aa r X i v : . [ m a t h . L O ] A ug Dp-minimal profinite groups andvaluations on the integers
Tim Clausen ∗ August 21, 2020
We study dp-minimal infinite profinite groups that are equipped with auniformly definable fundamental system of open subgroups. We show thatthese groups have an open subgroup A such that either A is a direct productof countably many copies of F p for some prime p , or A is of the form A ∼ = Q p Z α p p × A p where α p < ω and A p is a finite abelian p -group for each prime p . Moreover, we show that if A is of this form, then there is a fundamentalsystem of open subgroups such that the expansion of A by this family ofsubgroups is dp-minimal. Our main ingredient is a quantifier eliminationresult for a class of valued abelian groups. We also apply it to ( Z , +) and weshow that if we expand ( Z , +) by any chain of subgroups ( B i ) i<ω , we obtaina dp-minimal structure. This structure is distal if and only if the size of thequotients B i /B i +1 is bounded. A profinite group G together with a fundamental system { K i : i ∈ I } of open subgroupscan be viewed as a two-sorted structure ( G, I ) in the two-sorted language L prof . In thesestructures the fundamental system of open subgroups is definable. Since a fundamentalsystem of open subgroups is a neighborhood basis at the identity, this implies that thetopology on G is definable.These structures have been studied by Macpherson and Tent in [19]. They mainlyconsidered full profinite groups, i.e. profinite groups G where the family { K i : i ∈ I } consists of all open subgroups. Their main result states that a full profinite group ( G, I )is NIP if and only if it is NTP if and only if it is virtually a finite direct product ofanalytic pro- p groups. ∗ The author is partially supported by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster:Dynamics-Geometry-Structure. p groups can be described as products of copies of Z p with a twistedmultiplication, profinite NIP groups are composed of “one-dimensional” profinite NIPgroups. In the setting of full profinite groups the combinatorial structure of the latticeof open subgroups is visible in the model theoretic structure. This plays an importantrole in the classification.Without the fullness assumption, only a portion of this lattice is visible. In generalthe family { K i : i ∈ I } could simply consist of a chain of open subgroups. In this moregeneral setting, we will restrict ourselves to the “one-dimensional”, i.e. dp-minimal case.A profinite group ( G, I ) is dp-minimal if it has NIP and is dp-minimal in the group sort.We prove the following classification result:
Theorem.
Let ( G, I ) be a dp-minimal profinite group. Then G has an open abeliansubgroup A such that either A is a direct product of countably many copies of F p forsome prime p , or A is isomorphic to Q p Z α p p × A p where α p < ω and A p is a finiteabelian p -group for each prime p . Moreover, every abelian profinite group A of the aboveform admits a fundamental system of open subgroups such that the corresponding L prof -structure is dp-minimal. The main ingredient of this theorem is a quantifier elimination result which is alsoapplicable in other settings. We apply it to this situation: Consider the structure ( Z , +).If we expand it by the full lattice of subgroups, then the expanded structure interpretsPeano Arithmetic and hence is not tame in any sense. However, if we only name a chainin this lattice, we obtain a tame structure. A chain of subgroups Z = B > B > . . . isthe same as a valuation v : Z → ω ∪ {∞} defined by v ( a ) = max { i : a ∈ B i } . Theorem.
Let ( B i ) i<ω be a strictly descending chain of subgroups of Z , B = Z , andlet v : Z → ω ∪ {∞} be the valuation defined by v ( x ) = max { i : x ∈ B i } . Then ( Z , , , + , v ) is dp-minimal. Moreover, ( Z , , , + , v ) is distal if and only if thesize of the quotients B i /B i +1 is bounded. This stays true if we expand the value sort by unary predicates and monotone binaryrelations. There has been recent interest in dp-minimal expansions of ( Z , +) (e.g. [2],[21], [1], and [22]). Alouf and d’Elbée showed in [2] that if p is a prime and v p denotes the p -adic valuation, then ( Z , , , + , v p ) is a minimal expansion of ( Z , , , +) in the sensethat there are no proper intermediate expansions. We show that this does not hold truefor all valuations and we conjecture that the p -adic valuations are essentially the onlyexamples with this property among valuations v such that ( Z , , , + , v ) is distal.The proof of the classification theorem for dp-minimal profinite groups consists of threeparts: We analyze the algebraic structure of dp-minimal profinite groups in Section 3.This will imply the first part of the theorem. It then remains to show that these groupsappear as dp-minimal profinite groups. This is done in Section 4. The case where2he group is given by an F p -vector space has already been done by Maalouf in [10].We explain this result in Section 4.1. The remaining case is handled by a quantifierelimination result (see Section 4.2). This quantifier elimination result allows us to showthat a certain class of profinite groups as L prof -structures is dp-minimal (Theorem 4.29)and we are able to characterize distality in this class (Theorem 4.32).We will also apply the quantifier elimination result to valuations on ( Z , +). This willbe done in Section 5 where we discuss the second theorem and its consequences for thestudy of dp-minimal expansions of ( Z , + , , p -adic valuationshave a limit theory (Proposition 5.9) and we consider expansions of ( Z , + , ,
1) given bymultiple valuations.Section 6 contains a few results which are related to dp-minimal profinite groups. Weshow that our main result implies some structural consequences for uniformly definablefamilies of finite index subgroups in dp-minimal groups (Proposition 6.4). Jarden andLubotzky [8] showed that two elementarily equivalent profinite groups are isomorphic ifone of them is finitely generated. This was generalized to strongly complete profinitegroups by Helbig [7]. We will give an alternative proof for these results in Section 6.2.Finally, we prove a result about uniformly definable families of normal subgroups inNTP groups (Proposition 6.13) : If such a family is closed under finite intersections,then it must be defined by an NIP formula. Acknowledgments
I would like to thank my supervisor, Katrin Tent, for her help and support during the lastyears and for giving me the opportunity to work on these exciting topics. I would alsolike to thank Pierre Simon for interesting and useful discussions and valuable suggestionswhile I visited UC Berkeley.
We assume that the reader is familiar with both profinite groups and model theory. Wewill give a quick overview about the notions and tools that are used to prove the mainresult.
A topological group is profinite if it is the inverse limit of an inverse system of (discrete)finite groups. This condition is equivalent to the group being Hausdorff, compact, andtotally disconnected. If G is a profinite group, then G ∼ = lim ←− G/N where N ranges over all open normal subgroups. Thanks to Pierre Simon for bringing this question to my attention. G , i.e. every open set is a union of cosetsof open subgroups. A fundamental system of open subgroups is a family F consisting ofopen subgroups which generate the topology on G . Equivalently, every open subgroup of G contains a subgroup in F . If P is a property of groups, we will say that G is virtually P if G has an open subgroup H which satisfies P .We will use a number of results about the structure of abelian profinite groups. Recallthat a profinite group is pro- p if it is the inverse limit of finite p -groups. A free abelianpro- p group is a direct product of copies of Z p . Proposition 2.1 (Theorem 4.3.4 of [13]) . Let p be a prime.(a) If G is a torsion free pro- p abelian group, then G is a free abelian pro- p group.(b) Let G be a finitely generated pro- p abelian group. Then the torsion subgroup tor( G ) is finite and G ∼ = F ⊕ tor( G ) where F is a free pro- p abelian group of finite rank. Proposition 2.2 (Corollary 4.3.9 of [13]) . Let G be a torsion profinite abelian group.Then there is a finite set of primes π and a natural number e such that G ∼ = Y p ∈ π ( e Y i =1 ( Y m ( i,p ) C p i )) where each m ( i, p ) is a cardinal and each C p i is the cyclic group of order p i . In particular, G is of finite exponent. Proposition 2.3 (Proposition 1.13 and Proposition 1.14 of [6]) . Let G be a pro- p group.Then G is (topologically) finitely generated if and only if the Frattini subgroup Φ( G ) = G p [ G, G ] is open in G . Proposition 2.4.
Let A be an abelian profinite group. Then nA ≤ A is an open subgroupfor all n ≥ if and only if A ∼ = Y p Z α p p × A p where α p < ω and A p is a finite abelian p -group for each prime p .Proof. An abelian profinite group is the direct product of its p -Sylow subgroups. Let P be a p -Sylow subgroup of A . If pP ≤ P has finite index, then P is finitely generatedby Proposition 2.3. Then by Proposition 2.1 the p -Sylow subgroup P has the desiredform.We will also need the following result by Zelmanov: Theorem 2.5 (Theorem 2 of [24]) . Every infinite compact group has an infinite abeliansubgroup.
4e will view profinite groups as two-sorted structures in the following language whichwas introduced in [19]:
Definition 2.6. L prof is a two-sorted language containing the group sort G and theindex sort I . The language L prof then consists of: • the usual language of groups on G , • a binary relation ≤ on I , and • a binary relation K ⊆ G × I . Remark . A profinite group G together with a fundamental system of open subgroups { K i : i ∈ I } can be viewed as an L prof structure ( G, I ) as follows: • we set i ≤ j if and only if K i ⊇ K j , and • the relation K is defined by K ( G, i ) = K i for all i ∈ I . We will mostly work in the context of an NIP theory. We use [16] as our main referencefor this section.
An important class of model theoretic theories is the class of NIP (or dependent) theories,i.e. the class of theories which cannot code the ∈ -relation on an infinite set. This notionwas introduced by Shelah. Definition 2.8.
A formula ϕ ( x, y ) has the independence property (IP) if there are se-quences ( a i ) i<ω and ( b J ) J ⊆ ω such that | = ϕ ( a i , b J ) ⇐⇒ i ∈ J. We say that ϕ ( x, y ) has NIP if it does not have IP. This notion is symmetric in thesense that a formula ϕ ( x, y ) has NIP if and only if the formula ψ ( y, x ) ≡ ϕ ( x, y ) hasNIP (see Lemma 2.5 of [16]).We will make use of the following characterization of IP: Lemma 2.9 (Lemma 2.7 of [16]) . A formula ϕ ( x, y ) has IP if and only if there existsan indiscernible sequence ( a i ) i<ω and a tuple b such that | = ϕ ( a i , b ) ⇐⇒ i is odd . We call a theory NIP if all formulas have NIP.
Definition 2.10.
A subset X ⊆ M | = T is externally definable if there is a formula ϕ ( x, y ), an elementary expansion M ∗ of M , and an element b ∈ M ∗ such that X = ϕ ( M, b ). 5y a result of Shelah, naming all externally definable sets in an NIP structure preservesNIP:
Theorem 2.11 (Proposition 3.23 and Corollary 3.24 of [16]) . Let M be a model ofan NIP theory and let M Sh be the Shelah expansion, i.e. the expansion of M by allexternally definable sets. Then M Sh has quantifier elimination and is NIP. Theorem 2.12 (Baldwin-Saxl, Theorem 2.13 of [16]) . Let G be an NIP group and let { H i : i ∈ I } be a family of uniformly definable subgroups of G . Then there is a constant K such that for any finite subset J ⊆ I there is J ⊆ J of size | J | ≤ K such that \ { H i : i ∈ J } = \ { H i : i ∈ J } . As an easy consequence we obtain:
Corollary 2.13. If ( G, I ) is an NIP profinite group, then { K i : i ∈ I } can only containfinitely many subgroups of any given finite index. By a result of Shelah, abelian subgroups of NIP groups have definable envelopes givenby centralizers of definable sets:
Theorem 2.14 (Proposition 2.27 of [16]) . Let G be an NIP group and let X be aset of commuting elements. Then there is a formula ϕ ( x, y ) and a parameter b (insome elementary extension G ∗ ) such that Cen(Cen( ϕ ( G ∗ , b ))) is an abelian (definable)subgroup of G ∗ and contains X . NIP theories admit a notion of dimension given by dp-rank:
Definition 2.15 (Definition 4.2 of [16]) . Let p be a partial type over a set A , and let κ be a cardinal. We define dp-rk( p, A ) < κ if and only if for every family ( I t ) t<κ of mutually indiscernible sequences over A and b | = p , one of these sequences is indiscernible over Ab .A theory is called dp-minimal if dp-rk( x = x, ∅ ) = 1 where x is a singleton. Wecall a multi-sorted theory with distinguished home-sort dp-minimal if it is NIP and itis dp-minimal in the home-sort, i.e. dp-rk( x = x, ∅ ) = 1 where x is a singleton in thehome-sort. Remark . As a consequence of the quantifier elimination in Theorem 2.11 the Shelahexpansion of a dp-minimal structure is dp-minimal.We will use the fact that definable subgroups in a dp-minimal group are always com-parable in the following sense:
Lemma 2.17 (Claim in Lemma 4.31 of [16]) . Suppose G is dp-minimal and H and H are definable subgroups. Then | H : H ∩ H | or | H : H ∩ H | is finite. .2.3 Distality Distality is a notion introduced by Simon to describe the unstable part of an NIP theory.The general definition of distality is slightly more complicated than the definitions ofNIP and dp-minimality (see Definition 2.1 in [15] or Chapter 9 in [16]). In case of adp-minimal theory distality can be described as follows:
Proposition 2.18.
A dp-minimal theory T is distal if and only if there is no infinitenon-constant totally indiscernible set of singletons.Proof. This characterization follows from Example 2.4 and Lemma 2.10 in [15].By Exercise 9.12 of [16] distality is preserved under going to T eq : Proposition 2.19. If T is distal, then so is T eq . Recall that a theory T has quantifier elimination if every formula is equivalent to aquantifier free formula modulo T . The proof of Theorem 3.2.5 in [20] gives the followinguseful criterion for quantifier elimination: Proposition 2.20.
Let T be a theory and let ϕ ( x ) be a formula. Then ϕ ( x ) is equivalentto a quantifier free formula modulo T if and only if for all M , M | = T with commonsubstructure A and all a ∈ A we have M | = ϕ ( a ) = ⇒ M | = ϕ ( a ) . If T is a two-sorted theory and the only symbols that connect the two sorts arefunctions from one sort to the other, then it suffices to check quantifier elimination forvery specific formulas: Lemma 2.21.
Let T be a theory in a two-sorted language L = L ∪ L ∪ { f j : j ∈ J } with sorts S and S where L is purely in the sort S , L is purely in the sort S , andeach f j is a function from sort S to sort S . Suppose(a) every L -formula is equivalent to a quantifier free formula modulo T and(b) every formula of the form ∃ x ∈ S ^ r ∈ R ϕ r ( x, ¯ y r , ¯ z r ) is equivalent to a quantifier free formula modulo T where x is a singleton, ¯ y r ⊆ S , ¯ z r ⊆ S , and each ϕ r is either a basic L -formula or is of the form f j ( t ( x, ¯ y r )) = z where t is an L term and z is one of the variables in the tuple ¯ z r .Then T eliminates quantifiers. roof. To show quantifier elimination it suffices to consider simple existential formulas.Consider a formula of the form ∃ γ ∈ S ^ r ∈ R ϕ r ( γ, ¯ y r , ¯ z r )where γ is a singleton, ¯ y r ⊆ S , ¯ z r ⊆ S , and each ϕ r is a basic formula. We may assumethat γ appears non-trivially in each formula ϕ r . Then each ϕ r is a basic L -formulawhere the variables ¯ y r only appear as terms of the form f ( t ( ¯ y r ))where f is a function symbol and t is an L -term. Now the S -quantifier can be eliminatedby (a).Now consider a formula of the form ∃ x ∈ S ^ r ∈ R ϕ r ( x, ¯ y r , ¯ z r )where x is a singleton, ¯ y r ⊆ S , ¯ z r ⊆ S , and each ϕ r is a basic formula.Let ˜ R ⊆ R be the set of all r ∈ R such that ϕ r is a basic L -formula. If r ∈ ˜ R , thenwe may write ϕ r as ϕ r ≡ ψ r ( ¯ f r ( ¯ t r ( x, ¯ y r )) , ¯ z r )where ψ r is a basic L -formula such that all variables of ψ r are in S . Then ϕ r isequivalent to ∃ ¯ ξ ∈ S : ( ¯ ξ = ¯ f r ( ¯ t r ( x, ¯ y r )) ∧ ψ r ( ¯ ξ, ¯ z r )) . Now we may rewrite ∃ x ∈ S ^ r ∈ R ϕ r ( x, ¯ y r , ¯ z r )as a formula of the form ∃ ( ¯ ξ r ) r ∈ ˜ R ∈ S : (( ^ r ∈ ˜ R ψ r ( ¯ ξ, ¯ z r )) ∧ ( ∃ x ∈ S ^ r ∈ ˜ R ¯ ξ r = ¯ f r ( ¯ t r ( x, ¯ y r )) ∧ ^ r ∈ R \ ˜ R ϕ r ( x, ¯ y r ))) . We can now eliminate the S -quantifier by (b) and then eliminate the S -quantifiers asin the first step. We view a profinite group G together with a fundamental system of open subgroups { K i : i ∈ I } as an L prof -structure ( G, I ) (as in Remark 2.7). The aim of this chapter isto prove the first part of the main theorem: If (
G, I ) is a dp-minimal profinite group,then G has an open abelian subgroup A such that either A is a vector space over F p forsome prime p , or A ∼ = Q p Z α p p × A p where α p < ω and A p is a finite abelian p -group foreach prime p . 8imon showed in [14] that all dp-minimal groups are abelian-by-finite-exponent. Anexample of a dp-minimal group that is not abelian-by-finite was given by Simonetta in[17].We will show that all dp-minimal profinite groups have an open abelian subgroup. Wewill then analyze the structure of this abelian profinite group. For dp-minimal profinitegroups the fundamental system of open subgroups can always be replaced by a chain ofopen subgroups: Lemma 3.1.
Let ( G, I ) be a dp-minimal profinite group. Then the subgroups H i := \ { K j : | G : K j | ≤ | G : K i |} are uniformly definable open subgroups and hence the topology on G is generated by adefinable chain of open subgroups.Proof. The H i are open subgroups by Corollary 2.13. By Lemma 2.17 and compactnesswe can find a constant K such that for all i, j | K i : K i ∩ K j | < K or | K j : K i ∩ K j | < K .Given i, j ∈ I we have | G : K i | ≤ | G : K j | ⇐⇒ | K i : K i ∩ K j | ≥ | K j : K i ∩ K j | . Moreover, we have | K i : K i ∩ K j | < K or | K j : K i ∩ K j | < K . Therefore this is adefinable condition and hence the subgroups H i are uniformly definable.In a dp-minimal profinite group we cannot find infinite definable subgroups of infiniteindex: Lemma 3.2.
Let ( G, I ) be a dp-minimal profinite group. Let ( G ∗ , I ∗ ) be an elementaryextension and let H < G ∗ be a definable subgroup. If G ∩ H is infinite, then | G ∗ : H | isfinite.Proof. If | K ∗ i : K ∗ i ∩ H | is finite for some i ∈ I , then clearly | G ∗ : H | < ∞ .Now assume | K ∗ i : K ∗ i ∩ H | is infinite for all i ∈ I . We aim to show that | H : K ∗ i ∩ H | must be unbounded: Since G ∩ H is infinite and T i ∈ I K i = 1, | G ∩ H : K i ∩ H | must be unbounded. Therefore | H : K ∗ i ∩ H | must be unbounded. This contradictsLemma 2.17.As a consequence of Zelmanov’s theorem (Theorem 2.5) and the existence of definableenvelopes for abelian subgroups (Theorem 2.14) we get that a dp-minimal profinite groupmust be virtually abelian: Proposition 3.3.
Let ( G, I ) be a dp-minimal profinite group. Then G is virtuallyabelian.Proof. By Theorem 2.5 G has an infinite abelian subgroup A . By Theorem 2.14, we canfind an elementary extension ( G ∗ , I ∗ ), a formula ϕ ( x, y ), and a parameter b ∈ ( G ∗ , I ∗ )such that Cen(Cen( ϕ ( G ∗ , b ))) is an abelian subgroup of G ∗ and contains A . ThereforeCen(Cen( ϕ ( G ∗ , b ))) has finite index in G ∗ by Lemma 3.2. By elementarity there is some9 ′ ∈ ( G, I ) such that Cen(Cen( ϕ ( G, b ′ ))) is an abelian group and has finite index in G . Moreover, Cen(Cen( ϕ ( G, b ′ ))) is closed since it is a centralizer. Closed subgroups offinite index are open and therefore Cen(Cen( ϕ ( G, b ′ ))) is an open abelian subgroup of G .We are now able to prove the first part of the main theorem: Theorem 3.4.
Let ( A, I ) be an abelian dp-minimal profinite group. Then either A isvirtually a direct product of countably many copies of F p for some prime p , or A ∼ = Q p Z α p p × A p where α p < ω and A p is a finite abelian p -group for each prime p .Proof. Consider the closed subgroup A [ n ] := { x ∈ A : nx = 0 } . Suppose there is aminimal n such that A [ n ] is infinite. Then A [ n ] has finite index in A (by Lemma 3.2)and hence is an open subgroup of A . Therefore we may assume A = A [ n ]. Now theminimality of n and Proposition 2.2 imply that n must be prime and therefore A is adirect product of copies of F p (again by Proposition 2.2). Since A admits a countablefundamental system of open subgroups, this direct product must be a direct product ofcountably many copies of F p .Now assume A [ n ] is finite for all n . Then the closed subgroup nA must be open in A for all n (by Lemma 3.2). Now Proposition 2.4 implies the theorem. If A is an abelian group and ( A i ) i<ω is a strictly descending chain of subgroups suchthat A = A and T i<ω A i = { } , then we can define a valuation map v : A → ω ∪ {∞} by setting v ( x ) = max { i : x ∈ A i } . We have v ( x ) = ∞ if and only if x = 0, and this valuation satisfies the inequality v ( x − y ) ≥ min { v ( x ) , v ( y ) } where we have equality in case v ( x ) = v ( y ).The valued group ( A, v ) can be seen as a two-sorted structure consisting of the group A , the linear order ( ω ∪ {∞} , ≤ ), and the valuation v : A → ω ∪ {∞} .Our goal is to classify dp-minimal profinite groups up to finite index. We know byLemma 3.1 that the fundamental system of open subgroups can be assumed to be achain. Moreover, by Theorem 3.4 we only need to consider groups of the form Y i<ω F p or Y p Z α p p × A p where α p < ω and A p is a finite abelian p -group for each prime p .If A is such a group and { B i : i < ω } is a fundamental system of open subgroups whichis given by a strictly descending chain, then the above construction yields a definable10aluation v : A → ω ∪ {∞} . Conversely, given such a valuation v , we can recover thefundamental system of open subgroups by setting B i = { a ∈ A : v ( a ) ≥ i } . Hence the valuation and the fundamental system are interdefinable.We will show that if A is of the above form, then A admits a fundamental systemgiven by a chain of open subgroups such that the expansion of A by the correspondingvaluation (and hence the corresponding L prof -structure) is dp-minimal. If A = Q i<ω F p ,this follows from results by Maalouf in [10] and will be explained in Section 4.1. Definition 4.1. (a) The subgroups B i = { a ∈ A : v ( a ) ≥ i } are called the v -balls ofradius i . We will also denote them by B vi to emphasize that they correspond tothe valuation v .(b) A valuation v : A → ω ∪ {∞} is good if for all i < ω the subgroup B i is of the form B i = nA for some positive integer n .In case A ∼ = Q p Z α p p × A p , we will prove a quantifier elimination result for goodvaluations. Note that by Proposition 2.4 each such group can be equipped with a goodvaluation such that { B i : i < ω } is a fundamental system of open subgroups. We willshow the following theorem: Theorem 4.2.
Let A ∼ = Q p Z α p p × A p as above and let v be a good valuation. Then thestructure ( A, + , v ) is dp-minimal. Moreover, it is distal if and only if the size of thequotients B i /B i +1 is bounded. This theorem will be proven in this chapter. If π is a set of primes, a natural number n ≥ π -number if the prime decomposition of n only contains primes in π .An immediate consequence of the above theorem is the following: Corollary 4.3.
Let ( π i ) i<ω be a sequence of finite non-empty disjoint sets of primes.For each i < ω fix a finite non-trivial abelian group A i such that | A i | is a π i -number.Set A = Y i<ω A i and let v be the valuation defined by v (( a i ) i<ω ) = min { i : a i = 0 } . Then ( A, + , v ) is dp-minimal but not distal.Proof. We have B vk = ( Q i Proposition 4.4. The valued abelian profinite group ( A, v ) is dp-minimal.Proof. Set B = L i<ω F p and let w : B → ω ∪ {∞} be the valuation given by w (( x i ) i<ω ) = min { i : x i = 0 } . By Proposition 4 of [10] the valued vector space ( B, w ) is C-minimal and hence dp-minimal (by Theorem A.7 of [16]).Théorème 1 of [10] implies that ( A, v ) and ( B, w ) are elementarily equivalent. Hence( A, v ) is dp-minimal. Remark . The last step of the previous proof also follows from results in Section 6.1.Let ( B, w ) be as in the proof of Proposition 4.4 and set B i = { x ∈ B : w ( x ) ≥ i } . Then A ∼ = lim ←− i<ω B/B i and hence ( A, v ) is dp-minimal by Lemma 6.2. We denote the set of primes by P . For each prime p ∈ P we fix an integer α p ≥ p -group A p . Let Z ≺ Y p ∈ P Z α p p × A p be an abelian group that is (as a pure group) an elementary substructure. We willalways assume Z to be infinite. We fix a set of constants { c j : j < ω } ⊆ Z containing 0such that the set is dense with respect to the profinite topology on Z and contains everytorsion element. It follows from Proposition 2.4 that the set of constants is also densewith respect to the profinite topology on Q p ∈ P Z α p p × A p . Definition 4.6. If π is a set of primes, we set Z π = Z ∩ ( Y p ∈ P Z α p p × Y p ∈ P \ π A p ) and A π = Z ∩ Y p ∈ π A p . Z = Z π × A π for any set π ⊆ P . The group A π is the π -torsionpart of Z and the group Z π has no π -torsion.Let v be a good valuation and set I := ω ∪ { + ∞ , −∞} . For each l ≥ v l : Z → I by v l ( a ) = + ∞ iff a = 0 i iff a ∈ lB i \ lB i +1 −∞ iff a lZ. Note that if a ∈ lZ , then v l ( a ) = v ( a/l ). Now Z together with the valuation v maybe viewed as a two-sorted structure with group sort Z and value sort I in the language L − = L Z ∪ L v ∪ L −I where • L Z = { + , − , c j : j < ω } is the obvious language on Z , • L v = { v l : l ≥ } consists of symbols for the functions v l , and • L −I = {≤ , , + ∞ , −∞} is the obvious language on I .Since we consider the group Z and the constants c j to be fixed, this structure onlydepends on the valuation and we denote it by ( Z, v ).We define the following binary relations on I : • Ind π,lk ( i, j ) ⇐⇒ i ≤ j and | Z π ∩ lB i : Z π ∩ lB j | ≥ k , • Div π,lq k ( i, j ) ⇐⇒ i ≤ j and q kα q divides | Z π ∩ lB i : Z π ∩ lB j | ,where π is a finite set of primes, q ∈ π is a prime, and k ≥ 0. We set Ind π,lk ( i, + ∞ ) andDiv π,lq k ( i, + ∞ ) to be always true. Observation 4.7. (a) If q ∈ π then q kα q divides | Z π ∩ lB i : Z π ∩ lB j | if and onlyif ( Z π ∩ lB i ) / ( Z π ∩ lB j ) has an element of order q k . In particular, the predicateDiv π,lq k is definable.(b) In the standard model Div π,lq k ( i, j ) is equivalent to the statement that q k divides | Z q ∩ lB i : Z q ∩ lB j | . In that sense the expression | Z q ∩ lB i : Z q ∩ lB j | makes senseeven in non-standard models.(c) We have x ∈ nZ if and only if v n ( x ) ≥ 0. Hence the subgroups nZ are quantifierfree 0-definable. Since the subgroups nZ generate the profinite topology on Z , thisimplies that the open subgroups Z π are quantifier free 0-definable for finite subsets π ⊆ P . Moreover, in that case A π is also quantifier free 0-definable since it is afinite set of constants.Let V be the set of good valuations on Z . We set T Z := T v ∈ V Th(( Z, v )) to be thecommon L − -Theory of structures ( Z, v ), v ∈ V . The following quantifier eliminationresult will be shown in the next sections: 13 heorem 4.8. Let L I ⊇ L −I be an expansion on the sort I and let T ⊇ T Z be anexpansion of T Z to the language L = L Z ∪ L v ∪ L I . Suppose that:1. The relations Div π,lq k and Ind π,lk are quantifier free 0-definable modulo T .2. The successor function on I is contained in L I .3. Every L I -formula is equivalent to a quantifier free L -formula modulo T .Then T eliminates quantifiers. To prove the quantifier elimination result we will need to understand formulas thatdescribe systems of linear congruences. Therefore we will need to understand linearcongruences in models of the theory T . Z We will need generalizations of the following well-known fact: Fact 4.9. A linear congruence nx ≡ a mod m in Z has a solution if and only if d =gcd( n, m ) divides a . In that case it has exactly d many solutions modulo m . If s is asolution, then a complete system of solutions modulo m is given by s + tm/d, t = 0 , . . . d − . Observation 4.10. Fact 4.9 has two important consequences:(a) If nx ≡ a mod m has a solution and n = gcd( n, m ), then n divides a and hence a/n is a solution.(b) If nx ≡ a mod m has a solution, then all solutions agree modulo m/d .Part (a) will be important since in that case a solution will be determined by theconstant a . Part (b) tells us that solutions of linear congruences can “collapse”. We willneed to understand this collapsing of solutions.We now fix a group A of the form A = Q p ∈ P Z α p p , α p < ω . If n is a positive integer,let n ( p ) be the unique integer such that n = Q p n ( p ) . Note that Fact 4.9 can be appliedto Z p because Z p /k Z p = Z /p k ( p ) Z .We consider linear congruences nx ≡ a mod m in A where n and m are positive integers and a ∈ A . Note that solving the above linearcongruence is equivalent to solving it in each copy of Z p in the product A = Q p ∈ P Z α p p : Lemma 4.11. (a) Let nx ≡ a mod m be a linear congruence in A . Write a = ( a p,i ) p ∈ P ,i<α p ∈ A = Y p ∈ P Z α p p . The solutions for nx ≡ a mod m in A are exactly the tuples s = ( s p,i ) p ∈ P ,i<α p where each s p,i is a solution for nx ≡ a p,i mod m in Z p . b) Set d = gcd( n, m ) . Then the linear congruence nx ≡ a mod m has a solution ifand only if d divides a in A (i.e. a ∈ d A ). In that case it has exactly Q p | d p α p d ( p ) many solutions modulo m in A . We call a finite family of linear congruences (and negations of linear congruences) a system of linear congruences . Recall Bézout’s identity: Fact 4.12 (Bézout’s identity) . If a , . . . a n are integers, then gcd( a , . . . a n ) is a Z -linearcombination of a , . . . a n .We will look at systems of linear congruences where the modulus is fixed: Proposition 4.13. Let n r x ≡ a r mod m, r ∈ R, be a system S of linear congruencesin A (where R is a finite set). Set n = gcd( n r : r ∈ R ) and d = gcd( n, m ) . By Bézout’sidentity we can find integers z r such that n = P r ∈ R z r n r . Put a = P r ∈ R z r a r .(a) If the system S has a common solution, the solutions of S are exactly the solutionsof nx ≡ a mod m .(b) Set k = n/d and d r = n r /k . Then the system S has a solution if and only if thesystem T : d r x ≡ a r mod m, r ∈ R, has a solution. Moreover, the systems S and T have the same number of solutionsmodulo m .Proof. (a) It suffices to show this for each factor in the product A = Q p ∈ P Z α p p . Hencewe may assume A = Z p and m = p m ( p ) . Clearly any common solution of the system S solves nx ≡ a mod m .Now suppose s is a solution of S (and hence a solution of nx ≡ a mod m ). Then byFact 4.9 all solutions of nx ≡ a mod m are of the form s + tm/d where d = gcd( n, m ).Fix r ∈ R . Now d divides d r = gcd( n r , m ), say d r = k r d . Therefore s + tm/d = s + tk r m/d r solves n r x ≡ a r mod m for all t = 0 , . . . d − nx ≡ a mod m solves S .(b) We have n = gcd( n r : r ∈ R ) = P r ∈ R z r n r . If we divide by k , we get that d = gcd( d r : r ∈ R ) = P r ∈ R z r d r . We aim to show that S has a solution if and only if T has a solution. If s is a solution for S , then ks solves T . Now assume that T has asolution. Then by (a) the system T has the same solutions as the linear congruence dx ≡ a mod m. Since we assume that T has a solution, this implies that d = gcd( d, m ) divides a (bypart (b) of Lemma 4.11). Then the linear congruence nx ≡ a mod m d = gcd( n, m ) divides a . If s solves nx ≡ a mod m , then ks solves dx ≡ a mod m and hence is a solution for T . Thisimplies that s solves S . Hence S has a solution if and only if T has a solution. Moreover,if S and T have solutions, then by (a) the solutions of S are exactly the solutions of nx ≡ a mod m and the solutions of T are exactly the solutions of dx ≡ a mod m .Hence they have the same number of solutions modulo m by part (b) of Lemma 4.11.We will now consider systems of linear congruences where we vary the modulus: Lemma 4.14. Let nx ≡ a mod p m be a linear congruence in Z p . Set d = gcd( n, p m ) and suppose l > divides p m such that d divides p m /l . Then nx ≡ a mod p m and nx ≡ a mod dl have exactly d solutions modulo p m respectively dl and all these solutions agree modulo l .Proof. The assumption implies that dl divides p m . Therefore d = gcd( n, p m ) = gcd( n, dl ) . By Fact 4.9 the congruences have a solution if and only if d divides a . In that case nx ≡ a mod p m has exactly d solutions modulo p m and the congruence nx ≡ a mod dl has exactly d solutions modulo dl . Moreover, by part (b) of Observation 4.10 all thesesolutions agree modulo l . Proposition 4.15. Let nx ≡ a mod m be a linear congruence in A . Set d = gcd( n, m ) .Suppose l > divides m and is such that for all p | d we have p d ( p ) | ( m/l ) or p does notdivide ( m/l ) . Set k = Y p | d,p | ( m/l ) p d ( p ) . Then the linear congruences nx ≡ a mod m and nx ≡ a mod kl have the same number of solutions modulo m respectively kl . Moreover, if X is the setof solutions modulo m of nx ≡ a mod m , Y is the set of solutions modulo kl of nx ≡ a mod kl , and X/l and Y /l are the images of X and Y in A /l A , then X/l = Y /l and eachelement in X/l (resp. Y /l ) has exactly Q p | d,p | ( m/l ) p α p d ( p ) many preimages in X (resp. Y ).Proof. By an application of Lemma 4.11 it suffices to show this in case A = Z p . Hencewe will assume A = Z p .If p does not divide d , then d is a unit in Z p and hence each of the congruences has aunique solution in Z p . 16ence we may assume p divides d . If p does not divide m/l , then m ( p ) = l ( p ) and k ( p ) = 0. Then m Z p = kl Z p = l Z p and therefore the linear congruences nx ≡ a mod m, nx ≡ a mod kl, and nx ≡ a mod l have the same solutions (in Z p ). Since solutions modulo m (resp. modulo kl ) are thesame as solutions modulo l , each element of X/l (resp. Y /l ) has a unique preimage in X (resp. Y ).Now assume p divides m/l . Then by assumption p d ( p ) divides m/l . In that case theresult follows by Lemma 4.14. Note that each element in X/l (resp. Y /l ) has exactly d preimages in X (resp. Y ). Z Fix T ⊃ T Z as in Theorem 4.8 (for a group Z ≺ Q p ∈ P Z α p p × A p as in the beginning ofSection 4.2). We have v l ( a ) ≥ i if and only if a ∈ B v l i = lB i . Therefore we will considercertain formulas as linear congruences: v l ( nx − a ) ≥ i ⇐⇒ nx ≡ a mod lB i ,v l ( nx − a ) < i ⇐⇒ nx a mod lB i . Here x will be a variable and a will be a constant. The integer n will be part of theformula. In particular, it will always be a standard integer. Recall that for a subset π ⊆ P a natural number n ≥ π -number if the prime decomposition of n onlycontains primes in π .We will often work in the π -torsion free group Z π defined in Definition 4.6. If weassume that π is finite, then by part (c) of Observation 4.7 the subgroup Z π is quantifierfree 0-definable. If M | = T is any model, then we set Z π ( M ) to be the subgroup definedby the formula which defines Z π in Z . The subgroup A π ( M ) is defined analogously.If we use the notation in part (b) of Observation 4.7, thengcd( n, lB i ) := gcd( n, Y { p : α p > } | Z p : Z p ∩ lB i | )is well-defined even if lB i has infinite index because n is always a standard integer.Therefore the results in Section 4.2.1 can be formulated using the divisibility predicatesand they will hold true for models of T . Proposition 4.16. Let M be a model of T and let nx ≡ a mod lB i be a linear congru-ence in Z ( M ) . Let π be a finite set of primes such that n is a π -number and a ∈ Z π ( M ) .Then the linear congruence has a solution in Z π ( M ) if and only if d = gcd( n, lB i ) divides a (i.e. a ∈ dZ π ( M ) ). In that case there are exactly Q p α p d ( p ) many solutions modulo lB i in Z π ( M ) .Proof. This is essentially part (b) of Lemma 4.11. Since this is a first-order statement,it suffices to consider good valuations v on Z . Since the statement only affects the17uotients Z/lB i , we may assume that Z is of the form Z = Y p ∈ P Z α p p × A p . Put A = Q p ∈ P Z α p p and H = Q p π A p . Then Z π = A × H and H has no π -torsion. Write a = a h for a ∈ A and h ∈ H . Then we can applyLemma 4.11 to the linear congruence nx ≡ a mod lB i in A . Note that the linear congruence nx ≡ h mod lB i has a unique solution modulo lB i in H (namely h/n ). This shows the proposition. Proposition 4.17. Let M be a model of T and let n r x ≡ a r mod lB i , r ∈ R, bea system of linear congruences. Let π be a finite set of primes such that all n r are π -numbers and all a r are contained in Z π ( M ) . Set n = gcd( n r : r ∈ R ) and d =gcd( n, lB i ) . By Bézout’s identity we can find integers z r such that n = P r ∈ R z r n r . Put a = P r ∈ R z r a r .(a) If the system has a common solution in Z π ( M ) , the solutions of the system in Z π ( M ) are exactly the solutions of nx ≡ a mod lB i in Z π ( M ) .(b) Set k = n/d and d r = n r /k . Then the system has a solution in Z π ( M ) if and onlyif the system d r x ≡ a r mod lB i , r ∈ R, has a solution in Z π ( M ) . Moreover, these systems have the same number of solu-tions modulo lB i in Z π ( M ) .Proof. This follows from Proposition 4.13 by the same arguments that are used in Propo-sition 4.16. Proposition 4.18. Let M be a model of T and let nx ≡ a mod lB i be a linear con-gruence. Let π be a finite set of primes such that n is a π -number and a ∈ Z π ( M ) . Set d = gcd( n, lB i ) . Fix u > and j > i such that ulB j < lB i is a subgroup and is suchthat for all p | d we have that p d ( p ) divides | Z p ∩ lB i : Z p ∩ ulB i | or p does not divide | Z p ∩ lB i : Z p ∩ ulB i | . Set k = Y { p d ( p ) : p divides d and | Z p ∩ lB i : Z p ∩ ulB i |} . Then the linear congruences nx ≡ a mod lB i and nx ≡ a mod kulB j ave the same number of solutions modulo lB i respectively kulB j in Z π ( M ) . Moreover,if X is the set of solutions modulo lB i of nx ≡ a mod lB i , Y is the set of solutionsmodulo kulB j of nx ≡ a mod kulB j , and X/ulB j and Y /ulB j are the images of X and Y modulo ulB j , then X/ulB j = Y /ulB j and each element in X/ulB j (resp. Y /ulB j )has exactly Q p | d,p | ( m/l ) p α p d ( p ) many preimages in X (resp. Y ).Proof. This follows from Proposition 4.15 by the same arguments that are used in Propo-sition 4.16.The following lemma will often be useful: Lemma 4.19. Fix a model M | = T , let π be a finite set of primes, let a ∈ Z π ( M ) , andlet t be a π -number. Then the linear congruences nx ≡ a mod lB i and tnx ≡ ta mod tlB i have the same solutions in Z π ( M ) .Proof. Multiplying by t is injective since Z π ( M ) does not have t -torsion. Z Fix T as in Theorem 4.8. Note that we assume that the successor function S (on I ) iscontained in L I . Lemma 4.20. Let M and M be models of T and let ( A, J ) be a common substructure.Let π be a finite set of primes and let S : n r x ≡ a r mod lB i , r ∈ R, be a system of linear congruences where each n r is a π -number, a r ∈ Z π ( A ) , and i ∈ J .Suppose there is a π -number t and a constant b ∈ A such that t divides b and b/t solves S in Z π ( M ) . Then b/t solves S in Z π ( M ) .Proof. We have t divides b if and only if v t ( b ) ≥ 0. This does not depend on themodel. Moreover, b/t solves n r x ≡ a r mod lB i if and only if v l ( n r ( b/t ) − a r ) ≥ i . ByLemma 4.19 we have v l ( n r ( b/t ) − a r ) = v tl ( n r b − ta r ) . Therefore this value does not depend on the model. Lemma 4.21. Let M and M be models of T and let ( A, J ) be a common substructure.Let π be a finite set of primes and let n r x ≡ a r mod lB i , r ∈ R, be a system of linear congruences where each n r is a π -number, a r ∈ Z π ( A ) , and i ∈ J .Then the system has the same number of solutions modulo lB i in Z π ( M ) and Z π ( M ) . roof. Set n := gcd( n r : r ∈ R ), say n = P r ∈ R z r n r (by Bézout’s identity), and put a := P r ∈ R z r a r . Set d := gcd( n, Q { p : α p > } | Z p : Z p ∩ lB i | ), k = n/d , and d r = n r /k .Then by Proposition 4.17 (b) the system n r x ≡ a r mod lB i , r ∈ R, has the same number of solutions modulo lB i in Z π ( M ) (resp. Z π ( M )) as the system d r x ≡ a r mod lB i , r ∈ R. We have d = gcd( d r : r ∈ R ) = P r ∈ R z r d r . By Proposition 4.17 (a) any solution ofthe system d r x ≡ a r mod lB i , r ∈ R, is a solution of dx ≡ a mod lB i . Now by Proposition 4.16 the linear congruence dx ≡ a mod lB i has a solution if and only if d divides a . In that case a/d must be a solutionand we can apply Lemma 4.20 to see that this must hold true in both models.Hence Z π ( M ) contains a solution if and only if Z π ( M ) contains a solution. In thatcase the solutions are exactly the solutions of nx ≡ a mod lB i and by Proposition 4.16the number of solutions modulo lB i does not depend on the model. Lemma 4.22. Let M and M be models of T and let ( A, J ) be a common substructure.Let π be a finite set of primes and let S be a system n r x ≡ a r mod lB i r , r ∈ R, of linear congruences where l and each n r is a π -number, a r ∈ Z π ( A ) , i r ∈ J . Supposemoreover, that the index | B i r : B i r ′ | is a π -number whenever it is finite. Fix r max ∈ R such that i r max is maximal. Then S has the same number of solutions modulo lB i r max in Z π ( M ) and Z π ( M ) .Proof. If | lB i r : lB i r ′ | is finite, then there is a π -number t such that lB i r = tlB i r ′ .Lemma 4.19 allows us to replace the linear congruence n r ′ x ≡ a r ′ mod lB i r ′ by the linear congruence tn r ′ x ≡ ta r ′ mod lB i r . Hence we may assume that the index | lB i r : lB i r ′ | is infinite whenever i r < i r ′ .For r ∈ R set R [ r ] = { r ∈ R : i r = i r } and consider the system S r : n r x ≡ a r mod lB i r , r ∈ R [ r ] . By Lemma 4.21 the system S r has the same number of solutions modulo lB i r in Z π ( M )and Z π ( M ). If S r has no solution, then S has no solution and we are done. Henceassume that S r has a solution (and hence has the same number of solutions in bothmodels by Lemma 4.21). 20hen by Proposition 4.17 we can replace the system S r by a single linear congruencewithout changing the solutions.Hence we may assume i r = i r ′ ⇐⇒ r = r ′ for all r, r ′ ∈ R . Now we may write R = { r , . . . r m } such that i r > · · · > i r m . Weprove the lemma by induction on m . The case m = 0 is done by Lemma 4.21. Hence weassume m > S has the form n r x ≡ a r mod lB i r , ... n r m x ≡ a r mod lB i rm . Now set d := gcd( n r , Q p ∈ P ,α p > | Z p : Z p ∩ lB i r | ) and put u = Y p | d,α p > {| Z p ∩ lB i r : Z p ∩ lB i r | : | Z p ∩ lB i r : Z p ∩ lB i r | is finite } . Set k = Q { p d ( p ) : α p > u } and consider the system S ′ : n r x ≡ a r mod kulB i r ,un r x ≡ ua r mod ulB i r ,un r x ≡ ua r mod ulB i r , ... un r m x ≡ ua r m mod ulB i rm . By Proposition 4.18 the linear congruences n r x ≡ a r mod lB i r and n r x ≡ a r mod kulB i r have the same number of solutions modulo lB i r respectively kulB i r andthe sets of solutions agree modulo ulB i r . The statement about the number of preimagesin Proposition 4.18 implies that S and S ′ have the same number of solutions modulo lB i r respectively kulB i r . By Lemma 4.19 we can rewrite S ′ as follows: n r x ≡ a r mod kulB i r ,kun r x ≡ kua r mod kulB i r ,kun r x ≡ kua r mod kulB i r , ... kun r m x ≡ kua r m mod kulB i rm . By induction the system S ′ has the same number of solutions modulo kulB i r in Z π ( M ) and Z π ( M ). Hence S has the same number of solutions modulo lB i r in Z π ( M )and Z π ( M ). 21o deal with the general case we will make use of the following: Fact 4.23 (Inclusion-exclusion priciple) . Let A , . . . A n be finite sets. Then | n [ i =1 A i | = X ∅6 = J ⊆{ ,...n } ( − | J | +1 | \ j ∈ J A j | . Proposition 4.24. Let M and M be models of T and let ( A, J ) be a common sub-structure. Let π be a finite set of primes and let S be a system n r x ≡ a r mod lB i r , r ∈ R,n s x a s mod lB i s , s ∈ S, of linear congruences where each n t is a π -number, a t ∈ Z π ( A ) , i t ∈ J for all t ∈ R ∪ S .Assume there is r ∈ R such that i r is maximal in { i t : t ∈ R ∪ S } . Suppose moreover,that the index | B i t : B i t ′ | is a π -number whenever it is finite. Then S has the samenumber of solutions modulo lB i r in Z π ( M ) and Z π ( M ) .Proof. For pairwise distinct s , . . . s n ∈ S let A s ,...s n be the set of solutions modulo lB i r of the system S s ,...s n : n r x ≡ a r mod lB i r , r ∈ R,n s x ≡ a s mod lB i s , ... n s n x ≡ a s n mod lB i sn . By Lemma 4.22 the system S s ,...s n has the same number of solutions in Z π ( M ) and Z π ( M ). In particular, this holds true for the system S ∅ : n r x ≡ a r mod lB i r , r ∈ R. Moreover, S s ∈ S A s is exactly the set of solutions modulo lB i r for S ∅ that do not solve S .Note that A s ∩ · · · ∩ A s n = A s ,...s n and hence by an application of the inclusion-exclusion principle the number | S s ∈ S A s | (which is finite since we only count solutionsmodulo lB i r ) does not depend on the model.Now the system S is solved by exactly | A ∅ | − | [ s ∈ S A s | many solutions modulo lB i r and this number does not depend on the model.22 .2.4 Proof of quantifier elimination Proof of Theorem 4.8. By Lemma 2.21 it suffices to show that every formula of the form ψ (¯ z, ¯ i ) ≡ ∃ x ∈ Z ^ r ∈ R ϕ r ( x, ¯ z, ¯ i )is equivalent to a quantifier free formula modulo T where ¯ z ⊆ Z , ¯ i ⊆ I and each ϕ r iseither a basic L Z -formula or is of the form v l r ( t r ( x, ¯ z )) = i r where t r is an L Z -term and i r is one variable in the tuple ¯ i .Write R = R ∪ R ∪ R such that ϕ r ( x, ¯ z, ¯ i ) ≡ n r x − t r (¯ z ) = 0 , for r ∈ R ,ϕ r ( x, ¯ z, ¯ i ) ≡ n r x − t r (¯ z ) = 0 , for r ∈ R , and ϕ r ( x, ¯ z, ¯ i ) ≡ v l r ( n r x − t r (¯ z )) = i r , for r ∈ R . Now let π be a finite set of primes such that n r , l r , and the cardinalities of all finitequotients | lB i r : lB i r ′ | are π -numbers. Fix two models M , M of T and let ( A, J ) bea common substructure, ¯ a ⊆ A, ¯ η ⊆ J . Set a r := t r (¯ a ). We have A = Z π ( A ) × A π ( A )(since A π is a finite set of constants) and hence each a r can be written as a r = a πr b πr with a πr ∈ Z π ( A ) and b πr ∈ A π ( A ). Now suppose the formula ψ (¯ a, ¯ η ) has a solution in M . By Proposition 2.20 it suffices to show that it has a solution in M .If r ∈ R , then ϕ r must be satisfied in Z π ( M ) and A π ( M ). If r ∈ R , then it sufficesif ϕ r is satisfied in Z π ( M ) or A π ( M ). If r ∈ R , then we have ϕ r ( x, ¯ a, ¯ η ) ≡ v l r ( n r x − a r ) = i r . This is satisfied if we have “=” in Z π ( M ) or A π ( M ) and “ ≥ ” in the other subgroup.Hence there are subsets R π ⊆ R and R π ⊆ R such that the formulas ψ π ≡ ∃ x ∈ Z π ^ r ∈ R n r x − a πr = 0 ∧ ^ r ∈ R π n r x − a πr = 0 ∧ ^ r ∈ R π v l r ( n r x − a πr ) = i r ∧ ^ r ∈ R \ R π v l r ( n r x − a πr ) ≥ i r ψ π ≡ ∃ x ∈ A π ^ r ∈ R n r x − b πr = 0 ∧ ^ r ∈ R \ R π n r x − b πr = 0 ∧ ^ r ∈ R \ R π v l r ( n r x − b πr ) = i r ∧ ^ r ∈ R π v l r ( n r x − b πr ) ≥ i r have a solution in Z π ( M ) respectively A π ( M ). Since A π is a finite set of constants,this implies that ψ π has a solution in A π ( M ). It remains to show that ψ π has a solutionin Z π ( M ).If R = ∅ , then we are done since the formulas x ∈ nZ are quantifier free 0-definableand hence the result follows from the usual quantifier elimination for abelian groups.Therefore we assume R = ∅ .If R = ∅ , say r ∈ R , then a πr /n r is the solution of ψ π in Z π ( M ). Lemma 4.20implies that a πr /n r also solves ψ π in Z π ( M ). Hence we may assume R = ∅ .If i r = + ∞ for some r , then we have v l r ( n r x − a πr ) ≥ i r ⇐⇒ n r x − a πr = 0 . Hence we may assume i r < + ∞ for all r ∈ R .Given l ′ ≥ C l ′ in the language such that the formula v l ′ ( t ( x, ¯ z )) = −∞ is equivalent to _ c ∈ C l ′ v l ′ ( t ( x, ¯ z ) − c ) ≥ . Thus we may also assume i r > −∞ for all r ∈ R .Note that each formula of the form n r x − a πr = 0 excludes only a single solution. Sincewe assume R = ∅ and all formulas of the form v l r ( n r x − a πr ) = i r or v l r ( n r x − a πr ) ≥ i r are solved by cosets of l r B i r +1 , we may moreover assume R = ∅ .By Lemma 4.19 we have v l r ( n r x − a πr ) = v ml r ( mn r x − ma πr ) for all π -numbers m .Thus we may use Lemma 4.19 to replace each l r ′ by l := lcm( l r : r ∈ R ).We consider formulas as linear congruences: v l ( n r x − a πr ) = i r ⇐⇒ ( n r x − a πr ≡ lB i r ∧ n r x − a πr lB i r +1 ) ,v l ( n r x − a πr ) ≥ i r ⇐⇒ n r x − a πr ≡ lB i r . Hence it suffices to show that the system of linear congruences n r x − a πr ≡ lB i r , r ∈ R ,n r x − a πr lB i r +1 , r ∈ R π , Z π ( M ). After slightly adjusting the system (by using Lemma 4.19)and renaming, we get a system n s x − b s ≡ lB i s , s ∈ S,n t x − b t lB i t , t ∈ T, where S = ∅ and every index | B i r : B i r ′ | , r, r ′ ∈ S ∪ T is infinite or trivial. If there isan element s ∈ S such that i s is maximal in { i r : r ∈ S ∪ T } , then we are done byProposition 4.24. Hence suppose there is t ∈ T such that i t > i s for all s ∈ S . Then | B i s : B i t | is infinite for all s ∈ S . In particular, the congruence n t x − b t lB i t can be ignored, since each lB i s -class consists of infinitely many lB t classes. Hence weremoved one linear congruence from the system. After iterating this, we can find s ∈ S such that i s is maximal. Theorem 4.8 gives quantifier elimination up to a suitable language on I . The followinggives a tame expansion of L −I which allows us to analyze the definable sets.A binary relation R on a linear ordering is called monotone if and only if it satisfies x ′ ≤ xRy ≤ y ′ implies x ′ Ry ′ . The following result by Simon states that expanding a linear ordering by monotonebinary relations is tame: Proposition 4.25 (Proposition 4.1 and Proposition 4.2 of [14]) . Let ( I, ≤ , R α , C β ) α,β be a linear order equipped with monotone binary relations and unary predicates suchthat every ∅ -definable monotone binary relation is given by one of the R α and every ∅ -definable unary predicate is given by one of the C β . Then ( I, ≤ , R α , C β ) α,β has quantifierelimination and is dp-minimal. Fix a theory T Z as in the quantifier elimination statement and let M | = T Z be a model.Note that the definable relations ≤ , Div π,lq k , and Ind π,lk are monotone. Definition 4.26. Let S be a set of unary predicates and monotone binary relations onthe value set of M .(a) We define L S I , mon to be the monotone hull of L S I := L −I ∪ { Div π,lq k , Ind π,lk } q,π,l,k ∪ S, i.e. the expansion of L S I by all 0-definable (in L S I ) unary relations and all 0-definablemonotone binary relations on the value sort.25b) Set L S mon = L Z ∪ L v ∪ L S I , mon and define L mon = L ∅ mon .Note that L mon ⊇ L − is an expansion by definitions. Proposition 4.27. Let S be as in Definition 4.26. Then Th( M ) admits quantifierelimination in the language L S mon .Proof. The successor function and its inverse are 0-definable. If R ∈ L S I , mon is a mono-tone binary relation, then so is R m,n ( x, y ) ⇐⇒ R ( x + m, y + n ) for all m, n ∈ Z .The same holds true for 0-definable unary predicates. Therefore adding the successorfunction to the language does not add any new definable sets in I . Hence Theorem 4.8and Proposition 4.25 imply quantifier elimination in L S mon . Let T be a complete L S mon -theory as in Proposition 4.27. Lemma 4.28. Let ( Z, I, v ) | = T be a sufficiently saturated model and let ( a j ) j ∈ J and ( b j ) j ∈ J be mutually indiscernible sequences in the group sort. Let γ ∈ I be a singleton.Then one of the sequences is indiscernible over γ .Proof. We may assume that both sequences are indexed by a dense linear order. Suppose( a j ) j ∈ J is not indiscernible over γ . By the quantifier elimination result this must bewitnessed by a formula of the form R ( v l ( t (¯ x )) , γ )where t is an L Z -term, R is a monotone binary relation on I , and l ≥ 1. Hence we canfind tuples ¯ j , ¯ j ⊆ J of the same order type such that | = R ( v l ( t ( a j )) , c ) and = R ( v l ( t ( a j )) , c )where a j i = ( a j ) j ∈ j i is the tuple corresponding to j i ⊆ J .After replacing ¯ j or ¯ j if necessary, we may assume that ¯ j and ¯ j have disjoint convexhulls in J . We can extend to a sequence ( ¯ j i ) i<ω such that ( a j i ) i<ω is an indiscernablesequence. Then ( v l ( t ( a j i ))) i<ω is a non-constant indiscernible sequence in the value sort that is not indiscernible over γ . By Proposition 4.25 the value sort is dp-minimal. Therefore ( b j ) j ∈ J must be indis-cernible over γ : Otherwise we could apply the above argument to the sequence ( b j ) j ∈ J to get a second non-constant indiscernible sequence in the value sort which is not indis-cernible over γ . Since these two sequences would be mutually indiscernible, this wouldcontradict dp-minimality of the value sort. Theorem 4.29. T is dp-minimal. roof. Let M = ( Z, I, v ) | = T be a sufficiently saturated model and let J and J bemutually indiscernible sequences. We will assume that both of them are indexed by adense linear order. Let z ∈ Z be a singleton. We aim to show that one of the sequencesis indiscernible over z .Since I is essentially an imaginary sort, we may assume that the sequences J and J live in the sort Z . Note that equality on the value sort can be expressed usingthe monotone binary relation ≤ . By the quantifier elimination result, the failure ofindiscernibility must be witnessed by formulas of the following form:1. t (¯ x ) − nz = 0,2. C ( v l ( t (¯ x ) − nz )),3. R ( v l ( t (¯ x ) − n z ) , v l ( t (¯ x ) − n z )),where t is an L Z -term, C is a coloring on I , R is a monotone binary relation on I , l ≥ n ∈ Z . One of the terms in the third case could also be a quantifier free 0-definableconstant in the value sort. This case is analogous to case (b) below and therefore wewill not consider it separately.Note that a formula of the first type would imply that z is algebraic over the parametersplugged in for ¯ x . Hence it suffices to consider the other two types of formulas. If anindiscernible sequence J is not indiscernible over z , then this must be witnessed by¯ a, ¯ a ′ ⊆ J of the same order type such that we are in one of the following cases:(a) We have v l ( t (¯ a ) − nz ) = v l ( t (¯ a ′ ) − nz ) and v l ( t ′ (¯ a ) − n ′ z ) = v l ( t ′ (¯ a ′ ) − n ′ z )and | = R ( v l ( t (¯ a ) − nz ) , v l ( t ′ (¯ a ) − n ′ z )) and = R ( v l ( t (¯ a ′ ) − nz ) , v l ( t ′ (¯ a ′ ) − n ′ z ))for some choice of t, t ′ , n = 0 , n ′ = 0 , and a relation R .(b) We have v l ( t (¯ a ) − nz ) = v l ( t (¯ a ′ ) − nz )and | = R ( v l ( t (¯ a ) − nz ) , v l ( t ′ (¯ a ))) and = R ( v l ( t (¯ a ′ ) − nz ) , v l ( t ′ (¯ a ′ )))for some choice of t, t ′ , n = 0 , and a relation R .(c) We have v l ( t (¯ a ) − nz ) = v l ( t (¯ a ′ ) − nz ) and v l ( t ′ (¯ a )) < v l ( t ′ (¯ a ′ ))and = R ( v l ( t (¯ a ) − nz ) , v l ( t ′ (¯ a ))) and | = R ( v l ( t (¯ a ′ ) − nz ) , v l ( t ′ (¯ a ′ )))27r | = R ( v l ( t ′ (¯ a ) , v l ( t (¯ a ) − nz ))) and = R ( v l ( t ′ (¯ a ′ )) , v l ( t (¯ a ′ ) − nz ))for some choice of t, t ′ , n = 0 , and a monotone binary relation R .(d) We have v l ( t (¯ a ) − nz ) = v l ( t (¯ a ′ ) − nz ) and v l ( t ′ (¯ a ) − n ′ z ) < v l ( t ′ (¯ a ′ ) − n ′ z )and = R ( v l ( t (¯ a ) − nz ) , v l ( t ′ (¯ a ) − n ′ z )) and | = R ( v l ( t (¯ a ′ ) − nz ) , v l ( t ′ (¯ a ′ ) − n ′ z ))or | = R ( v l ( t ′ (¯ a ) − n ′ z ) , v l ( t (¯ a ) − nz ))) and = R ( v l ( t ′ (¯ a ′ ) − n ′ z ) , v l ( t (¯ a ′ ) − nz ))for some choice of t, t ′ , n = 0 , n ′ = 0 , and a monotone binary relation R .The case corresponding to a coloring is essentially the same as (b) so we will not do itexplicitly.We will use Lemma 4.19 to assume that all the l i coincide: Let π be a finite set ofprimes. We want to be able to work in Z π ( M ). Fix a term v l ( t (¯ a ) − nz )and write t (¯ a ) = b (¯ a ) + b (¯ a ), z = c + c for b (¯ a ) , c ∈ Z π ( M ), b (¯ a ) , c ∈ A π ( M ).Since A π ( M ) is a finite set of constants, the value of b (¯ a ) only depends on the ordertype of ¯ a . Therefore γ = v l ( b (¯ a ) + nc ) ∈ A π ( M )also only depends on the order type of ¯ a . We have v l ( t (¯ x ) − nz ) = min { v l ( b (¯ x ) − nc ) , γ } because Z = Z π ( M ) × A π ( M ). If v l ( t (¯ a ′ ) − nz ) = γ for all ¯ a ′ of the same order type as¯ a , then this value is a constant. If v l ( t (¯ a ′ ) − nz ) = v l ( b (¯ a ′ ) − nc ) for all ¯ a ′ of the sameorder type as ¯ a , then this value can always be calculated in Z π ( M ). If we are not in oneof these two cases, then the quantifier free 0-definable coloring C <γ ( i ) ⇐⇒ i < γ witnesses (in Z π ( M )) that J is not indiscernible over z . Hence we can work in Z π ( M )and therefore we can assume that all the l i coincide (by Lemma 4.19). Moreover, tosimplify the notation we will assume that all the l i are equal to 1.We say that an indiscernible sequence J has an approximation for z over α ∈ I if thereis a set D such that J is indiscernible over D , α is definable over D , and the residueclass of z modulo B α is algebraic (in Z/B α ) over parameters in D .We now assume that the mutually indiscernible sequences J and J both fail to beindiscernible over z . Then this must be witnessed as in (a) to (d). Such a witness for J (resp. J ) is good if J (resp. J ) has an approximation for z for a suitable α defined asfollows: 28 If the witness is given as in (a), then we set α = max { v ( t (¯ a ) − t (¯ a ′ )) , v ( t ′ (¯ a ) − t ′ (¯ a ′ )) } + 1 . If (for example) α = v ( t (¯ a ) − t (¯ a ′ )) + 1 (= v ( t (¯ a ) − nz ) + 1 < v ( t (¯ a ′ ) − nz ) + 1),then t ( ¯ a ′ ) ≡ n ′ z mod B α . Therefore the residue class of z modulo B α is algebraicover t (¯ a ′ ). • If the witness is given as in (b), then we set α = v ( t (¯ a ) − t (¯ a ′ )) + 1 = min { v ( t (¯ a ) − nz ) , v ( t (¯ a ′ ) − nz ) } + 1 . If v ( t (¯ a ) − nz ) < v ( t (¯ a ′ ) − nz ), then t ( ¯ a ′ ) ≡ n ′ z mod B α and therefore the residueclass of z modulo B α is algebraic over t (¯ a ′ ). • If the witness is given as in (c), we set α = v ( t (¯ a ) − nz ) . • Now assume the witness is given as in (d). We set α = v ( t (¯ a ) − nz ) and α = v ( t ′ (¯ a ) − n ′ z ) + 1 . Now put α = max { α , α } .In particular, every witness of type (a) or (b) is good because J and J are mutuallyindiscernible. Recall that if v ( x ) < v ( y ), then v ( x − y ) = v ( x ). We aim to show that wecan always find a good witness:Suppose the witness is given as in (c). Choose ¯ a ⊆ J of the same order type as ¯ a and ¯ a ′ such that all indices involved in ¯ a are smaller than the indices in ¯ a and ¯ a ′ (fromnow on, we will write ¯ a ≪ ¯ a, ¯ a ′ in that case). If v ( t (¯ a ) − nz ) = v ( t (¯ a ) − nz ), theneither the pair ( ¯ a , ¯ a ) or the pair (¯ a , ¯ a ′ ) gives a good witness as in case (b).Hence we will assume v ( t (¯ a ) − nz ) = v ( t (¯ a ) − nz ). Let J > ¯ a be the sequence consistingof all elements of J with index larger than all indices in ¯ a and set J ∪ ¯ a to be thesequence J where each tuple is expanded by ¯ a . Then J > ¯ a and J ∪ ¯ a are mutuallyindiscernible. Moreover, J > ¯ a is not indiscernible over α = v ( t (¯ a ) − nz ) (as witnessed by¯ a and ¯ a ′ ). Hence J ∪ ¯ a is indiscernible over α by Lemma 4.28. Now J is indiscernibleover the set { ¯ a , α } and we have ¯ a ≡ nz mod B α . Therefore the residue class of z modulo B α is algebraic over ¯ a and hence the witness is good.Now suppose the witness is given as in (d). We set α = v ( t (¯ a ) − nz ) and α = v ( t ′ (¯ a ) − n ′ z ) + 1 . If α ≥ α , then we have a good witness by the same arguments as in (a) and (b). Henceassume α := α > α . Suppose for all ¯ a ≪ ¯ a, ¯ a ′ we have v ( t (¯ a ) − nz ) = α . Fix¯ a ≪ ¯ a ≪ ¯ a, ¯ a ′ . J > ¯ a and J ∪ ¯ a .Assume that J > ¯ a is indiscernible over α . Then the residue class of z modulo B α is algebraic over t (¯ a ). Since t ′ (¯ a ) n ′ z mod B α , we get t ′ (¯ a ′ ) n ′ z mod B α byindiscernibility (applied to α and ¯ a ). Therefore v ( t ′ (¯ a ) − n ′ z ) and v ( t ′ (¯ a ′ ) − n ′ z ) onlydepend on the residue class of z modulo B α (and can be calculated in Z/B α ) and hencecannot witness the failure of indiscernibility over z .Hence J > ¯ a is not indiscernible over α . Then J ∪ ¯ a is indiscernible over α byLemma 4.28. Therefore J is indiscernible over { ¯ a , α } and the residue class of z modulo B α is algebraic over ¯ a . Hence we have a good witness.Hence we assume that there is ¯ a ≪ ¯ a, ¯ a ′ such that v ( t (¯ a ) − nz ) = α. If v ( t (¯ a ) − nz ) > α , then α = v ( t (¯ a ) − t (¯ a )) and we have a good witness as in cases(a) and (b). Hence we assume v ( t (¯ a ) − nz ) < α .If v ( t ′ (¯ a ) − n ′ z ) 6∈ { v ( t ′ (¯ a ) − n ′ z ) , v ( t ′ (¯ a ′ ) − n ′ z ) } , then (¯ a , ¯ a ) or (¯ a , ¯ a ′ ) gives a goodwitness as in case (a). If v ( t ′ (¯ a ) − n ′ z ) = v ( t ′ (¯ a ) − n ′ z ), then either (¯ a , ¯ a ′ ) gives awitness as in case (a) or the new witness is given by (¯ a , ¯ a ) and we have v ( t (¯ a ) − nz ) > v ( t (¯ a ) − nz ) and v ( t ′ ¯ a ) − n ′ z ) = v ( t ′ (¯ a ) − n ′ z ) . Hence we are again in case (d) but J is indiscernible over v ( t ′ (¯ a ) − n ′ z ) = v ( t ′ (¯ a ) − t ′ (¯ a ′ ))and hence this witness given by (¯ a , ¯ a ) must be good.Now only the case v ( t ′ (¯ a ) − n ′ z ) = v ( t ′ (¯ a ′ ) − n ′ z ) is left. We then have v ( t (¯ a ) − nz ) = v ( t (¯ a ′ ) − nz ) > v ( t (¯ a ) − nz ) ,v ( t ′ (¯ a ) − n ′ z ) < v ( t ′ (¯ a ′ ) − n ′ z ) = v ( t ′ ( ¯ a ) − n ′ z ) . Assume the witnessing formula was of the form R ( v ( t (¯ x ) − nz ) , v ( t ′ (¯ x ) − n ′ z ))for a monotone binary relation R (the other case is done analogously).We then have the following implications by monotonicity: | = R ( v ( t (¯ a ) − nz ) , v ( t ′ (¯ a ) − n ′ z ))= ⇒ | = R ( v ( t (¯ a ′ ) − nz ) , v ( t ′ (¯ a ′ ) − n ′ z ))= ⇒ | = R ( v ( t (¯ a ) − nz ) , v ( t ′ (¯ a ) − n ′ z )) . Hence R ( v ( t (¯ a ) − nz ) , v ( t ′ (¯ a ) − n ′ z )) must be false and R ( v ( t (¯ a ′ ) − nz ) , v ( t ′ (¯ a ′ ) − n ′ z ))must be true (since this was a witness for the failure of indiscernibility over z ). Then R ( v ( t (¯ a ) − nz ) , v ( t ′ (¯ a ) − n ′ z )) must be true. But then ¯ a and ¯ a give a witness as in(a). Hence we can always find a good witness.30ince we assume that both J and J fail to be indiscernible over z , we can find agood witness for each of them. Let α be the constant for the witness in J and let β bethe constant for the witness in J . We assume α ≤ β . Then J is indiscernible over β and over the residue class of z in Z/B β .Suppose the witness for J is given as in (a) or (b). If we have v ( t (¯ a ) − nz ) < v ( t (¯ a ′ ) − nz ) , then v ( t (¯ a ) − nz ) < β and indiscernibility (and algebraicity of z modulo B β over asuitable parameter) imply that v ( t (¯ a ′ ) − nz ) < β . Hence those values only depend onthe residue class of z modulo B β (and can be calculated in Z/B β with the restrictedvaluation). Therefore they cannot witness the failure of indiscernibility over z .Now suppose the witness for J is given as in case (c). If α = β , then this cannot bea witness for the failure for indiscernibility. Hence we must have α < β . But then v ( t (¯ a ) − nz ) = v ( t (¯ a ′ ) − nz )only depends on the residue class of z in B β and we can argue as before. The samearguments work if the witness for J is given as in case (d).Hence J or J must be indiscernible over z .To characterize distality we will show that the quotients B i /B i +1 are stable. We willmake use of the following lemma: Lemma 4.30 (Lemma 5.13 of [2]) . Let L be any language and let T be an unstable L -theory. Let L − ⊆ L be such that T | L − is stable. Then there exists an L -formula ϕ ( x, y ) , | x | = 1 , over ∅ and a parameter b such that ϕ ( x, b ) is not L − -definable. Proposition 4.31. Suppose B i /B i +1 is infinite. Then the induced structure on B i /B i +1 is stable.Proof. Suppose B i /B i +1 is infinite. By Lemma 4.30 it suffices to show that for everyformula ˜ ϕ (˜ x, ˜ y ) (in B i /B i +1 ) and every constant ˜ b ∈ B i /B i +1 the formula ˜ ϕ (˜ x, ˜ b ) isdefinable in the pure group ( B i /B i +1 , +).Given such a formula there is an L S mon -formula ϕ ( x, y ) such that ϕ is the preimage of˜ ϕ under the natural projection π i : B i → B i /B i +1 . Now fix a preimage b of ˜ b . Note that ϕ ( B i , b ) is a union of cosets of B i +1 .By the quantifier elimination result ϕ is equivalent to a boolean combination of atomic L S mon -formulas. We aim to show the following: Claim. There is a formula ψ ( x, y ) which is defined in the pure abelian group ( B i , +)such that ϕ ( B i , b ) and ψ ( B i , b ) coincide on all but finitely many cosets of B i +1 .It suffices to prove the claim for atomic formulas. Therefore we may assume that ϕ isatomic. Then we are in one of the following cases:31a) ϕ ( x, b ) ≡ nx − t ( b ) = 0,(b) ϕ ( x, b ) ≡ R ( v l ( n x − t ( b )) , v l ( n x − t ( b ))),(c) ϕ ( x, b ) ≡ C ( v l ( nx − t ( b ))).In case (a) there is nothing to show. Therefore we consider the cases (b) and (c) whichinclude valuations. We show that sets of the form( a + lB j ) ∩ B i are definable in the pure abelian group ( B i , +) up to a unique coset of B i +1 :If j < i , then B j > B i and lB j has finite index in B j . Hence lB j ∩ B i has finite indexin B i . Moreover, lB j ∩ B i is of the form lB j ∩ B i = l ′ B i for a positive integer l ′ because this holds true for the standard models. The cosets of l ′ B i are definable in the pure group language. If j = i , then lB i ∩ B i = lB i is definablein the pure group language. Now assume j > i . Then lB j < B j ≤ B i +1 and therefore B i ∩ ( a + lB j ) is trivial outside of a single coset of B i +1 .This also shows that there are only finitely many intersections of the form( a + lB j ) ∩ ( B i \ ( a + B i +1 ))where everything except j is fixed. Therefore the restriction of v l ( x − a ) to B i \ ( a + B i +1 )is given by a finite chain of definable subgroups (in the pure abelian group ( B i , +)).Since nx ≡ B i +1 has only finitely many solutions modulo B i +1 the same holdstrue for the valuation v l ( nx − a ) restricted to B i : Outside of finitely many cosets of B i +1 it is given by a finite chain of ( B i , +)-definable subgroups. In that sense v l ( nx − a ) is( B i , +)-definable outside of finitely many cosets of B i +1 .Therefore the formula ϕ in (a) or (b) is definable in ( B i , +) outside of finitely manycosets of B i +1 . This shows the claim.Hence we can find such a formula ψ ( x, y ) defined in the pure abelian group ( B i , +)such that ϕ ( B i , b ) and ψ ( B i , b ) coincide on all but finitely many cosets of B i +1 . Theusual quantifier elimination result for abelian groups shows that ψ ( B , b ) is a booleancombination of cosets of the trivial subgroup and groups of the form lB i for l ≥ 1. Eachsubgroup lB i has finite index in B i and the family { lB i : l ≥ } is closed under finiteintersections. Hence a boolean combination of such groups is a union of finitely manycosets of lB i for a suitable l .Since ϕ ( B i , b ) and ψ ( B i , b ) agree on all but finitely many cosets of B i +1 and ϕ ( B i , b )is a union of cosets of B i +1 , the same must be true for ˜ ϕ ( B i /B i +1 , ˜ b ), i.e. ˜ ϕ ( B i /B i +1 , ˜ b )is a boolean combination of cosets of ( B i /B i +1 , +)-definable subgroups. Therefore˜ ϕ ( B i /B i +1 , ˜ b ) is definable in ( B i /B i +1 , +). Theorem 4.32. T is distal if and only if there is a constant k < ω such that | B i /B i +1 | ≤ k holds for all i < ∞ . roof. Suppose | B i /B i +1 | is unbounded. Then there is some i such that | B i /B i +1 | is infinite. By Proposition 4.31 the induced structure on B i /B i +1 is stable. Hence itfollows from Proposition 2.19 that T is not distal.Now let X be a non-constant totally indiscernible set of singletons and fix x, y ∈ X .Put i = v ( x − y ). If x = y , then i < ∞ and hence xB i = yB i xB i +1 = xB i +1 . It follows easily from total indiscernibility that i does not depend on the choice of x = y .Hence | X | ≤ | B i /B i +1 | . The most well-known example of a dp-minimal expansion of ( Z , +) is ( Z , + , ≤ ). Based onwork by Palacín and Sklinos [12], Conant and Pillay [5] proved the remarkable result that( Z , + , 0) has no proper stable expansions of finite dp-rank. Hence any proper dp-minimalexpansion must be unstable. The other known examples of dp-minimal expansions are: • ( Z , + , v p ) where v p is the p -adic valuation on Z . This was shown by Alouf andd’Elbée in [2]. • ( Z , + , C ) where C is cyclic order. These were found by Tran and Walsberg in [21]. • Proper dp-minimal expansions of ( Z , + , S ), where S is a dense cyclic order, and( Z , + , v p ) were very recently found by Walsberg in [22].An overview about the current research on dp-minimal expansions of ( Z , +) is given byWalsberg in Section 6 of [22]. We add the following family of examples which generalize the p -adic examples by Aloufand d’Elbée: Theorem 5.1. Let ( B i ) i<ω be a strictly descending chain of subgroups of Z , B = Z ,let v : Z → ω ∪ {∞} be the valuation defined by v ( x ) = max { i : x ∈ B i } , and let S be a set of unary predicates and monotone binary relations on the value set.Then ( Z , , , + , v, S ) admits quantifier elimination in the language L S mon (with and as constants) and is dp-minimal. Moreover, ( Z , , , + , v, S ) is distal if and only if thesize of the quotients B i /B i +1 is bounded.Proof. Note that any infinite strictly descending chain of subgroups of Z must havetrivial intersection. Moreover, every non-trivial subgroup of Z is of the form n Z forsome n ≥ v is a good valuation in the sense of Definition 4.1.33oreover, Z ≺ ˆ Z = Q p Z p and h i = Z is dense. Hence Proposition 4.27 impliesthe quantifier elimination result. Dp-minimality follows by Theorem 4.29 and the claimabout distality follows by Theorem 4.32.In case of the p -adic valuation Alouf and d’Elbée proved in Theorem 1.1 of [2] that( Z , + , v p ) has quantifier elimination in the language L Ep = { + , − , , , | p , D n ) n ≥ where x | p y ⇐⇒ v p ( x ) ≤ v p ( y ) and D n = n Z . Conant [4] showed that the structure ( Z , + , , , ≤ ) is a minimal proper expansion of( Z , + , , Z , + , , , v p ). We will show that this does not hold true for arbitraryvaluations. Proposition 5.2. Fix distinct primes p , p , q ∈ P and put s = p p q . For i < ω fix σ i ∈ Sym( { , } ) , set n = 1 , and recursively define n l + m = n l − q iff m = 0 ,n l p σ l (0) iff m = 1 ,n l +1 p σ l (1) iff m = 2 . Set v σ to be the valuation corresponding to ( n i Z ) i<ω and let w be the valuation corre-sponding to ( s i Z ) i<ω . Then w is definable in ( Z , + , v σ ) .Proof. If a ∈ Z \ { } , then there is a unique t a ∈ { p , p , q } such that v σ ( a ) < v σ ( t a a ).Let a, b ∈ Z \ { } . If | v σ ( a ) − v σ ( b ) | ≥ 3, then w ( a ) < w ( b ) ⇐⇒ v σ ( a ) < v σ ( b ) . If | v σ ( a ) − v σ ( b ) | < 3, then w ( a ) ≤ w ( b ) can be determined using t a and t b . Corollary 5.3. Let w be as in Proposition 5.2. Then there are ℵ many valuations v such that w is definable in ( Z , + , v ) . Only countably many of those can be definable in ( Z , + , w ) .Proof. There are 2 ℵ many valuations v σ as in Proposition 5.2 and w is definable in each( Z , + , v σ ). On the other hand, ( Z , + , w ) has only countably many definable sets. Remark . Note that by Theorem 4.32 all these structures are distal. Hence not evenall dp-minimal distal expansions by valuations are minimal expansions.The fact that expansions by arbitrary valuations are dp-minimal allows us to constructother non-trivial examples: For k ≥ v k denote the valuation corresponding to thesequence ( k i Z ) i<ω . Proposition 5.5. Let r and s be coprime positive integers. Then the expansion ( Z , + , v r ( x ) < v s ( x )) is dp-minimal. roof. We have v r ( x ) < v s ( x ) ⇐⇒ v rs ( x ) < v rs ( rx ) . Hence the relation v r ( x ) < v s ( x ) is definable in the dp-minimal structure ( Z , + , v rs ).It seems unlikely that v rs is definable from v r ( x ) ≤ v s ( x ).The induced structure on the index set ω ∪ {∞} seems to be important. If it is noto-minimal and X ⊆ ω ∪ {∞} is a definable infinite and co-infinite subset, then the set A = { a ∈ Z : w ( a ) ∈ X } ⊆ Z is definable. It is not clear if w is definable in ( Z , + , , , A ).If the induced structure on ω ∪ {∞} is o-minimal, then k = | B i /B i +1 | ∈ N ∪ {∞} mustbe constant for all sufficiently large i (in some elementary extension). If k is finite, thenthe size of the quotients B i /B i +1 is bounded and hence we are in the distal case. Conjecture 5.6. Let ( Z , + , , , v ) be distal. Then the following are equivalent:(a) ( Z , + , , , v ) is a minimal expansion of ( Z , + , , p such that | B i /B i +1 | = p for almost all i < ω ,(c) v is interdefinable with a p -adic valuation for some prime p ,(d) the ( Z , + , v )-induced structure on the value set of v ′ is o-minimal for all ( Z , + , v )-definable valuations v ′ . Proposition 5.7. If (a) implies (d), then Conjecture 5.6 holds.Proof. We already know (b) = ⇒ (c) = ⇒ (a) and by assumption (a) = ⇒ (d) holds.Hence (d) = ⇒ (b) remains to be shown.Let ( Z , + , , , v ) be distal and assume (d). Then there is k > | B i : B i +1 | = k for almost all i < ω . Therefore v and v k are interdefinable and we may assume v = v k .If k = st where s and t are coprime, then v is interdefinable with the valuation w suchthat | B wi /B wi +1 | alternates between s and t . Then the induced structure on the value setof w is not o-minimal. This contradicts (d).Hence we may assume k = p n for some prime p and n ≥ 1. If n > 1, then the p -adicvaluation v p is definable by v p ( x ) ≤ v p ( y ) ⇐⇒ n − ^ r =0 v p n ( p r x ) ≤ v p n ( p r y ) . Now the set v p ( { a ∈ Z : v p n ( pa ) > v p n ( a ) } ) contradicts (d).Hence k = p must be a prime. This shows (b).There are non-distal candidates for minimal expansions:35 uestion . Let ( p i )
For each prime p let v p denote the p -adic valuation on Z . Then thecorresponding limit theory exists, i.e. Th( Y p ( Z , + , v p ) / U ) does not depend on the choice of the non-principal ultrafilter U ⊆ P ( P ) .Proof. We fix the common L − -theory T = \ U Th L − ( Y p ( Z , + , v p ) / U )of these ultraproducts. Note that the predicate Div lq k ( i, j ) fails for all 0 < i < j andthe predicate Ind lk holds true for all 0 < i < j . Thus they are quantifier free 0-definableafter naming the successor function S on I . Therefore T has quantifier eliminationafter naming S by Theorem 4.8 (because ( ω, , ≤ , S ) has quantifier elimination). Theconstants in L Z generate a subgroup that is isomorphic to Z . An element a ∈ Z \ { } must have valuation 0 in all models of T . Therefore all models of T have isomorphicsubstructures and hence T is complete. If P ⊆ P is a non-empty set of primes, then Alouf and d’Elbée proved that the structure( Z , + , v p ) p ∈ P has dp-rank exactly | P | . We will generalize this result to expansions of( Z , +) by arbitrary valuations which involve disjoint sets of primes.Let V be a non-empty family of non-trivial valuations v : Z → ω ∪ {∞} . For each v ∈ V set π v = { p ∈ P : p divides | B vi /B vi +1 | for some i < ω } . We view ( Z , +) together with these valuations as a multi-sorted structure with groupsort Z and with a distinct value sort I v for each valuation v ∈ V . Now put • L Z = { , , + , −} , • L v = { v l : l ≥ } , and • L −I v = {−∞ , , + ∞ , ≤ , Div lq k , Ind lk } q,k,l for each valuation v ∈ V . Let L v mon be the monotone hull of L −I v as in Section 4.3 andset L mon = L Z ∪ ( [ v ∈ V L v ) ∪ ( [ v ∈ V L v mon )to be the disjoint union of these languages.36 roposition 5.10. Suppose the sets π v are pairwise disjoint. Then ( Z , + , v ) v ∈ V hasquantifier elimination in the language L mon .Proof. This is very similar to the proof of Theorem 4.8. Note that a multi-sorted versionof Lemma 2.21 holds true in this setting. As in the proof of Theorem 4.8 it suffices toshow the back-and-forth property for systems of linear congruences. Let S be the system n s x − b s ≡ l s B vi s , s ∈ S v ,n t x − b t l t B vi t , t ∈ T v , where S v = ∅ and T v are finite index sets for each v ∈ V for a finite subset V ⊆ V . Byan application of Lemma 4.19 we may assume that all the l s and l t have the same valuewhich we denote by l .Let a be a solution. We will show that we can assume that a and all constants b s and b t are contained in lB : If a is in lB , then all b s must be contained in lB sinceotherwise the congruences can not be satisfied. If b t is not contained in lB , then thecongruence n t x − b t lB vi t does not impose any restrictions on lB and we can ignore it without changing thesolutions in lB .If a is not contained in lB , then there is a constant c ∈ Z (and hence in the language)such that a − c ∈ lB . In that case the shifted system S c : n s ( x + c ) − b s ≡ lB vi s , s ∈ S v ,n t ( x + c ) − b t lB vi t , t ∈ T v , is solved by a − c ∈ lB and all the constants n s c − b s and n t c − b t can be assumed tolie in lB . Thus we can replace S by S c .Hence we may assume that S is a system of linear congruences in the subgroup lB .We have lB ≡ ˆ Z = Y p Z p and the valuations v l involve disjoint sets of primes. Therefore the system S can be solvedindependently for each valuation v ∈ V . This is done as in the proof of Theorem 4.8. Theorem 5.11. Suppose the sets π v are pairwise disjoint. Thendp-rk (( Z , + , v ) v ∈ V ) = | V | . Proof. ≥ is shown exactly as in the case of the p -adic valuations which was done byAlouf and d’Elbée (Theorem 1.2 of [2]).Now assume κ := dp-rk(( Z , + , v ) v ∈ V ) > | V | . As in the proof of Theorem 4.29 thisis witnessed by mutually indiscernible sequences ( I i ) i<κ (in the group sort) and a sin-gleton c in the group sort such that no sequence is indiscernible over c . As argued in37heorem 4.29, the fact that a sequence I is not indiscernible over c must be witnessedby an atomic L mon -formula which involves a valuation.Since κ > | V | , there must be two sequences I and I for which this witnessing formulainvolves the same valuation v . This is a contradiction because ( Z , + , v ) is dp-minimalby Theorem 4.29. The classification of NIP profinite groups by Macpherson and Tent in [19] yields informa-tion about uniformly definable families of finite index subgroups in arbitrary NIP groups(see Theorem 8.7 in [3]). We will do the same in the dp-minimal case. The argumentsare almost identical to those in Section 8 of [3] (see also Remark 5.5 in [19]), we onlyneed to make sure that the construction presented there preserves dp-minimality.Let H be a group and let ( N i : i ∈ I ) be a family of normal subgroups of finite indexsuch that ∀ i, j ∃ k : N k ≤ N i ∩ N j . We view H as an L prof -structure H = ( H, I ). Let f j : lim ←− H/N i → H/N j be theprojection maps. Then { ker f j : j ∈ I } is a neighborhood basis at the identity. Thereforewe may view lim ←− H/N i as an L prof -structure (lim ←− H/N i , I ). Lemma 6.1. Let H ∗ = ( H ∗ , I ∗ ) be an | I | + -saturated elementary extension of H . Then H ∗ / \ i ∈ I N ∗ i ∼ = lim ←− i ∈ I H/N i and ( N ∗ j / T i ∈ I N ∗ i : j ∈ I ) is a neighborhood basis for the identity consisting of opennormal subgroups.Proof. By elementarity we have | H ∗ : N ∗ i | = | H : N i | for all i ∈ I . Using this andelementarity it is easy to see thatlim ←− i ∈ I H ∗ /N ∗ i = lim ←− i ∈ I H/N i . Now write lim ←− i ∈ I H ∗ /N ∗ i = { ( g i N ∗ i ) i : ∀ i ≥ j : g i N ∗ j = g j N ∗ j } and let f : H ∗ → lim ←− i ∈ I H ∗ /N ∗ i , g ( gN ∗ i ) i be the natural homomorphism. Clearlyker f = T i ∈ I N ∗ i . It remains to show that f is surjective.Fix ( g i N i ) i ∈ lim ←− i ∈ I H ∗ /N ∗ i and consider the partial typeΣ( x ) = { x ∈ g i N ∗ i : i ∈ I } . I o ⊆ I finite, there is j ∈ I such that N j ≤ T i ∈ I N i . Then g i N j = g i ′ N j forall i, i ′ ∈ I . Hence Σ( x ) is finitely satisfiable and as H ∗ is | I | + -saturated, there exists g ∈ H ∗ such that g ∈ g i N ∗ i for all i ∈ I and hence f ( g ) = ( g i N ∗ i ).The family ( N ∗ j / T i ∈ I N ∗ i : j ∈ I ) is a neighborhood basis for the identity consistingof open normal subgroups. Lemma 6.2. If H = ( H, I ) has NIP, then (lim ←− i ∈ I H/N i , I ) has NIP. If moreover ( H, I ) is dp-minimal, then (lim ←− i ∈ I H/N i , I ) is dp-minimal.Proof. Since ( H, I ) has NIP, every uniformly definable family of subgroups containsonly finitely many subgroups of each finite index. Let ( H ∗ , I ∗ ) be an | I | + -saturatedelementary extension. Then I is externally definable (since I = { i ∈ I ∗ : | H ∗ : K ∗ i | < ∞} ). If ( H, I ) is dp-minimal, then ( H ∗ , I ∗ , I ) is dp-minimal by Remark 2.16. By theabove lemma the structure (lim ←− i ∈ I H/N i , I ) is interpretable as a quotient in ( H ∗ , I ∗ , I )and hence is NIP (resp. dp-minimal).Let G be a group and let ϕ ( x, y ) be a formula. Set N ϕ = { N i : i ∈ I } to be thefamily of all normal subgroups which are finite intersections of conjugates of ϕ -definablesubgroups of finite index. Note that every ϕ -definable subgroup of finite index containssome N ∈ N ϕ . The profinite group lim ←− i ∈ I G/N i naturally becomes an L prof -structure G ϕ = (lim ←− i ∈ I G/N i , I ). Proposition 6.3. Let G and ϕ be as above. If G is NIP, then G ϕ is NIP. If moreover G is dp-minimal, then G ϕ is dp-minimal in the group sort.Proof. By Baldwin-Saxl finite intersections of conjugates of ϕ -definable subgroups areuniformly definable by some formula ψ ( x, z ). The set J = { b : ψ ( G, b ) E G } is definable.Put J = { b : | G : ψ ( G, b ) | < ∞} . Then N ϕ = { ψ ( G, b ) : b ∈ J } . Since N ϕ is closedunder intersections, it follows that J is externally definable. Let E be the equivalencerelation defined by aEb ⇐⇒ ψ ( G, a ) = ψ ( G, b ). Now apply the previous lemma to thestructure ( G, J /E ).By Proposition 3.3 every dp-minimal profinite group ( G, I ) has an open abelian sub-group. Now Proposition 6.3 implies the following: Proposition 6.4. Let G be a dp-minimal group and let ϕ ( x, y ) be a formula. Let N ϕ be the family of all normal subgroups which are finite intersections of conjugates of ϕ -definable subgroups of finite index. If N ϕ is infinite, then there is N ∈ N ϕ such that forall M ∈ N ϕ the quotient N/ ( N ∩ M ) is abelian.Proof. The profinite group (lim ←− N ∈N ϕ G/N, N ϕ ) is dp-minimal and therefore is virtuallyabelian by Proposition 3.3. Since the quotients G/N are preserved, this implies theproposition. Remark . By Theorem 3.4 there are essentially two types of dp-minimal profinitegroups. This will also be seen in the abelian quotients in the statement of Proposition 6.4.39 emark . By Proposition 5.1 of [19] every profinite NIP group ( G, I ) has an open pro-solvable subgroup. Hence if we only assume NIP in the previous theorem, the quotientswill be solvable instead of abelian (see Theorem 8.7 of [3]). Jarden and Lubotzky [8] showed that two elementarily equivalent profinite groups areisomorphic if one of them is finitely generated. This was generalized to strongly completeprofinite groups by Helbig [7]. A profinite group is strongly complete if all subgroups offinite index are open. The tools used by Helbig and the construction in Section 6.1 givea proof for strong homogeneity.Let G be a profinite group and suppose ( N i : i ∈ I ) is a neighborhood basis at theidentity consisting of open normal subgroups. Let L P be the group language expandedby a family of unary predicates ( P i : i ∈ I ). We consider G as an L p structure bysetting P i ( G ) = N i . Note that if G ∗ is an elementary expansion, then there is a natural L P -structure on the quotient G ∗ / ( T i ∈ I P i ( G ∗ )). Lemma 6.7. Let G a profinite group equipped with an L P structure as above. Let G ∗ be an elementary extension of G in the language L P . Then the composition G → G ∗ → G ∗ / ( \ i ∈ I P i ( G ∗ )) is an L P -isomorphism.Proof. The lemma follows from the same arguments as Lemma 6.1. Proposition 6.8. Let G and H be profinite groups as L P structures such that thepredicates ( P i : i ∈ I ) encode neighborhood bases at the identity consisting of opennormal subgroups in both groups. Suppose A ⊆ G is a subset and f : A → H is anelementary map with respect to the language L P . Then f extends to an L P -isomorphismbetween G and H .Proof. Let G ∗ be a common strongly | A | + -homogeneous elementary extension of G and H . We can find ˜ f ∈ Aut( G ∗ ) such that ˜ f | A = f . Since ˜ f is an L P -automorphism, itinduces an automorphism of G ∗ / ( T i ∈ I P i ( G ∗ )). Now use Lemma 6.7 to get the desiredisomorphism between G and H .The following observation in Remark 3.12 in [7] is a consequence of Theorem 2 in [18]and Corollary 52.12 in [11]: Theorem 6.9. Let G be a profinite group. Then the following are equivalent:(a) G is strongly complete.(b) For each finite group A there exists a group word w such that w ( A ) = 1 and w ( G ) is open in G . finite width if h w ( G ) i = w ( G ) n for some n > ω . We willmake use of the following result: Proposition 6.10 (Proposition 5.2(b) of [23]) . Let G be a profinite group. If w is agroup word, then w ( G ) is closed in G if and only if w has finite width in G . Proposition 6.11. Let G and H be profinite groups. Let A ⊆ G be a subset of G andlet f : A → H be an elementary map. If one of the groups is strongly complete, then f extends to an isomorphism.Proof. By Theorem 6.9 and Proposition 6.10 strong completeness is a first-order propertyamong profinite groups. For each finite group A there is a group word w A such that w A ( A ) = 1, w A ( G ) is open in G , and w A ( H ) is open in H . Note that by Proposition 6.10and elementary equivalence of G and H , w A ( G ) and w A ( H ) are definable by the sameformula without parameters.If N is an open normal subgroup of G then w G/N ( G/N ) = 1 and hence w G/N ( G ) ⊆ N .Therefore the family ( w B ( G ) : B a finite group) is a neighborhood basis at the identity.Hence we may consider G and H as L P -structures where the predicates are given by P B ( G ) = w B ( G ). By Proposition 6.8 f extends to an isomorphism. groups By Theorem 1.1 of [19] a full profinite group is NIP if and only if it is NTP . Since thestructure of these groups is determined by the lattice of subgroups, this only dependson a single formula. We will show a version for formulas in NTP groups. We will usethe following lemma by Macpherson and Tent on groups in NTP : Lemma 6.12 (Lemma 4.3 in [19]) . Let G be an ∅ -definable group in a structure withNTP theory, and ϕ ( x, ¯ y ) a formula implying x ∈ G . Then there is k = k ϕ ∈ N suchthat the following holds. Suppose that H is a subgroup of G , π : H −→ Π i ∈ J T i is anepimorphism to the Cartesian product of the groups T i , and π j : H −→ T j is for each j ∈ J the composition of π with the canonical projection Π i ∈ J T i → T j . Suppose also thatfor each j ∈ J , there is a subgroup ¯ R j ≤ G and group R j < T j with ¯ R j ∩ H = π − j ( R j ) ,such that finite intersections of the groups ¯ R j are uniformly definable by instances of ϕ ( x, ¯ y ) . Then | J | ≤ k . Proposition 6.13. Let G be an NTP group and let ϕ ( x, y ) be a formula such that | x | = 1 . Suppose that the family { ϕ ( G, b ) : b ∈ G } consists of normal subgroups of G and is closed under finite intersections. Then ϕ ( x, y ) has NIP.Proof. We aim to show that ϕ satisfies the Baldwin-Saxl condition. Let N , . . . N n beinstances of ϕ and fix k ϕ as in Lemma 6.12. Now set C k = T { N i : 1 ≤ i ≤ n, i = k } and C = T { N i : 1 ≤ i ≤ n } . Note that C i ∩ N i = C and we have C i ⊆ N j if i = j . Now set H = h C i : 1 ≤ i ≤ n i . 41e then have H/C ∼ = n Y i =1 C i /C. Now set T i = C i /C and assume that all T i are non-trivial. Let π : H → H/C = n Y i =1 T i be the natural projection and let π j : H → T j be the projection on T j . Set R j = 1 < T j and put ¯ R j = N j . Then ¯ R j ∩ H = N j ∩ H = π − j ( R j ) . Hence Lemma 6.12 implies n ≤ k ϕ .If n > k ϕ , then there must be 1 ≤ i ≤ n such that T i = 1 and hence C = C i canbe written as in intersection of n − ϕ . Inductively this shows that anyintersection of instances of ϕ is an intersection of at most k ϕ instances. Hence ϕ satisfiesthe Baldwin-Saxl condition.This implies that ϕ has NIP: Otherwise we can find a constant a and an indiscerniblesequence ( b i ) i<ω such that | = ϕ ( a, b i ) ⇐⇒ i is odd.Set n = k ϕ and take i , . . . i n < ω , all of them odd. By the Baldwin-Saxl lemma we mayassume H i ∩ . . . H i n = H i ∩ . . . H i n . By indiscernibility this implies H i +1 ∩ . . . H i n = H i ∩ . . . H i n . But this is a contradiction since a H i +1 .While it is clearly sufficient to assume that the ϕ -definable subgroups normalize eachother, the above proof requires some normality assumption. Question . 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