i (x, y) where θ i (x, y) is an L-formula and each t i (x, y) is a L 0 -term. By Lemma 2.8 the formulas t i (x, y) ∈ G are stable for i ≤ n and by hypothesis θ i (x, y) is also stable, thus t(x, y) ∈ G is stable. Thus if T is stable so is T G . The same argument works for preservation of N IP .
We even get some easy properties about strong dependence and the stability spectrum based on properties of the subgroups of G: Corollary 2.10. Assume that for infinitely many primes {p i : i ∈ I} we have that the index of p i G in G is infinite. Then if T is stable we have that T G is strictly stable. If T is NIP then T G is not strongly dependent.
Proof. One can build an array using the collection of groups {p i G : i ∈ I} and their cosets (see for example [4]) to show that the expansion does not preserve strong dependence. Similarly, if we order the primes in the list as p 1 , p 2 , . . . then the chain of definable groups p 1 G, (p 1 p 2 )G, . . . is strictly descending and the expansion can not be superstable.
We end this section with some examples and comments relating our work with the general perspective from [8].
Example 2.11. Let T is the theory of a pure vector space over a field F of characteristic zero. This is a strongly minimal theory and thus geometric and it satisfies that acl = span F . As we saw before the corresponding theory T G is stable and we will see below how the stability spectrum of the expansion will vary according to the choice of R.
Assume first that R = F = Q, then the corresponding theory T G is the theory of lovely pairs and it will be ω-stable of Morley rank equal to 2. It is the model companion of the theory of pairs of models of T .
On the opposite end, consider the case R = Z. By Corollary 2.10 the corresponding expansion T G will be stable, not superstable. It is the model companion of the theory of T expanded by a subgroup. Note that this is vector space-like phenomenon, the theory of a field of characteristic 0 with a predicate for an additive subgroup does not have a model-companion [14].
Assume now that R = Z 2Z , the rationals whose denominator is relatively prime with 2. Then the collection of definable groups {2 n G : n ∈ ω} is a descending chain and T G is not superstable. Question. ¿Is T G dp-minimal? Proposition 2.12. Let (V, G) |= T G and assume that V , seen as an L-structure is an abelian structure. Then (V, G) is an abelian structure.
Proof. We show that all definable subsets in the pair are again boolean combinations of cosets of ∅-definable groups. By Theorem 2.4 the expansion has quantifier elimination and it suffices to show that atomic formulas in the pair (V, G) give rise to definable sets that also have this property. Since L-definable sets have the desired property, we only need to consider definable sets given by G(t( x, a)) where t( x, y) is an L-term and for a predicates of the form P φ (x, b). Consider first the case of a formula of the form G(t( x, a)). Assume first that there are λ 1 , . . . , λ n , µ 1 , . . . , µ k ∈ F such that t( x, y) = λ 1 x 1 + . . . λ n x n + µ 1 y 1 + · · · + µ k y k . Choose b = b 1 , . . . , b n ∈ V such that λ 1 b 1 + · · · + λ n b n = µ 1 a 1 + · · · + µ k a k . Then G(t( x, a)) holds if and only if λ 1 (x 1 − b 1 ) + · · · + λ k (x k − b k ) ∈ G. Now observe that H = { x ∈ V n : λ 1 x 1 + · · · + λ n x n ∈ G} is a ∅-definable group and the coset H − b agrees with G(t(V n , a)).
Assume now that the term t( x) agrees with t i ( x) for some L 0 -terms {t i ( x) : i ≤ n} and some partition {θ i ( x) : i ≤ n} given by L-formulas. Then G(t( x)) holds if and only if G(t i ( x)) holds when θ i ( x) holds. Since both θ( x) and G((t i ( x)) are boolean combinations of cosets, so is also their conjuntion as well as disjuntions of families of this form.
On the other hand, all definable sets in a module are boolean combination of cosets of ∅-definable groups, in particular this will be the case for P φ (x, b).
Corollary 2.13. Let T be the theory of a pure vector space over a field F and let R be a subring. Then T G is 1-based.
Proof. Since V is a 1-based group, it is an abelian structure. Now apply the previous Proposition.
Example 2.14. In this example we deal with the base structure V = (V, +, 0, < , {λ r } r∈F ) and we assume that F is an ordered field and V is an ordered vector space over F. The theory of V is o-minimal, has quantifier elimination and acl = span F . In particular it is geometric.
We first consider the case where R = Z and F = Frac(Z) = Q. Since T has NIP, so does T G . Moreover since nG has infinite index in G for n > 1 the expansion is not strongly dependent. Now consider the case where F = R = Q, then the theory T G agrees with the theory of lovely pairs (dense pairs) and it is strongly dependent of dp-rank two.
Finally, we point a few connections with the setting introduced in [8] that can also be used to analize this family of expansions. Following the terminology introduced in [8], we can define the languages L β = L = {+, 0, {λ} λ∈F , . . . } and L α = {+, 0, {λ} λ∈R } and define T β = T = T h(V, +, 0, {λ} λ∈F , . . . ) and T α to be the theory of R-modules. The theory T β is geometric and by the results we proved in this section T G is an example of a Mordell-Lang theory of pairs in the sense of Definition 2.6 [8]. In particular, by [8, Corollary 3.6] the theory T G is near model complete (we prove in Proposition 2.7 that it is actually the model-completion of T U ) and Corollary 2.9 follows from [8, section 4]. There are some other properties of the pair that are studied [8] that are not addressed in this paper; among others, it follows by [8,Theorem 4.5] that the family considered in Example 2.14 has o-minimal open core.

V -structures: Back-and-forth and first properties
We now turn to the general case, we do not assume anymore that F = F rac(R), instead we suppose that F is a field of characteristic zero and that R is a subring of F. Let L 0 = +, 0, {λ·} λ∈F and let L ⊃ L 0 be an extension. Let T be an L-theory expanding the theory of vector spaces over F which has quantifier elimination in L for which dcl = acl = span F and such that it eliminates the quantifier ∃ ∞ . Notation 3.1. Throughout the rest of the paper, we will denoteR = F rac(R).
Let G be a unary predicate and for each formula φ( x) in the language L R−mod = {+, 0, (r·) r∈R } of R-modules, let P φ ( x) be a new predicate. Let L G be the expansion of L by G and P φ for all formula φ in L R−mod . We will consider pairs (V, G) in the language L G such that V |= T and that also satisfy the following first order conditions (A) G is a proper R-submodule of the universe, and for all a ∈ V , P φ ( a) if and only if a ∈ G and G |= φ( a) as an R-module. (B) If λ 1 , . . . , λ n ∈ F areR-linearly independent, then for all g 1 , . . . , g n ∈ G (C) (Density Property) for all r ∈ R \ {0}, rG is dense in the universe. This is a first order property that can be axiomatized through the scheme: for every L-formula φ(x, y), add the sentence ∃ ∞ xφ(x, y) → ∃x(φ(x, y) ∧ rG(x)); (D) (Extension/co-density property) for any L-formulas φ(x, y) and ψ(x, y, z) and n ≥ 1, the following sentence Remark 3.2. Since our ambient structure is a vector space V over F satisfying dcl = acl = span F , we can be a little more explicit in axiom scheme (D). Instead of listing all ψ(x, y, z) such that ∃ ≤n xψ(x, y, z)) holds, we could simply list all finite disjuntions of linear equations with coefficients in F in the variables x, y, z with nontrivial coefficient in x. In order to understand definable sets in T G , we will consider an expansion by definition of T U and T G , see Definition 3.5.
Remark 3.4. Assume that λ ∈ F n isR-independent, and that in a model of T U we have a = λ 1 g 1 + · · · + λ n g n for some g i ∈ G. Then, axiom (B) implies that such collection (g 1 , . . . , g n ) is unique and depends only on a and λ, hence the following definition.
Definition 3.5. For each finiteR-independent tuple λ, we add a new unary predicate G λ . For each finiteR-independent tuple λ = λ 1 , . . . , λ n and 1 ≤ i ≤ n we also add new a unary function symbol f λ,i . Let Let T G+ be the expansion of T G to the language L + G by the following sentences: We will show that T G+ has quantifier elimination.
The following notion is a straightforward modification of the corresponding notions from [20] and [8].
Definition 3.6 (Mordell-Lang property). A model (V, G) of T U has the Mordell-Lang property if for all definable sets X of the form λ 1 x 1 + · · · + λ n x n = 0 with λ 1 , . . . , λ n ∈ F, the trace X ∩ G n is equal to the trace Y ∩ G n for Y defined by a conjunction of formulas of the form r 1 x 1 + . . . + r n x n = 0 where r i ∈ R. In particular, X ∩ G n is ∅-definable from the R-module structure in G, i.e., by a formula of the form P ψ (x 1 , . . . , x n ) for some L R−mod pp-formula ψ. Proof. Let λ 0 , λ 1 , . . . , λ k ∈ F and let X be the definable set given by We may assume that for some 0 ≤ m ≤ k, λ 0 , λ 1 , . . . , λ m are linearly independent overR as elements of the field F, while λ m+1 , . . . , λ k ∈ spanR(λ 0 , λ 1 , . . . , λ m ).
Then for each m + 1 ≤ i ≤ k we have where q i,j ∈R. Now let g 0 , . . . , g k ∈ G be a realization of X. Collecting terms with λ 0 , . . . , λ m , we have: By multiplying by a common multiple of the denominators of the q i,j , there are r i,j ∈ R such that By axiom (B), each of the terms in parentheses above is equal to 0, that is r 0,0 g 0 + r m+1,0 g m+1 + r m+2,0 g m+2 + . . . + r k,0 g k = 0.
The collection of equations listed (as a formula in the elements g 0 , . . . , g k ) is ∅definable in the R-module structure in G. It is easy to see also that any solution (g 0 , . . . , g k ) ∈ G k+1 of this set of equations is also a realization of X ∩ G k+1 and vice versa.
Remark 3.8. We actually proved a stronger version of the Mordell-Lang property: given a definable set X of the form λ 1 x 1 + · · · + λ n x n = 0 with λ 1 , . . . , λ n ∈ F, the trace X ∩ G n is quantifier free definable by a positive formula in the R-module G. We could have also restated the Modell-Lang property as any F-linear dependence in G is witnessed by a linear combination with coefficients inR Recall that L 0 = {(λ·) λ∈F , +, 0}. We want to express L + G -terms using only the F-vector space language together with the f λ,i -functions. So we introduce the auxiliary language Lemma 3.9. Let (V, G) |= T G+ and let t( x) be an L + G -term. Then there exists Lformulas θ 1 ( x), . . . , θ n ( x) forming a partition of V | x| and L + 0 -terms t 1 ( x), . . . , t n ( x) such that Proof. We prove it by induction on the complexity of t( x). If t( x) = g(t ′ 1 ( x), . . . , t ′ n ( x)) for some L-function g, then by induction hypothesis, for each 1 ≤ k ≤ n there exists Putting together, we have It is easy to see that for any finite family of sets We now have to check that the family We now give a description of L + 0 -terms. Lemma 3.10. In a model (V, G) of T G+ , any term t( a) in the language L + 0 is equivalent to a term of the form: Proof. First, observe the following: By the claim, the result follows if every term is equivalent to one of the form which we prove now by induction. By linearity of the expression, the only step to check is the one where t( a) = f γ,i (t ′ ( a)). For convenience, we assume i = 1.
If t( a) = 0, then there exists g 1 , . . . , g n such that We denote by B the L + 0 -substructure spanned by B. Using Lemma 3.10, If on the other hand a / ∈ span F (BG), then f µ,i (αa + α b) = 0 for all α, α, µ, so the result follows. (2) Let α 1 , . . . , α n ∈ F be such that for some g 1 , . . . , g n ∈ G one have a = α · g + c for c ∈ span F (B). By extracting anR-basis of α and replacing a by ra for some r ∈ R, we may assume that α is From the claim in the proof of Lemma 3.10, we have that and of an F-linear combination of f µ,k (βa + γ b) which is also of the desired form by the Claim.
We will use the term 'locally' to mean that something holds on a finite definable partition of the universe.
is locally equivalent to a formula of the form P ψ ( x, h) for ψ( x, z) an L R−modformula and a tuple h in G( B ). Equivalently, there is a finite L-partition (θ i ( x)) i of the universe and L R−mod -formulas ψ i such that Proof. By Lemma 3.9, t( x, b) is locally an L + 0 -term, and by Lemma 3.11 (1), we may assume that t( x, b) is locally of the form α · x + b for some b ∈ B . We show that the formula α · x + b ∈ G λ is equivalent to a formula P ψ ( x, h) for h ∈ G( B ) and ψ an L R−mod -formula. Let a ∈ G be a realisation of α · x + b ∈ G λ and let g ′ ∈ G be such that α · a + b = λ · g ′ . Then for some basis δ of spanR( α λ) we have b ∈ G δ and hence there exists Remark 3.13. Note that the proof above also proves that whenever t( x, b) is a term, there exists a finite L-partition (θ i ( x)) i≤n of the universe and L R−mod -formulas where φ(x, y) is an L R−mod -formula.
. As L-formulas and formulas of the form P ψ are closed by boolean combinations, every formula t(x, b) ∈ G λ is also equivalent to a finite disjunction of expressions the form j P ψj (x, h) ∧ ϑ j (x) (note that here ϑ j do not necessarily form a partition of the universe). More generally, . Then by Lemma 3.12 there is an L R−mod -formula ψ(x, y) and an L-formula ϑ(x) such that a |= P ψ (x, g) ∧ ϑ(x) and Let p(x) be a non-algebraic L-type. We show that q 1 (x) is consistent with p(x), with infinitely many realisations. Let p sh (x) = p(x + a). The type p sh is also non-algebraic, hence by density of R-divisible elements (Lemma 2.2), there exists infinitely many d's such that . By quantifier elimination in R-modules, every formula in q 1 (x) is a boolean combination of conditions of the form ra ′ + b ∈ s 1 G + · · · + s n G for r, s ∈ R and b ∈ G(B). As d ∈ G div , we have that rd ∈ s 1 G + · · · + s n G for all r, s ∈ R, hence ra ′ + b ∈ s 1 G + · · · + s n G if and only if ra + b ∈ s 1 G + · · · + s n G, so a ′ |= q 1 (x). It follows that q 1 (x) ∪ p(x) has infinitely many realisations.
Let q( x) be any set of boolean combinations of formula of the form t( x, b) ∈ G λ , consistent in G. Then p( x) ∪ q( x) is consistent in G.
Step 1. If a ∈ G(V ). From Lemma 3.11 (1), the quantifier-free type of a over B is implied by tp L (a/B), the set q(x) of boolean combinations of conditions of the form t(x, b) ∈ G λ satisfied by a in V , for b ∈ B, t(x, y) an L + G term and λ ∈ F, and inequations αx = b for b ∈ B. Equations do not appear as a / ∈ B (by Lemma 3.11 (1)). By compactness and Proposition 3.14, it is enough to show that σ[q(x) ∪ tp L (a/B)] has infinitely many solutions. Let q 1 (x) be as in Proposition 3.14. In particular, for any φ(x, g) ∈ q 1 (x) we have (V, G) |= P ψ ( g) and g ∈ G(B) for ψ( y) = ∃xφ(x, y), so (V ′ , G ′ ) |= P ψ (σ( g)). By compactness, this shows that σ(q 1 ) = q σ 1 (x) is a consistent partial type in (V ′ , G ′ ), and q σ 1 is the set of boolean combinations of formulas of the form t(x, b ′ ) satisfied by a realisation of q σ 1 in V ′ . Let p(x) = σ(tp L (a/B)). The fact that q σ 1 (x) ∪ p(x) has infinitely many realisations follows from Proposition 3.14, as p(x) is non-algebraic. Using again Proposition 3.14, we have that q σ 1 (x) ∪ p(x) |= q σ (x), so the type q σ (x) ∪ p(x) is realised and non-algebraic, so we can extend σ by σ(a) = a ′ for some a ′ |= q σ (x) ∪ p(x).
Step 2. If a ∈ span F (BG), then by Lemma 3.11 (2) there existsR-independent α ∈ F and c ∈ B such that a = α· g + c, and so g i = f α,i (a− c). Lemma 3.11 (2) also implies that every L + G -term t(a, b) is locally an F-linear combination of g 1 , . . . , g n . By Step 1 we can extend σ to g 1 , . . . , g n hence we can extend σ to a.
Step 3. If a ∈ V \ span F (BG). By Lemma 3.11 (1), every term t(a, b) is locally of the form αa + b, for some b ∈ B. So it is enough to find in V ′ an element a ′ such that a ′ + b ′ / ∈ G λ for all b ′ ∈ B ′ , λ ∈ F, and such that σ(tp L (a/B)) = tp L (a ′ /B ′ ). As tp L (a/B) is non-algebraic, such a ′ exists by axiom (D) and |F| + -saturation.
Recall that T U is the theory consisting of T together with the schemes (A) and (B) and T G the theory consisting of adding (A), (B), (C), (D), hence T G is the restriction of T G+ to the language L G . Corollary 3.17. If T U is inductive, then T G is model-complete an is the modelcompletion of T U .
Proof. T G+ has quantifier elimination by Theorem 3.16, and every function f λ,i is existentially definable in the language L G , to T G is model-complete. Note that the proof of Lemma 2.6 did not use thatR = F, hence we have that T U has SAP. It remains to prove that every model of T U extends to a model of T G , which is similar to the proof of Corollary 2.7.
We can now characterize the algebraic closure in the extended language.
Proof. It is clear that B ⊆ dcl G (B) ⊆ acl G (B). Let a / ∈ B . Case 1. If a ∈ G, then from quantifier elimination tp G (a/B) is determined by tp L (a/B) ∪ q(x) for q(x) as in Proposition 3.14, and is non-algebraic from the conclusion of Proposition 3.14, since tp L (a/B) is non-algebraic.
Case 2. If a ∈ span F (BG(V )). By Lemma 3.11 (2) there is an F-independent tuple α, g ∈ G and c ∈ B such that a = α · g + c and g i = f α,i (a − c) and such that every term in a b is equal to an element in span F ( gB). By quantifier elimination, tp G (a/B) is determined by tp G ( g/B) ∪ {g 1 = f α,1 (x − c), . . . , g n = f α,n (x − c)}. As a / ∈ B , there is some i such that g i / ∈ B , hence by case 1 g i / ∈ acl(B) hence a / ∈ acl(B). Case 3. If a ∈ span F (BG(V )). By Lemma 3.11 and quantifier elimination, By codensity (D), as tp L (a/B) is non algebraic, this type has infinitely many realisations, so a / ∈ acl G (B).
From Theorem 3.16, every L G -formula φ( x) is a boolean combination of quantifier free L + G -formulas, so there exists an L-formula ψ( x), an L R−mod -formula θ( x) Note that P θ ( x) is equivalent to a formula such that quantifiers only occur in the predicate G. Similarly, using the definition of G λi and f λ,i , we see that quantification only occurs in the group G. This particular instance of formulas is called bounded in the sense of [12], hence every formula in T G is bounded.
For subsets of G we get a cleaner description.
We now give a characterization of G-independent sets. Proof. Clearly span F (A) ⊂ A holds for any set A In order to proof the lemma, we will show that A | ⌣G(A) G(V ) if and only if span F (A) ⊃ A . We may assume that A = span F (A). Suppose first that A | ⌣G(A) G(V ), then there is a ∈ A \ span F (G(A)) such that a ∈ span F (G(V )). Let λ 1 , . . . , λ n ∈ F and g 1 , . . . , g n ∈ G(V ) be such that a = λ 1 g 1 + · · · + λ n g n . We may choose λ 1 , . . . , λ n ∈ F to beR-independent. Since a ∈ span F (G(A)), then A is not closed under the function f λ,i for some i and thus span F (A) A . Now assume that A | ⌣G(A) G(V ) and A = span F (A). Using Lemma 3.10, we get that only if f λ,i (a) ∈ A for all a ∈ span F (A), λ ∈ F aR-independent tuple and i ≤ | λ|.
Let λ 1 , . . . , λ n ∈ F beR-independent, let a ∈ A and consider the function f λ,i (a). If f λ,i (a) = 0, since A = span F (A), we have that 0 ∈ A as desired. If f λ,i (a) = 0, then a ∈ span F (G) ∩ A, which equals span F (G(A)) by assumption, so a ∈ span F (G(A)). Then there exists anR-independent tuple α such that a = i α i f α,i (a), with f α,i (a) ∈ G(A). By Lemma 3.11 (2) (with B = ∅), every L + 0 -term in a is a linear combination of f α,i hence belongs to span F (A) so f λ,i (a) ∈ G(A).
The next corollary shows quantifier elimination for G-independent tuples down to L ∪ {G}-formulas and R − mod-formulas (the latter restricted to elements of G), similar to Proposition 3.4 of [4]. Proof. Note that the L-elementary map taking a to b extends uniquely to an elementary map τ : span F ( a) → span F ( b). We claim that τ (G(span F ( a))) = G(span F ( b)). It suffices to show that τ (G(span F ( a))) ⊂ G(span F ( b)).
Indeed, if a = (a 1 , . . . , a n ), and g ∈ G(span F ( a)), then g ∈ G(V ) and g = λ 1 a 1 + . . . λ n a n for some λ i ∈ F. On the other hand, a is G-independent, so assuming for simplicity that G( a) = (a 1 , . . . , a k ), we have g = α 1 a 1 + . . . α k a k for some α i ∈ F. By the Mordell-Lang property, we may assume that α i ∈R. The fact that is witnessed by tp R−mod (G( a)), and since tp R−mod (G( a)) = tp R−mod (G( b)), we also have that τ (g) = α 1 τ (a 1 ) + . . . + α k τ (a k ) ∈ G(W ), as needed. By Lemma 3.20, τ preserves the action of the functions f λ,i . The rest follows by Theorem 3.16. Proof. This is a direct consequence of quantifier elimination (Theorem 3.16), Lemma 3.12 and Remark 3.13.
Recall from [5] that a unary expansion (M, P ) of a model M of geometric theory T satisfies Type Equality Assumption (TEA) if whenever a, b, c ∈ M are such that a is P -independent (i.e. a | ⌣P ( a) P (M )), b | ⌣ a P (M ), c | ⌣ a P (M ), and tp( b a) = tp( c a), then tp P ( b a) = tp P ( c a) (where tp P refers to the type in the language expanded by the unary predicate symbol).
The following corollary follows directly form Corollary 3.21. Note that the conclusion of TEA also holds for the full language L + G . Remark 3.24. In the context of our construction, one can define the notion of "G-basis" analogous to the one used in [4]. Namely, given a G-independent set C and tuple a in V , we are looking for a "canonical" subset of G(V ) such that adding it to aC makes the set G-independent. We claim that the appropriate notion of GB( a/C) is given by aC ∩ G(V )\ spanR(G(C)).
Indeed, GB( a/C) ∪ a ∪ C is G-independent, and for any b ∈ G(V ) such that a | ⌣C b G(V ), we have GB( a/C) ⊂ spanR( bG(C)).
We will now explicitly show that T G is consistent by showing how to construct a model using the notion of H-structure [6]. We will assume the reader is familiar with the definition of H-structure, but we will not require any deeper knowledge of its theory. Let (V, H) be a sufficiently saturated H-structure and let G(V) be the R-submodule of (V, +, 0) generated by H(V ). We will show that T h(V, G) = T G . Lemma 3.26. Let λ 1 , . . . , λ k ∈ F be linearly independent overR, g 1 , . . . , g k ∈ G(V ) and assume that λ 1 g 1 + . . . + λ k g k = 0. Then g 1 = . . . = g k = 0, i.e. (V, G) satisfies axiom (B) of T G .
For each 1 ≤ i ≤ k let r ij ∈ R be such that Then we have: Hence, for any 1 ≤ j ≤ m we have k i=1 r ij λ i = 0. Since λ 1 , . . . , λ k are linearly independent overR, we conclude that r ij = 0 for all 1 ≤ i ≤ k and 1 ≤ j ≤ m. Thus, g 1 = . . . = g k = 0, as needed.
We can now deduce the following result.

Preservation of stability and NIP
Now we will start by proving that NIP is preserved in the expansion by using the ideas of Chernikov and Simon [12]. Let (V, G) be a G-structure. Recall [12] that a formula ϕ( x, y) ∈ L G is said to be NIP over G if there is no L G -indiscernible sequence { a i : i ∈ ω} of tuples of G and an element b ∈ M such that ϕ( a i , b) ↔ i is even.
Proposition 4.1. Assume that T is NIP. Then no formula ϕ( x, y) ∈ L G is NIP over G.
Proof. Assume otherwise, so there is a L G -indiscernible sequence { a i : i ∈ ω} of tuples of G and an element b ∈ M such that ϕ( a i , b) ↔ i is even. We may enlarge b if necessary and assume that b is G-independent and let h = G( b). By Corollary 3.22 there are θ( x) an R-module formula and ψ( x, y) an L-formula such that ∀x(ϕ( x, b) ∧ G( x) ↔ θ( x, h) ∧ ψ( x, b)). But every R-module formula is stable, so the cofinal value of θ( a i , h) is fixed. Similarly, since T is NIP, the cofinal value of ψ( a i , b) is fixed, a contradiction.

Proposition 4.2.
Assume that T is NIP, then T G is also NIP.
Proof. By quantifier elimination in T G+ every L G formula is equivalent to a bounded formula and by Proposition 4.1 no formula has NIP over G, by Theorem 2.4 in [12] we have that T P is also NIP.
A similar argument studying the induced structure of (V, G) in G and using the ideas of Casanovas and Ziegler [9] shows that stability is also preserved in the pair: Proposition 4.3. Assume that T is stable, then T G is also stable.
We will now consider some examples. Let V be a pure vector space over a field F, which is strongly minimal. It is proved in [9] that the stability spectrum of the expansion depends on the stability spectrum of V and the stability spectrum induced by the pair in the predicate. In our setting, since V is ℵ 0 -stable, it depends on the relation between R and F.
We will concentrate in the case where the pair is ω-stable and we will construct expansions with Morley rank n for every n ≥ 2. Assume that R and F are fields and [F : R] = n. We can see F as a vector space of dimension n over R. Choose λ 1 , . . . , λ n a basis of F over R. Consider the map f : G n → V given by f (g 1 , . . . , g n ) = λ 1 g 1 + · · · + λ n g n . This map is definable and generically one to one. Since the structure of G is that of a pure vector space over R, we have that M R(G) = 1 and thus M R(f (G n )) = n. By the extension property V contains properly the set f (G n ), so M R(V, G) = n + 1. Note that f (G n ) is a vector space over F and that the structure (V, f (G n )) is a lovely pair of the theory vector spaces over F. Proof. We follow the same strategy as we did in section 2. Since V is a 1-based group and T G + has quantifier elimination in the extended language, it is enough to show that atomic formulas define boolean combinations of cosets of ∅-definable subgroups. By the proof of Lemma 2.13 it suffices to check that the result is true for formulas of the form G λ (t( x, a)) and t( x, a) = 0 where t( x, a) is a term. Since G λ (x) defines a group, a similar computation to the one done in Lemma 2.13 gives the desired result.
Example 4.6. (Ordered vector spaces, example 2.14 revisited) In this example we deal with the base structure V = (V, +, 0, <, {λ r } r∈F ) and we assume that F is an ordered field and V is an ordered vector space over F. The theory of V is dense o-minimal, so it is geometric.
As in the previous example, we can consider the case where R and F are ordered fields and [F : R] = n. Then the expansion is strongly dependent and has dp − rk(V, G) = n + 1. This example can also be studied from the perspective of [8]; in particular by Theorem 4.8 [8] it was already known that this expansion is NIP. The authors of [8] also show this expansion has open core and characterize when it is decidable.

5.
Preservation of tree properties: NTP 2 , NTP 1 and NSOP 1 In this section, we prove the expansion preserves other nice properties such as N T P 2 , NTP 1 and NSOP 1 . To prove the preservation of N T P 2 and NTP 1 , we will follow the approach from [2], [18], but will need to modify several parts of the arguments. The main difference is that in our setting the definable subsets in G involve not only the induced structure from V (as is the case in [2,18]) but also the structure that it carries as an R-module. We will follow a similar approach to show the preservation of NSOP 1 using as a guide the ideas from [23]. Let us start by recalling the basic definitions for NTP 2 .
Definition 5.1. A theory T has k-TP 2 (for some integer k ≥ 2) if there exist a formula ϕ( x, y) and a set of tuples { a i,j | i, j < ω} (in some model of T ) such that {ϕ( x, a i,f (i) ) | i < ω} is consistent for every function f : ω → ω and {ϕ( x, a i,j ) | j < ω} is k-inconsistent for every i < ω. A theory has TP 2 if it has 2-TP 2 and a theory has NTP 2 if it does not have TP 2 .
Definition 5.2. Let M be a structure in a language L. A set of parameters { a µ | µ ∈ ω × ω} in M is called an indiscernible array if the L-type of any finite tuple ( a µ1 , · · · , a µn ) is determined by the quantifier-free array-type of the tuple (µ 1 , · · · , µ n ).
Just as sequences can be enlarged in order to extract indiscernible sequences, for arrays we have the following result: Fact 5.3. If a formula ϕ( x, y) witnesses k-TP 2 then it may do so with an indiscernible array. Moreover, for such any such indiscernible array and any function f : ω → ω, the collection of formulas {ϕ( x, a i,f (i) ) | i < ω} has infinitely many realizations.
Using a result of Chernikov [11] reducing the property T P 2 to formulas of the form ϕ(x, y) (where x is a single variable) and the fact that we can witness T P 2 with indiscernible arrays, we can reduce the problem to: Proposition 5.4. A theory has TP 2 if and only if there exist a formula ϕ(x, y) (where x is a single variable) and an indiscernible array { a i,j | i, j < ω} such that (1) i<ω ϕ(x, a i,0 ) has infinitely many realizations, (2) j<ω ϕ(x, a 0,j ) has at most finitely many realizations.
Proof. For details, the reader can see [2,Section 3]. Now let us consider the problem in the setting of this paper. As before, we write L for the language of the vector space (maybe with extra structure), T for its theory and L G and T G are the language and the theory of the associated G-structure. As mentioned before, we want to show that T has N T P 2 if and only if T G does. Our first result in this direction deals with the induced structure on the predicate. Proposition 5.5. Assume there exists some L G -formula ϕ(x, y) (where x is a single variable) such that ϕ(x, y) ∧ G(x) witnesses k-TP 2 for some k ≥ 2. Then T has TP 2 .
Proof. Assume that there exists such an L G -formula ϕ(x, y). By Proposition 5.4 we may assume that ϕ(x, y) ∧ G(x) witnesses TP 2 with some indiscernible array A := { a i,j i, j < ω} and that for every function f : ω → ω, the collection {ϕ( x, a i,f (i) ) | i < ω} is infinite. Furthermore, enlarging the indiscernible array if necessary, we may assume that each a i,j is G-independent (for details see [2,Section 4]). Then, by Corollary 3.22, there exists some L-formula ψ(x, y) and a L R−mod -formula θ(x, y) such that for all i, j < ω, Since i<ω ϕ(x, a i,0 ) ∧ G(x) has infinitely many realizations, the conjunction i<ω θ(x, a i,0 ) also has infinitely many realizations in G. Claim 1. j<ω θ(x, a 0,j ) has at infinitely many realizations. Otherwise it has finitely many realizations and by Proposition 5.4 the R-module formula θ(x, y) has TP 2 . Since the theory of R-modules is stable, we obtain a contradiction.
Let us now analyze the formula ψ(x, y). As before, it is easy to see that the conjunction i<ω ψ(x, a i,0 ) also has infinitely many realizations.
Otherwise, it has infinitely many solutions. By Proposition 3.14 and Claim 1 we have that j<ω ψ(x, a 0,j ) ∧ j<ω θ(x, a 0,j ) also has infinitely many solutions, a contradiction.
Since the conjunction i<ω ψ(x, a i,0 ) has infinitely many realizations. and the conjunction j<ω ψ(x, a 0,j ) has finitely many realizations, by Proposition 5.4 we conclude that T has TP 2 .
As in [2, Section 4] we can extend the previous result to several variables: Corollary 5.6. If there exists some L G -formula ϕ( x, y) such that ϕ( x, y) ∧ G( x) witnesses k − T P 2 for some k ≥ 2 then T has T P 2 .
Proof. It follows from submultiplicity of burden, for details see [2] Definition 5.7. Let (V, G) |= T G be sufficiently saturated, let X ⊂ V n and let A ⊂ V be a set of parameters. We say that X is A-small if X ⊂ acl(AG). If X is A-small for some A, then we say X is small, otherwise we say the set X is large. Similarly, for b ∈ V , we write b ∈ scl(A) if b ∈ acl(AG) and we say that b belongs to the small closure of A.
In this section we need to approximate large sets by L-definable sets: Proof. By Theorem 3.16 we may assume Y = ∪ j≤n Y j where each Y j is the set of realizations of a formula of the form Assume for each Y j we can find a L-definable set X j such that Y j △X j is small.
is small. So it suffice to prove the result for the sets Y j . If the set Y j is small we can choose X = ∅. Otherwise we can choose X j = ψ j (V ).
Note that the above proposition also follows from the fact that T G has TEA, see Corollary 3.23 and [5, Proposition 2.6]. We are ready to prove the first main result.
Theorem 5.9. If T G has TP 2 then so does T .
Proof. Assume T G has TP 2 . So there exists some L G -formula ϕ(x, y) (where x is a single variable) witnessing TP 2 with some indiscernible array A := { a i,j | i, j < ω} and we may assume each element a i,j is G-independent. There are two possible cases: Case 1. i<ω ϕ(x, a i,0 ) is realized by some b ∈ scl(A). Such b is then in the algebraic closure (in the language L) of some tuples c = (c 1 , . . . , c n ) ∈ A and h = (h 1 , . . . , h k ) ∈ G(M ). Since in the old language the algebraic closure coincides with the vector space span, we have b ∈ dcl( c, h) and there are some coefficients in F such that b = n i=1 λ i c i + k j=1 λ ′ j h j . Let φ(z 1 , . . . , z k , y; c) be the formula Choose a finite N such that c is part of the sub-array { a i,j |i ≤ N, j < ω} and let A ′ := { a i,j | j < ω, N < i < ω}. It is then easy to show that the L G -formulâ φ(z 1 , . . . , z k , y; c) has TP 2 with respect to the array A ′ and the result follows from Proposition 5.5. Case 2. All the realizations of i<ω ϕ(x, a i,0 ) are in M \ scl(A). By Proposition 5.8 and indiscernibility of the array A, there exists a single L-formula ψ(x, y) such that, for each i, j < ω, ϕ(x, a i,j )△ψ(x, a i,j ) defines an a i,jsmall set. Since the realizations of the conjunction are not small, every realization of i<ω ϕ(x, a i,0 ) is also a realization of i<ω ψ(x, a i,0 ). In particular, i<ω ψ(x, a i,0 ) has infinitely many realizations.
Moreover, j<ω ψ(x, a 0,j ) has only finitely many realizations. (Otherwise, the co-density condition of the predicate G implies that ψ(x, a 0,0 )∧ψ(x, a 0,1 ) is realized by some d ∈ M \ scl( a 0,0 a 0,1 ). But then such d also realizes ϕ(x, a 0,0 ) ∧ ϕ(x, a 0,1 ), contradiction.) Hence, T has TP 2 by Proposition 5.4. Now we deal with NTP 1 (also called NSOP 2 ). We will now follow the strategy from [18], emphasizing the main differences that are needed to adapt the arguments to the new setting. We start with an appropriate notion of tree-indiscernability (see [18,Def. 3.3]) that will play the role of indiscernible array in the argument for NTP 2 .
Such b is in the L-definable closure of some tuples c in S and h in G(M ), so there are coefficients (λ i ) i , (ρ j ) j in F such that b = λ i h i + ρ j c j for some elements {s j } j in S. Let n < ω be such that c belongs to the subtree (a η ) η∈2 <ω ,|η|≤n and choose ν ∈ 2 <ω such that |ν| > n. Let ψ(x, x ′ , y, z) be the formula ϕ(x, y) ∧ x = λ i x ′ i + ρ j z j . It is then easy to check that the L G -formula ∃xψ(x, x ′ , y, c)∧G( x ′ ) has SOP 2 with respect to the strongly indiscernible tree (a η c) η∈2 <ω ,η>ν , and the result follows from Lemma 5.13. Case 2. All the realizations of i<ω ϕ(x, a 0 i ) are in M \ scl(S). By Proposition 5.8, there exists some L-formula ψ(x, y) such that, for each η ∈ 2 <ω , ϕ(x, a η )△ψ(x, a η ) defines an a η -small set. Since all realizations are not small over the tree S, this implies that every realization of i<ω ϕ(x, a 0 i ) is also a realization of i<ω ψ(x, a 0 i ). In particular, i<ω ψ(x, a 0 i ) has infinitely many realizations.
It follows that the L-formula ψ(x, y) witnesses that T has SOP 2 .
In particular, since simple theories are those that have at the same time NTP 2 and NSOP 2 we get the following Corollary.
Corollary 5.15. Assume T is simple, then so is T G . Now we deal with NSOP 1 . We will use the following characterization presented in the work by Ramsey [23]: Definition 5.16. Let T be a complete theory and let M |= T be sufficiently saturated. We say T has SOP 1 if there is a formula ϕ( x; y), possibly with parameters in a set C, and array ( c i,j ) i<ω,j<2 so that (a) c i,0 ≡ C c<i c i,1 for all i < ω.
By Lemma 2.5 and Theorem 2.7 [23] it suffices to check the definition above for formulas ϕ(x; y) where x is a single variable and we may choose an array j ) i<ω,j<2 satisfies these extra conditions, we say the array ( c i,j ) i<ω,j<2 is an indiscernible array.
We will need the following modified version of the property that is more suitable to dense pairs. The next result is due to Nicholas Ramsey: Proposition 5.17 (Ramsey). A theory T has SOP 1 if and only if there exists some set C, a formula ϕ(x, y) (where x is a single variable) and an indiscernible array ( c i,j ) i<ω,j<2 such that (a) c i,0 ≡ C c<i c i,1 for all i < ω; (b) {ϕ(x; c i,0 ) : i < ω} is consistent and non-algebraic; (c) Whenever i < j, ϕ(x; c i,1 ) ∧ ϕ(x; c j,1 ) is finite.
Proof. If T has SOP 1 , then by [23,Theorem 2.7], there is a formula in one variable witnessing SOP 1 , in particular, it satisfies conditions (a), (b) and (c). Conversely assume that C, ( c i,j ) i<ω,j<2 and φ(x, y) are given. Since the array is indiscernible, there is an N such that for any i < j the set ϕ(x; c i,1 ) ∧ ϕ(x; c j,1 ) has cardinality N . Consider the formula were consistent, there would be more than N different realisations of ϕ(x, c i,1 )∧ϕ(x, c j,1 ), so ψ( x, c i,1 ) ∧ ψ( x, c j,1 ) is inconsistent, and T has SOP 1 .
We follow the same strategy that as we did earlier in this section, first we check what happens for formulas in G(x) and then we extend the results to the general case.
Proposition 5.18. Assume there exists some L G -formula ϕ(x, y) (where x is a single variable) such that ϕ(x, y) ∧ G(x) witnesses SOP 1 . Then T has SOP 1 .
Proof. Assume that there exists such an L G -formula ϕ(x, y) that witnesses SOP 1 . We may assume that the sequence ( c i ) i<ω witnessesing SOP 1 is an indiscernible array and enlarging the tuples if necessary we may assume that each c i,j is Gindependent. Using Corollary 3.22, there exists some L-formula ψ(x, y) and a L R−mod -formula θ(x, y) such that for all i, j < ω, Since i<ω ϕ(x, c i,0 ) ∧ G(x) is consistent and non-algebraic, clearly the collection of L-formulas {ψ(x, c i,0 ) : i < ω} is consistent and non-algebraic. Similarly, {θ(x, c i,0 ) : i < ω} is consistent and non-algebraic. Also c i,0 ≡ L C c<i c i,1 for all i < ω. We just need to show the almost 2-inconsistency of the family {ψ(x; c i,1 ) : i < ω}. Claim 1. θ(x, c i,1 ) ∧ θ(x, c j,1 ) has infinitely many realizations whenever i < j. Otherwise whenever i < j we have that θ(x, c i,1 ) ∧ θ(x, c j,1 ) has finitely many realizations and by Proposition 5.17 the R-module formula θ(x, y) has SOP 1 . Since the theory of R-modules is stable, we obtain a contradiction.
Corollary 5.19. Assume there exists some L G -formula ϕ( x, y) (where x is a tuple) such that ϕ( x; y) ∧ G( x) witnesses SOP 1 . Then T has SOP 1 .
Proof. An easy modification of the proof of Theorem 2.7 in [23] shows that for some single variable x i in the tuple x and an appropriate tuple b, the formula ϕ(x i , b; y) ∧ G(x i , b) has SOP 1 . Now apply Proposition 5.18 to get the desired result.
With the previous results we are ready to show the last theorem of this section: Theorem 5.20. If T G has SOP 1 then so does T .
Proof. Assume that T G has SOP 1 . Let C, ϕ(x, y) and S = ( c i,j ) i<ω,j<2 be as in Proposition 5.17, in particular i<ω ϕ(x; c i,0 ) is consistent and non-algebraic. We may assume that ( c i ) i<ω is an indiscernible sequence and enlarging the tuples if necessary we may assume that each c i,j is G-independent. Also, to simplify the presentation we will assume that C = ∅.
Let d ∈ S and let g ∈ G be tuples such that b = i λ i d i + j ρ j g j for some Then for all n 0 ≤ i < ω, c i,0 ≡ C d c<i c i,1 . It is then easy to see that the formula ∃xψ(x, x ′ , y, d) ∧ G( x ′ ) witnesses SOP 1 , hence by Corollary 5.19, T has SOP 1 . Case 2. All realisations of i<ω φ(x, c i,0 ) are not in scl(S). By Proposition 5.8, there exists some L-formula ψ(x, y) such that, for each i < ω and j < 2, φ(x, c i,j )△ψ(x, c i,j ) defines a c i,j -small set. Since all realizations are not small over S, this implies that every realization of i<ω φ(x, c i,0 ) is also a realization of i<ω ψ(x, c i,0 ). In particular, i<ω ψ(x, c i,0 ) has infinitely many realizations.
We show that ψ(x, c i,1 ) ∧ ψ(x, c j,1 ) has only finitely many realizations, for i = j. Assume not, then by the codensity property for G, there is some , and similarly, d satisfies φ(x, c j,1 ). It follows that d satisfies φ(x, c i,1 )∧φ(x, c j,1 ). Repeating this process and using again the codensity property, there is an infinite sequence (d i : i < ω) of realisations of ψ(x, c i,1 ) ∧ ψ(x, c j,1 ) such that d i ∈ scl(c i,1 c j,1 d <i ). Hence φ(x, c i,1 ) ∧ φ(x, c j,1 ) has infinitely many realisation, a contradiction. It follows that the L-formula ψ(x, y) witnesses that T has SOP 1 .
Question. It may be interesting to check in this setting if other model theoretic properties are preserved. For example, is n-dependence preserved? Is NSOP n preserved for n ≥ 3?
We end this section with some examples. Since for pure vector spaces over F we have that acl L0 = dcl L0 = span F , by [10] the structure V has quantifier elimination and acl = dcl = span F . The completions of the theory depend on the truth value of S(0) and to simplify the example, assume that 0 ∈ S. By [10] this structure is simple unstable of SU -rk one. Regardless how we choose the submodule R, by Corollary 5.15 the theory of the pair (V, G) will always be simple, but as before κ T will depend on how we choose F and R.
Assume first that R = Z and F = Q. Then the theory of the pair (V, G) will be simple and not supersimple. Assume now that R = Q = F, then the expansion T G corresponds to the theory of a lovely pair of models of T (see [24]), which is known to be supersimple of SU -rank two. M A, B, C ≡ MA C 1 and C ≡ MB C 2 . Furthermore, in this case | ⌣ * is Kim-independence over models.
We will assume that | ⌣ 0 has the basic properties listed above: invariance, symmetry, existence, finite character, monotonicity, transitivity and extension. We will write | ⌣ R−mod for forking independence inside R-modules. We now introduce an independence relation in the expansion (V, G) |= T G+ and assume the expansion is sufficiently saturated.
In the above definition above we require the set D to be algebraically closed in T G+ and in particular it is G-independent. Note that for a, d tuples, we can define Proof. Invariance follows from the fact that | ⌣ 0 satisfies invariance and the fact that, | ⌣ R−mod being forking independene in R-modules also satisfies invariance. Let us consider now monotonicity. Given two sets A ⊂ B ⊂ V , we have that G( A ) ⊂ G( B ), since monotonicity holds for forking-independence, in stable theories, we have that item (2) in the definition of | ⌣ G is preserved after taking subsets of C and D. Since | ⌣ 0 satisfies monotonicity, then | ⌣ G will also satisfy monotonicity. Now we point out some observations on F-linear independence.
Proof. Fix tuples a, d. Clearly items (1) and (2) are symmetric conditions since | ⌣ 0 satisfies symmetry as well as forking-independence in modules.
Using the definition and the fact that | ⌣ 0 is transitive we get AV 1 | ⌣ G V0 B. Similarly, since transitivity holds for forking-independence in R-modules we also get the condition G( AV 1 ) | ⌣ R−mod G(V0) G( BV 0 ) as desired. As the next example shows, the codensity property, that holds for algebraic independence, may not hold for other independence relations: Example 6.7. Lovely pair of a modular SU-rank 1 expansion of a vector space: the case R = F. Let V = (V, . . .) be a supersimple SU-rank 1 expansion of a vector space over a field F, and we assume that acl = span F and T = T h(V) has quantifier elimination. Let T P be the theory of a lovely pair (V, P ). It has been shown in [24] that in this case the algebraic closure in the pair coincides with the algebraic closure in V, i.e. acl P = acl = span F and T P is supersimple of SU-rank 2. As shown in [25], for any SU-rank 1 theory T , the theory of lovely pairs of T has the weak nonfinite cover property. In particular, T P eliminates ∃ ∞ . Thus, in our setting, T P is geometric, with algebraic closure given by the linear span. Note also that due to the modularity of the pregeometry, T P has quantifier elimination (any closed set is P -independent).
Consider the dense-codense generic submodule expansion T P G of T P in the case when R = F. Thus, G(V ) is a subspace of V , and moreover, by the density property, an elementary submodel of V. Clearly, T P G is the theory of lovely pairs of T P viewed as a geometric theory. Note that T P Q is not the theory of lovely pairs of T P in the sense of [1]. In any model (V, P, G) of T P G , for any a ∈ V the formula P (x − a) is non-algebraic, and therefore is realized in G(V ). It follows that V = P (V ) + G(V ), whereas in a lovely pair T P Q of T P in the sense of [1] (where Q represents an elementary substructure of (V, P ) satisfying the coheir and extension properties), V has infinite dimension over P (V ) + Q(V ). Note also that in the context of supersimple theory T P , in (V, P, G), the type of a ∈ P (V ) over ∅ has no non-forking extension over G(V ). Proof. Let (V 0 , G) |= T G+ , let B, D be sets, and let p( x) be a L G+ -type over V 0 B such that whenever a |= p we have a | ⌣ G V0 B. After adding to a elements from aV 0 if necessary, we may assume that if a |= p, then we can write a = ( a 1 , a 2 , a 3 ), where a 1 | ⌣V 0 G(V ) and it is F-linearly independent over V 0 , a 2 ∈ G(V ) and a 2 is Flinearly independent over V 0 a 1 , and a 3 ∈ span F ( a 1 a 2 ). Since | ⌣ 0 satisfies extension we can find a ′ |= p such that a ′ = ( a ′ 1 , a ′ 2 , a ′ 3 ) | ⌣ 0 V0 BD. Since dim( a 1 /V 0 ) = | a 1 | and | ⌣ 0 extends algebraic independence, we also have By the codensity property in the pair (V, G) we may assume dim( a ′ 1 /G(V )BD) = | a ′ 1 |, so the tuple a ′ 1 can also be chosen to be F-linearly independent over BDG(V ). Now let p 2 ( x 2 ) = tp L ( a ′ 2 /V 0 BD a ′ 1 ) and let p R 2 ( x 2 ) = tp R−mod ( a 2 /G( V 0 B )). Since | ⌣ R−mod is a stable independence notion, it satisfies existence and we can G( a 1 , a 2 ) = a 2 and G( a ′ 1 , a ′ 2 ) = a ′ 2 we get by Corollary 3.21 that tp G+ ( a/V 0 ) = tp G+ ( a ′ /V 0 ) and a ′ is the desired tuple. Theorem 6.9. Assume that the independence theorem holds for | ⌣ 0 -independence over models of T . Then the independence theorem also holds for | ⌣ G .
Proof. Let (V 0 , G) |= T G+ and let B, C ⊃ V 0 be supersets such that B | ⌣ G V0 C. In particular by clause (1) of G-independence we get B | ⌣ 0 V0 C and, exchanging the sets for their L G -algebraic closure, we may assume that C = C and D = D .
Let p( x) be an L G -type over V 0 and let p B ( x) ∈ S(B), p C ( x) ∈ S(C) be independent extensions of p( x) in the sense of | ⌣ G . If a |= p( x), after adding elements of aV 0 if necessary, we may assume that any realization of p( x) is of the form a = a 1 a 2 a 3 , where a 1 is a V 0 ∪ G(V )-independent tuple, a 2 ∈ G(V ) is F-linearly independent over V 0 and a 3 ∈ dcl F ( a 1 a 2 V 0 ). Note that under this assumption a is G-independent over V 0 .
Let p L B ( x) be the restriction of p B ( x) to the language L. Similarly let p L C ( x) be the restriction of p C ( x) the language L. Since B | ⌣ 0 V0 C, we use the independence theorem for Using the codensity property, we can assume that a 1 is independent from G(V ) over B ∪ C.
We could use the density property and assume that a 2 ∈ G, but the module type of a 2 over V 0 may not be the desired one. Instead, let p L 2BC a1 ( x 2 ) = tp L ( a 2 /BC a 1 ); we will refine the way we choose the realization of this last type in order to take into account the module structure.
Let p 2B ( x 2 ) be the restriction of p B ( x) to the | a 2 |-coordinates considered over the parameters set B and let p R 2B ( x 2 ) be its restriction to the language of R-modules over the parameter set G(B). Similarly let p 2C ( x 2 ) be the restriction of p C ( x 2 ) ∈ S(C) to the | a 2 |-coordinates considered only over the parameters set C and let p R 2C ( x 2 ) be its restriction to the language of R-modules to G(C). Finally let p R 2 ( x 2 ) be the restriction of p( x) to the | a 2 |-coordinates considered only over the parameters set G(V 0 ) restricted to the language of R-modules.
Since C | ⌣ G V0 D we get using clause (2) that G(C) | ⌣ R−mod G(V0) G(D). Since the theory of R-modules is stable, p R 2C ( x 2 ) is the unique non-forking extension of p R 2 ( x 2 ) to G(C) and p R 2B ( x 2 ) is the unique non-forking extension of p R 2 ( x 2 ) to G(B) which can be amalgamated to the unique non-forking extension of p R 2 ( x 2 ) to G(B) ∪ G(C), which we will call p R 2BC ( x 2 ). Now we show that both extensions p L 2BC a1 and p R 2BC are compatible and can be extended to a type that is | ⌣ G -independent from B ∪ C over V 0 . Claim 1. p R 2BC ( x 2 )∪p L 2BC a1 ( x 2 ) is consistent and any realization is independent in the sense of R-modules from G(B) ∪ G(C) over G(V 0 ).
Since dim( a 2 /BC a 1 ) = | a 2 |, the consistency result follows from Corollary 3.15. Let a ′ 2 ∈ G | x2| be such that a ′ 2 |= p L 2BC a1 ( x 2 ) ∪ p R 2BC ( x 2 ), then a ′ 2 |= p R 2BC and we have that a ′ 2 is independent in the sense of R-modules from G(B) ∪ G(C) over G(V 0 ). Note that a 1 a ′ 2 ≡ L BC a 1 a 2 and thus a 1 a ′ 2 | ⌣ 0 V0 B ∪ C. Let a ′ 3 be such that a 1 a ′ 2 a ′ 3 ≡ L BC a 1 a 2 a 3 and write a ′ = a 1 a ′ 2 a ′ 3 for short. Claim 2. a ′ |= p B ( x) ∪ p C ( x) and it is | ⌣ G -independent from B ∪ C over V 0 .
Since a 1 a 2 a 3 | ⌣ 0 V0 B∪C, a 1 a ′ 2 a ′ 3 ≡ L BC a 1 a 2 a 3 we get that part (1) in the definition of G-independence holds.By the way we constructed a ′ 2 we know that part (2) in the definition of | ⌣ G -independence also holds.
Proof. Let a be a finite tuple and let κ > |T G+ | = |T | be a regular cardinal. Let {(V i , G i )} i<κ be a continuous chain of models of T G+ and let (V, G) = ∪ i<κ (V i , G i ).
Then {V i } i<κ is also a continuous chain and by hypothesis there is j < κ such that a | ⌣ 0 Vj V . Since forking independence in R-modules is stable, let κ R be the corresponding local character cardinal and note that κ R ≤ |T G+ |. Let B ⊂ G(V ) be of cardinality ≤ κ R be such that G( aB ) | ⌣ R−mod G(B) G(V ). Since κ is regular and κ R < κ, we may find j 2 < κ such that B ⊂ G(V j2 ) and thus G( aG(V j2 ) ) | ⌣ R−mod G(Vj 2 ) G(V ). Let j 3 = max(j, j 2 ), then we have a | ⌣ The results of this section together with Fact 6.1 can be summarized as: Theorem 6.11. Assume T is a NSOP 1 theory then T G+ is also NSOP 1 , and Kim-independence is given by | ⌣ G over models.

7.
Example: Lovely pair of vector spaces with a generic NSOP 1 structure on its factor-space In this section, we describe a candidate for an example of an NSOP 1 not simple theory which is modular pregeometric. This construction shall serve as a general method for constructing modular pregeometric theories of arbitrary combinatorial complexity (stable, simple, NIP, NSOP 1 ). We leave most of the details to the reader.
Let F be any field, and let us denote by F-vs the theory of infinite dimensional F-vector space in the usual one-sorted language. For V |= F-vs and a a tuple in V , we write tp V ( a) for it type in the vector space language. Similarly, when we add a new sort S with extra structure to V , we will write tp S ( a) for the type of the tuple seeing S with the induced structure, provided a belongs to the sort S. Proof. Since the theory F−vs is strongly minimal the pair (V, H) is ω-stable, H is stably embedded and there is no outside structure induced on H ind . Moreover, this last set just models the theory in the empty language with infinitely many elements. The rest follows from the work in [13,Section 3]. We check the following properties: (1) acl eq ( a) ∩ H eq = dcl feq (H(span F ( a))); (2) H is algebraically embedded in (V, H), which means: if a | ⌣ (V,H) c b, then acl eq ( a b c) ∩ H eq ⊆ acl eq (acl eq ( a c) ∩ H eq , acl eq ( b c) ∩ H eq ) (see [13,Definition 4.9]).
(1) follows from the fact that H ind and V have weak elimination of imaginaries, and the description of algebraic closure in H-structures (see Corollary 3.14 in [6] Fact 7.2. Let (V, W ) be a lovely pair of models of the theory F-vs. Consider the two sorted structure (V, V /W, π) with the quotient homomorphism π : V → V /W and let V * = V /W the quotient vector space.
(3) The induced structure on V * in (V, V * , π) is the one of a pure vector space.
Proof. This is left to the reader as an exercise, it is very similar to the work in [15, Section 3].
Also, for any relation R ⊆ H n in H ′ , define R π ⊆ (H π ) n so that x ∈ R π ⇐⇒ π(x) ∈ R.
For function symbol f and constant symbol c, define the relations R π f and R π c as follows: x ∈ R π f ⇐⇒ π(x) ∈ Graph(f ). x ∈ R π c ⇐⇒ π(x) = c. We also keep the predicate W in the structure V π .
Corollary 7.4. The structure V π is NSOP 1 . If H ′ is not simple, neither is V π .
Proof. Recall that (V, W ) is a lovely pair and π : V → V /W is the projection and W is kept as a predicate inside V π . The structure V π is interdefinable with (V, (V * , H ′ )) so V π is NSOP 1 . Also if H ′ is not simple, V π is not simple. Now even if the algebraic closure in (V * , H ′ ) might be larger than the vector space span, the algebraic closure should not change in V π because π has infinite fibers. Every formula φ(x) in (V * , H ′ ) has a dual formula φ π (x) in V π such that V π |= φ π (a) ⇐⇒ V * |= φ(π(a)) so no algebraicity should come from the extra structure in V π . For the same reason, uniform finiteness should be preserved. This leads us to state as a conjecture: Conjecture. In V π , the algebraic closure of a set A ⊂ V is given by the span F (A), and V π eliminates ∃ ∞ . In particular the theory of V π is geometric.