aa r X i v : . [ m a t h . L O ] J a n NOTES ON TRACE EQUIVALENCE
ERIK WALSBERG
Abstract.
We introduce trace definability, a weak notion of interpretability,and trace equivalence, a weak notion of equivalence for first order structuresand theories. In particular we get an interesting weak equivalence notion forNIP theories. We describe a close connection to indiscernible collapse. Wealso show that if Q is a divisible subgroup of ( R ; +) and Q is a dp-rank oneexpansion of ( Q ; + , < ) then exactly one of the following holds: Th( Q ) tracedefines RCF or Q is trace equivalent to a reduct of an ordered vector space. Introduction
Throughout M and O are structures, L, L ′ are languages, and T, T ′ is a complete L, L ′ -theory, respectively. Let τ : O → M m be an injection. We say that M tracedefines O via τ if for every O -definable subset X of O n there is an M -definablesubset Y of M mn such that for all a , . . . , a n ∈ O we have( a , . . . , a n ) ∈ X ⇐⇒ ( τ ( a ) , . . . , τ ( a n )) ∈ Y. and M trace defines O if M trace defines O via some injection τ : O → M m .We say that T trace defines T ′ if some T ′ -model is trace definable in a T -model.We will see that if T trace defines T ′ then every T ′ -model is trace definable in a T -model. We say that T trace defines O if some T -model trace defines O . We saythat T and T ′ are trace equivalent if T trace defines T ′ and vice versa and saythat M and O are trace equivalent if Th( M ) and Th( O ) are trace equivalent.If M interprets O then M trace defines O . We view trace definability as a weaknotion of interpretability. Trace equivalence is an interesting notion of weak equiv-alence for NIP theories: many NIP-theoretic properties are preserved under tracedefinability and there are nice examples of trace equivalences between particularNIP structures. For example a NIP structure is trace equivalent to its Shelah ex-pansion. It follows in particular that if M is a NIP expansion of a linear order and C is any collection of convex subsets of M then M is trace equivalent to ( M , C ). Asa special case we get that RCF is trace equivalent to RCVF.In previous notes we gave examples of NIP structures M such that M does notinterpret an infinite field (or group) and the Shelah expansion M Sh does. In Sec-tions 12.1 and 13 we repurpose these examples to give NIP structures which tracedefine but do not interpret some algebraic structure. These structures are notpathologies. For example let B be the set of balls in Q p and B be the structureinduced on B by Q p . Then Th( B ) is trace equivalent to Q p but does not, provideda standard conjecture holds, interpret an infinite field. Date : January 29, 2021.
We characterize several model-theoretic dividing lines in terms of trace definability.For example if M is ℵ -saturated then M is unstable if and only if M trace defines( Q ; < ), M is IP if and only if M trace defines the Erd¨os-Rado graph, and M is k -independent if and only if M trace defines the generic countable k -hypergraph.There is a close connection to indiscernible collapse: if M is a monster model and I is the Fra¨ıss´e limit of a Fra¨ıss´e class in a finite relational language with the Ramseyproperty then M trace defines I if and only if M admits an uncollapsed indiscerniblepicture of I . Given such I we let C I be the class of theories T such that the monstermodel of T does not admit an uncollapsed indiscernible picture of I . Theorem 1.1.
Suppose that I , J are both Fra¨ıss´e limits of Fra¨ıss´e classes in finiterelational languages with the Ramsey property. Then C I = C J if and only if I and J are trace equivalent. Therefore a classification of relational Fra¨ıss´e limits up to trace equivalence wouldalso classify model-theoretic properties that can be defined in terms of indiscerniblecollapse. We take a few steps towards this classification below. For example weshow that any IP theory which admits quantifier elimination in a binary languageis trace equivalent to the Erd¨os-Rado graph.We now describe the original motivation for trace definability. We first recallFact 1.2, a special case of the Peterzil-Starchenko o-minimal trichotomy [35]. Welet R Vec be the ordered vector space ( R ; + , <, ( t λt ) λ ∈ R ), recall that R Vec admitsquantifier elimination. A subset of R m is semilinear if it is definable in R Vec . Fact 1.2.
Suppose that R is an o-minimal expansion of ( R ; + , < ) .Then the following are equivalent:(1) R does not define an isomorphic copy of ( R ; + , × ) ,(2) R does not define an infinite field,(3) R is locally modular,(4) R is a reduct of R Vec . Fact 1.2 asserts the equivalence of (a) absence of some algebraic structure, (b)an abstract model-theoretic “linearity” notion, and (c) semilinearity of definablesets. Versions of Fact 1.2 hold for other o-minimal structures. At present weare interested in a different generalization. It is a theorem of Simon [41] that anexpansion of ( R ; + , < ) is o-minimal if and only it has dp-rank one. It is thereforenatural to ask for an analogue of Fact 1.2 for dp-rank one expansions of ( Q ; + , < ).In Section 12.2 we give an example which shows that we need different notions ofdefinable algebraic structure and model-theoretic linearity. We prove Theorem 1.3in Section 12.1. Theorem 1.3.
Suppose that Q is a divisible subgroup of ( R ; +) and Q is a dp-rankone expansion of ( Q ; + , < ) . Then the following are equivalent:(1) Th( Q ) does not trace define RCF ,(2)
Th( Q ) does not trace define an infinite field,(3) Q has near linear Zarankiewicz bounds,(4) any Q -definable X ⊆ Q n is of the form Y ∩ Q n for a semilinear Y ⊆ R n .(5) Q is trace equivalent to a reduct of R Vec .If Q = Q then T h ( Q ) does not trace define RCF if and only if Q is a reduct of thestructure induced on Q by R Vec . See Section 4.2.2 for a definition of “near linear Zarankiewicz bounds”. A theorywith this property is NIP and cannot trace define an infinite field. Ordered vectorspaces were recently shown to have near linear Zarankiewicz bounds in [1]. Wedon’t know if this is actually a good model-theoretic linearity notion for generalNIP theories, but it’s a start. We show in Section 14 that Presburger arithmetichas near linear Zarankiewicz bounds. One can probably modify that proof to showthat other ordered abelian groups have near linear Zarankiewicz bounds, but thesame method will not generalize to arbitrary ordered abelian groups.Outside of NIP trace equivalence is coarser. Any unstable theory trace defines DLO,so trace definability does not preserve any property in the NSOP hierarchy. We saythat a theory is trace maximal if it trace defines any theory and M is trace maximalwhen Th( M ) is. We show below that an ℵ -saturated structure M is trace maximalif and only if there is an infinite subset A of M m such that for every X ⊆ A k thereis a definable Y ⊆ M mk such that X = Y ∩ A k . Thus it is reasonable to view tracemaximality as “ ∞ -independence”, this seems to be an interesting notion in its ownright. In Proposition 7.5 below we show that a pseudofinite field is trace maximal,so any structure is trace definable in a simple structure. More generally we showthat a non-separably closed PAC field is trace maximal. We do not known of aninfinite model-theoretically tame IP field that is not trace maximal. We also showthat infinite boolean algebras are trace maximal, so every theory is trace definablein an ℵ -categorical theory.We now summarize the contents of this paper. In Section 2 we recall some back-ground information and set some terminology (not all of which is standard). InSection 3 we prove some very general results about trace definability. These re-sults will be used to show that certain structures trace define other structures. InSection 4 we characterize stability and NIP in terms of trace definability and showthat a number of stability and NIP-theoretic properties of interest are preservedunder trace definability. In Section 5 we discuss the general connection betweentrace definability and indiscernible collapse and prove Theorem 1.1. In Section 6we characterize k -dependence in terms of trace definability and prove some thingsabout trace definability in the generic countable k -hypergraph. In Section 7 wediscuss trace maximality. The rest of the paper is devoted to some examples. InSection 8 we give a few basic examples. We show that any complete theory of or-dered groups trace defines DOAG and any complete theory of ordered fields tracedefines RCF. In Section 9 we show that an infinite field L is trace definable in C ifand only if there is an elementary embedding L → C . (The right to left implicationfollows on general grounds, see Proposition 3.1 below.) We also show that if R isa real closed field and L is an infinite field trace definable in R via an injection L → R then L is either real closed or algebraically closed of characteristic zero. InSection 10 we discuss Shelah completions and associated examples. For examplewe apply work of Hrushovksi and Pillay to show that if M is the monster model ofa NIP theory and G is a definably amenable definable group and G/G is a Liegroup then M trace defines G/G . In Section 11 we discuss the minimal unstabletrace equivalence class, DLO. We show for example that all homogenous k -orders(sometimes called finite dimensional permutation structures) are trace equivalentto ( Q ; < ). We also show that DLO does not trace define an infinite group. Asa corollary we show that an ℵ -categorical and ℵ -stable theory trace defines an ERIK WALSBERG infinite group if and only if it interprets an infinite group. In Section 12 we proveTheorem 1.3 and discuss some related examples. In Section 13 we discuss the p -adicstructure described above.1.1. Acknowledgements.
Five years ago or so John Goodrick told me that hewould like to have an analogue of the Peterzil-Starchenko trichotomy for dp-rankone ordered structures. Theorem 1.3 came out of this.2.
Conventions and background
Notation.
Throughout n, m, k are natural numbers. We use M , N , O , . . . todenote monster models and M , N , O , . . . to denote their domains. Given an L -structure M and L ∗ ⊆ L we let M ↾ L ∗ be the L ∗ -reduct of M . Given a structure M , a ∈ M n , and A ⊆ M we let tp M ( a | A ) be the type of a over A and let tp M ( a ) =tp M ( a |∅ ). If M and M ∗ are two structures with domain M then we say that M and M ∗ are interdefinable if M is a reduct of M ∗ and vice versa.Let M be a structure and A ⊆ M m . The structure induced on A by M is thestructure A with an n -ary predicate P X defining X ∩ A n for every M -definable X ⊆ M mn . Note that the induced structure admits quantifier elimination if andonly if every A -definable subset of each A n is of the form X ∩ A n for some M -definable X ⊆ M mn .Suppose that M expands a linear order ( M ; < ). Then M is weakly o-minimal ifevery definable subset of M is a finite union of convex sets and the theory of M isweakly o-minimal if the same holds in every elementary extension of M . It is easyto see that Th( M ) is weakly o-minimal if and only if for every formula ϕ ( x, y ) with | y | = 1 there is n such that { a ∈ M : M | = ϕ ( b, a ) } is a union of ≤ n convex setsfor all b ∈ M | x | . Fact 2.1 follows from this by o-minimal cell decomposition Fact 2.1.
Suppose that M is o-minimal, A ⊆ M , and the structure A induced on A by M admits quantifier elimination. Then Th( A ) is weakly o-minimal. Homogeneous structures. A homogeneous structure is a countable re-lational structure such that every finite partial automorphism extends to a totalautomorphism. A finitely homogeneous structure is a homogeneous structurein a finite language. Recall that a finitely homogeneous structure is ℵ -categoricaland admits quantifier elimination. We let Age( M ) be the age of an L -structure M for relational L , i.e. the class of finite L -structures that embed into M . Recall that M is homogeneous if and only if Age( M ) is a Fra¨ıss´e class with Fra¨ıss´e limit M .Suppose k ≥
2. We say that a structure or theory is k -ary if every formula isequivalent to a boolean combination of formulas of airity ≤ k and is finitely k -ary if there are formulas ϕ , . . . , ϕ n of airity ≤ k such that any n -ary formula φ ( x , . . . , x n ) is a boolean combination of formulas of the form ϕ j ( x i , . . . , x i ℓ )for 1 ≤ i , . . . , i ℓ ≤ n . A finitely k -ary theory is ℵ -categorical and a finitelyhomogeneous structure is k -ary for some k .A k -hypergraph ( V ; E ) is a set V equipped with a symmetric k -ary relation E suchthat E ( a , . . . , a k ) implies that the a i = a j when i = j . Finite k -hypergraphs forma Fra¨ıss´e class, we refer to the Fra¨ıss´e limit of this class as the generic countable k -hypergraph . The generic countable 2-hypergraph is the Erd¨os-Rado graph . A bipartite graph ( V, W ; E ) consists of sets V, W and E ⊆ V × W . Let ( V, W ; E )be a bipartite graph. The generic countable bipartite graph is the Fra¨ıss´e limitof the class of finite bipartite graphs.2.3. Shelah completeness.
The Shelah completion is usually referred to as the“Shelah expansion”. I use “completion” because it is more suggestive. Let M ≺ N be | M | + -saturated. A subset X of M n is externally definable if X = M n ∩ Y for some N -definable subset Y of N n . An application of saturation shows that thecollection of externally definable sets does not depend on choice of N . Fact 2.2 iswell-known and easy. Fact 2.2.
Suppose that X is an M -definable set and < is an M -definable linearorder on X . Then any < -convex subset of X is externally definable. Lemma 2.3 is a saturation exercise.
Lemma 2.3.
Suppose that λ is a cardinal, M is λ -saturated, X ⊆ M m is externallydefinable, and A ⊆ M m satisfies | A | < λ . Then there is a definable Y ⊆ M m suchthat X ∩ A = Y ∩ A . Lemma 2.4 is an easy generalization of Fact 2.2.
Lemma 2.4.
Suppose that ( X a : a ∈ M n ) is an M -definable family of subsets of M m . If A ⊆ M n is such that ( X a : a ∈ A ) is a chain under inclusion then S a ∈ A X a and T a ∈ A X a are both externally definable. The
Shelah completion M Sh of M is the expansion of M by all externally definablesets, equivalently the structure induced on M by N . Fact 2.5 is due to Shelah [40]. Fact 2.5.
Suppose M is NIP . Then the structure induced on M by N admitsquantifier elimination. Equivalently: every M Sh -definable set is externally definable. Dp-rank.
We will use a few basic facts about dp-rank. We first recall thedefinition of the dp-rank dp-rk M X of an M -definable set X ⊆ M m . It suffices todefine the dp-rank of a definable set in the monster model. Let X be a definable setand λ be a cardinal. An ( M , X, λ )-array consists of a sequence ( ϕ α ( x α , y ) : α < λ )of parameter free formulas and an array ( a α,i ∈ M | x α | : α < λ, i < ω ) such that forany function f : λ → ω there is a b ∈ X such that M | = ϕ α ( a α,i , b ) if and only if f ( α ) = i for all α, i. Then dp-rk M X ≥ λ if there is an ( M , X, λ )-array. We declare dp-rk M X = ∞ ifdp-rk M X ≥ λ for all cardinals λ . We let dp-rk M X = max { λ : dp-rk M X ≥ λ } ifthis maximum exists and otherwisedp-rk M X = sup { λ : dp-rk M X ≥ λ } − . Of course this raises the question of what exactly λ − λ . But this doesn’t matter. We also defines dp-rk M = dp-rk M M .The first three claims of Fact 2.6 are immediate consequences of the definition ofdp-rank. The fourth is due to Kaplan, Onshuus, and Usvyatsov [26]. Fact 2.6.
Suppose
X, Y are M -definable sets. Then(1) dp-rk M < ∞ if and only if M is NIP ,(2) dp-rk M X = 0 if and only if X is finite,(3) If f : X → Y is a definable surjection then dp-rk M Y ≤ dp-rk M X , ERIK WALSBERG (4) dp-rk M X × Y ≤ dp-rk M X + dp-rk M Y . Fact 2.7 is proven in [43].
Fact 2.7.
Suppose that Q is a divisible subgroup of ( R , +) and Q expands ( Q, + , < ) .Then the following are equivalent:(1) Q has dp-rank one,(2) Q is weakly o-minimal,(3) Th( Q ) is weakly o-minimal.(4) there is an o-minimal expansion Q (cid:3) of ( R ; + , < ) such that the structureinduced on Q by Q (cid:3) eliminates quantifiers and is interdefinable with Q Sh . The implication (4) ⇒ (3) is not stated in [43] but follows from Fact 2.1.2.5. O-minimal interpretatons.
In this section we give some restrictions on or-dered structures interpretable in an o-minimal expansion of an ordered group to beuse below. I don’t know if this is original. Fact 2.8 is due to Ramakrishnan [39].
Fact 2.8.
Suppose that M is an o-minimal expansion of an ordered abelian groupand ( L ; ⊳ ) is a definable linear order. Then there is a definable embedding ( L ; ⊳ ) → ( M k ; < Lex ) , where < Lex is the lexicographic order on M k , for some k ≥ . Corollary 2.9.
Suppose that M is an o-minimal expansion of an ordered abeliangroup and ( L ; ⊳ ) | = DLO is interpretable in M . Then there are non-empty openintervals I ⊆ M and J ⊆ L and a definable isomorphism ι : ( J ; < ) → ( I ; ⊳ ) .Proof. We may suppose that ( L ; ⊳ ) is definable in M as M eliminates imaginaries.By Fact 2.8 we may suppose that L ⊆ M k and that ⊳ is the restriction of thelexicographic order to L . We apply induction on k . Suppose that k = 1. As L isinfinite L contains a nonempty open interval I ⊆ M , so we let ι be the identity I → I . Suppose k ≥
2. Let π : L → M k − be given by π ( x , . . . , x k ) = π ( x , . . . , x k − ).Note that each π − ( b ) is a < Lex -convex subset of L and that the restriction of < Lex to any π − ( b ) agrees with the usual ordering on the k th coordinate. Suppose that b ∈ M k − and | π − ( b ) | ≥
2. Then π − ( b ) is infinite as ( L ; ⊳ ) is dense, so thereis a nonempty open interval I ⊆ M such that { b } × I ⊆ π − ( b ). In this case wetake ι : I → { b } × I be ι ( x ) = ( b, x ). So we may suppose that π is injective. Notethat π is a monotone map ( L ; < Lex ) → ( M k − ; < Lex ), so π induces an isomorphism( L ; < Lex ) → ( π ( L ); < Lex ). Apply induction on k . (cid:3) Corollary 2.10 follows easily from Corollary 2.9.
Corollary 2.10.
Suppose that O is an expansion of a dense linear order ( O ; < ) , I is a nonempty open interval, and X ⊆ I is definable and dense and co-dense in I .Then O is not interpretable in an o-minimal expansion of an ordered group. Corollary 2.11.
Suppose that Q is a divisible subgroup of ( R ; +) and Q is a weaklyo-minimal expansion of ( Q ; + , < ) . If Q is not o-minimal then Q is not interpretablein an o-minimal expansion of an ordered group. Corollary 2.11 does not extend to weakly o-minimal expansions of non-archimedeanordered abelian groups. Let < Lex be the lexicographic order on ( R ; +). Then( R ; + , < Lex ) is a divisible ordered abelian group and is hence o-minimal. Let C = { (0 , . . . , , t ) : t ∈ R } and note that C is a convex subgroup of ( R ; + , < Lex ).Facts 2.2 and 2.5 together show that ( R ; + , < Lex , C ) is weakly o-minimal and( R ; + , < Lex , C ) is clearly interpretable in ( R ; + , < ). Proof.
Suppose that Q is not o-minimal. Then there is definable downwards closedsubset C of Q which does not have a supremum in Q . Note that for any non-emptyopen interval I ⊆ Q there is a ∈ Q such that I ∩ ( a + C ) and I \ ( a + C ) are bothnonempty. So for any nonempty open interval I there is a definable proper subset X of I such that X is downwards closed in I and does not have a supremum in I .Apply Corollary 2.9. (cid:3) Basic properties
In this section τ will denote an injection O → M m . We will also use τ to denote thefunction O n → M mn given by ( a , . . . , a n ) ( τ ( a ) , . . . , τ ( a n )). Proposition 3.1 isimmediate from the definitions. Proposition 3.1.
Let M , O and P be structures.(1) If M trace defines O and O trace defines P then M trace defines P .(2) If T trace defines T ′ and T ′ trace defines T ′′ then T trace defines T ′′ .(3) If O ≺ M then O is trace definable in M . Thus trace definability is a quasi-order on theories and trace equivalence is anequivalence relation.
Proposition 3.2. If M interprets O then M trace defines O . In the proof below π will denote a certain map X → O and we will also use π todenote the map X n → O n given by π ( a , . . . , a n ) = ( π ( a ) , . . . , π ( a n )). Proof.
Suppose O is interpretable in M . Let X ⊆ M m be M -definable, E be an M -definable equivalence relation on X , and π : X → O be a surjection such that(1) for all a, b ∈ E we have E ( a, b ) ⇐⇒ π ( a ) = π ( b ), and(2) if X ⊆ O n is O -definable then π − ( X ) is M -definable.Let τ : O → X be a section of π and A = τ ( O ). If Y ⊆ O n is O -definable then π − ( X ) is M -definable and X is the set of a ∈ O n such that τ ( a ) ∈ π − ( Y ). So M trace defines O via τ . (cid:3) Proposition 3.3.
Suppose that O is a substructure of M which admits quantifierelimination. Then M trace defines O via the identity O → M .Proof. Let L be the language of M . Let X ⊆ O n be O -definable and ϕ ( x ) be an L ( O )-formula which defines X . Let Y = { a ∈ M n : M | = ϕ ( a ) } . We may supposethat ϕ is quantifier free. This implies that Y ∩ O n = X . (cid:3) Proposition 3.4.
Suppose that L ′ is relational, O is an L ′ -structure with quantifierelimination, and τ : O → M m is an injection. Suppose that for every k -ary relationsymbol R ∈ L ′ there is an M -definable Y ⊆ M mk such that for all a ∈ O k we have O | = R ( a ) ⇐⇒ τ ( a ) ∈ Y. Then M trace defines O via τ .Proof. We suppose that O ⊆ M m and τ is the identity O → M m . For each k -ary R ∈ L ′ fix an M -definable subset Y R of M mk with Y R ∩ O k = { a ∈ O k : O | = R ( a ) } .Let P be the L ′ -structure with domain M m such that for every k -ary R ∈ L ′ and a ∈ M mk we have P | = R ( a ) ⇐⇒ a ∈ Y R . Then P is definable in M and byProposition 3.3 O is trace definable in P . (cid:3) ERIK WALSBERG
Corollary 3.5 is a reformulation of Proposition 3.4.
Corollary 3.5.
Suppose that L ′ is relational and O is an L ′ -structure with quan-tifier elimination. Then the following are equivalent:(1) M trace defines O , and(2) O is isomorphic to an L ′ -structure P such that P ⊆ M m and for every k -ary R ∈ L there is an M -definable Y ⊆ M mk such that for any a ∈ P k we have P | = R ( a ) ⇐⇒ a ∈ Y . Proposition 3.6.
Suppose τ : O → M m is an injection and for every subset X of O n which is definable in O without parameters there is M -definable Y ⊆ M mn suchthat for all a ∈ O n we have a ∈ X ⇐⇒ τ ( a ) ∈ Y . Then M trace defines O via τ .Proof. Let L ∗ be the language with a k -ary predicate R X for each parameter-freedefinable X ⊆ O n and let O ∗ be the L ∗ -structure with domain O where for each R ∈ L ∗ we have O ∗ | = R X ( a ) ⇐⇒ a ∈ X . Note that O ∗ has quantifier elimination.Apply Proposition 3.4 to M and O ∗ . (cid:3) Proposition 3.7.
Suppose that O is ℵ -categorical and τ : O → M m is an injec-tion. Then M trace defines O via τ if and only if for any p ∈ S k ( O ) there is an M -definable X ⊆ M mk such that if a ∈ O n then tp O ( a ) = p ⇐⇒ τ ( a ) ∈ X .Proof. By Ryll-Nardzewski a subset of O k is definable without parameters if andonly if it is a finite union of sets of the form { a ∈ O n : tp O ( a ) = p } for p ∈ S k ( O ).Apply Proposition 3.6. (cid:3) Recall our standing assumption that
T, T ′ is a complete L, L ′ -theory, respectively. Proposition 3.8.
Suppose that T trace defines T ′ . Let λ ≥ | L ′ | be an infinitecardinal, O | = T ′ , | O | < λ , and M | = T be λ -saturated. Then M trace defines O . So if T trace defines T ′ then every T ′ -model is trace definable in a T -model. Proof.
Suppose that P | = T ′ is trace definable in N | = T . We may suppose that P ⊆ N m and N trace defines P via the identity P → N m . By Morleyization we maysuppose that L ′ is relational and T ′ admits quantifier elimination. For each n -aryrelation symbol R ∈ L ′ we fix an L ( N )-formula δ R ( x ) such that for all a ∈ P n wehave N | = δ R ( a ) ⇐⇒ P | = R ( a ). Let A be the set of coefficients of the δ R for R ∈ L .Assume that L and L ′ are disjoint. Let L ∪ be L ( A ) ∪ L ′ ∪ { P } where P is a new m -ary predicate. Given an L ∪ -structure S we define the associated L ′ -structure S L ′ by letting S L ′ have domain { a ∈ S m : S | = P ( a ) } , and letting the interpretation ofeach n -ary R ∈ L ′ be { ( b , . . . , b n ) ∈ S mn : S | = P ( b ) ∧ . . . ∧ P ( b n ) ∧ δ R ( b , . . . , b n ) } . Let T ∪ be the L ∪ -theory where an L ∪ -structure S satisfies T ∪ if and only if:(1) S ↾ L ( A ) satisfies the L ( A )-theory of N , and(2) S L ′ | = T ′ .It is clear that T ∪ is consistent. Proposition 3.4 shows that if S | = T ∪ then S ↾ L trace defines S L ′ . Let S | = T ∪ be λ -saturated. So S L is also λ -saturated, hencethere is an elementary embedding O → S L . So we suppose that O is an elementarysubstructure of S L . Let T be an elementary substructure of S which contains O ∪ A and satisfies | T | ≤ λ . So T ↾ L trace defines T L , and hence trace defines O . Bysaturation there is an elementary embedding T L → M , so we may suppose that T L is an elementary submodel of M . Then M trace defines O . (cid:3) Corollary 3.9 follows from Proposition 3.8 and ℵ -saturation of ultrapowers. Corollary 3.9.
Suppose that M | = T and O | = T ′ are countable. Then the followingare equivalent:(1) T trace defines T ′ ,(2) every nonprinciple ultrapower of M trace defines O ,(3) some nonprinciple ultrapower of M trace defines O . Stability and
NIP4.1.
Stability.Proposition 4.1. If T is stable and T ′ is trace definable in T then T ′ is stable.Furthermore T is stable if and only if T trace defines DLO . The first claim of Proposition 4.1 is clear. The second claim follows from Lemma 4.2below.
Lemma 4.2.
Suppose that M is ℵ -saturated. Then M is unstable if and only if M trace defines ( Q ; < ) .Proof. The right to left implication follows from the first claim of Proposition 4.1.Suppose that M unstable. So there is a sequence ( a q : q ∈ Q ) of elements of some M m and a formula φ ( x, y ) such that for all p, q ∈ Q we have M | = φ ( a p , a q ) if andonly if p < q . Let τ : Q → M m be given by declaring τ ( q ) = a q for all q ∈ Q . As( Q ; < ) admits quantifier elimination an application of Proposition 3.4 shows that M trace defines ( Q ; < ) via τ . (cid:3) Given a structure M , A ⊆ M , and a cardinal λ , we let S n ( M , A ) be the set of n -types over A and S n ( M , λ ) be the supremum of {| S n ( M , A ) | : A ⊆ M, | A | ≤ λ } .Given a theory T we let S ( T, λ ) be the supremum of { S n ( M , λ ) : n ∈ N , M | = T } . Proposition 4.3.
Suppose that λ ≥ | L ′ | is an infinite cardinal and T ′ is tracedefinable in T . Then S ( T ′ , λ ) ≤ S ( T, λ ) . So if T is λ -stable then T ′ is λ -stable. Before proving Proposition 4.3 we give an immediate corollary.
Corollary 4.4.
Suppose that T ′ is trace definable in T . If T is superstable then T ′ is superstable. If L ′ is countable and T is ℵ -stable then T ′ is ℵ -stable. We now prove Proposition 4.3.
Proof.
By Proposition 3.8 it is enough to suppose that O is λ + -saturated, N is λ + -saturated, and N trace defines O via an injection O → N m , and show that S n ( O , λ ) ≤ S nm ( N , λ ) for any n . Fix n and A ⊆ O such that | A | ≤ λ . Let B bethe collection of all subsets of O n that are O -definable with parameters from A . As | L ′ | , | A | ≤ λ we have |B| ≤ λ . For each X ∈ B we let Y X be an N -definable subsetof N nm such that Y X ∩ O n = X , and let C = { Y X : X ∈ B} . So |C| ≤ λ . Fix B ⊆ N such that | B | ≤ λ and every Y X is definable with parameters from B .It suffices to show that | S n ( O , A ) | ≤ | S nm ( N , B ) | . We produce an injection ι : S n ( O , A ) → S nm ( N , B ). Applying λ + -saturation we fix for each p ∈ S n ( O , A ) an a p ∈ O n such that tp O ( a p | A ) = p . We declare ι ( p ) = tp N ( a p | B ). We show that ι isinjective. Fix distinct p, q ∈ S n ( O , A ). So there is X ∈ B such that p concentrateson X and q does not. So a p ∈ X and a q / ∈ X . Then a p ∈ Y X and a q / ∈ Y X , sotp N ( a p | B ) = tp N ( a q | B ). (cid:3) Corollary 4.5.
Suppose that T is totally transendental and T ′ is trace definablein T . Then T ′ is totally transendental.Proof. We apply the fact that an L ′ -structure O is totally transcendental if andonly if O ↾ L ∗ is ℵ -stable for every countable L ∗ ⊆ L ′ . Suppose that M | = T trace defines O | = T ′ . We may suppose that O ⊆ M m and M trace defines O viathe identity O → M m . Fix countable L ∗ ⊆ L ′ . For each n -ary parameter free L ∗ -formula φ ( x ) fix an mn -ary L ( M )-formula ϕ φ ( x ) such that for every a ∈ O n wehave O | = φ ( a ) ⇐⇒ M | = ϕ φ ( a ). Let L ∗∗ be a countable sublanguage of L such thateach ϕ φ is an L ∗∗ ( M )-formula. Proposition 3.6 shows that M ↾ L ∗∗ trace defines O ↾ L ∗ . By Corollary 4.4 O ↾ L ∗ is ℵ -stable. (cid:3) . Proposition 4.6 is easy and left to the reader.
Proposition 4.6. If T is NIP and T ′ is trace definable in T then T ′ is NIP . Proposition 4.7.
Suppose that M is ℵ -saturated. The following are equivalent:(1) M is IP ,(2) M trace defines the Erd¨os-Rado graph,(3) M trace defines the generic countable bipartite graph.So the following are equivalent for any theory T :(4) T is IP ,(5) T trace defines the Erd¨os-Rado graph,(6) T trace defines the generic countable bipartite graph.Proof. By Proposition 3.8 it is enough to show that (1) − (3) are equivalent. Propo-sition 4.6 shows that (2) and (3) both imply (1). We show that (1) implies both(2) and (3). Suppose that M is IP. We first show that M trace defines thegeneric countable bipartite graph. Fix a formula ϕ ( x, y ), a sequence ( a i : i ∈ N )of elements of M | x | , and a family ( b I : I ⊆ N ) of elements of M | y | such that M | = ϕ ( a i , b I ) ⇐⇒ i ∈ I for any I ⊆ N and i ∈ N . Let V = { a i : i ∈ N } , W = { b I : I ⊆ N } , and E = { ( a, b ) ∈ V × W : M | = ϕ ( a, b ) } . Any countablebipartite graph embeds into ( V, W ; E ), so in particular the generic countable bipar-tite graph embeds into ( V, W ; E ). Apply Proposition 3.4 and quantifier eliminationfor the generic countable bipartite graph. We now show that M trace defines theErd¨os-Rado graph. Laskowski and Shelah [32, Lemma 2.2] show that the Erd¨os-Rado graph embeds into a definable graph. Apply Proposition 3.4. (cid:3) Strong dependence and dp-finiteness are also preserved.
Theorem 4.8.
Suppose that T ′ is trace definable in T . If T has finite dp-rankthen T ′ has finite dp-rank, and if T has infinite dp-rank then dp-rk T ′ ≤ dp-rk T .In particular if T is strongly dependent then T ′ is strongly dependent. Note that the last claim follows from the previous as T is strongly dependent if andonly if dp-rk T < ℵ . Proof.
By Proposition 3.8 we may suppose that M | = T and O | = T ′ are both ℵ -saturated, O ⊆ M m , and M trace defines O via the identity O → M m . Supposethat λ is a cardinal and dp-rk T ′ ≥ λ . Hence dp-rk O O ≥ λ . Fix an ( O , O, λ )-array consisting of parameter free L ′ -formulas ( ϕ α ( x α , y ) : α < λ ) and tuples( a α,i ∈ O | x α | : α < λ, i < ω ) . For each α < λ we fix an L ( M )-formula θ α ( z α , w ), | z α | = m | x α | , | w | = m , such that for any a ∈ O | x α | , b ∈ O we have O | = ϕ α ( a, b ) ⇐⇒ M | = θ α ( a, b ). It is now easy to see that ( θ α ( z α , w ) : α < λ ) and ( a α,i : α < λ, i <ω ) forms an ( M , M m , λ ) array. Thus dp-rk M M m ≥ λ . We have shown thatdp-rk M M m ≥ dp-rk O .Suppose that dp-rk M is finite. By Fact 2.6 dp-rk M M m ≤ m dp-rk M M , so O hasfinite dp-rank. Suppose that dp-rk M is infinite. Again by Fact 2.6 dp-rk M M n =dp-rk M M , so O has dp-rank ≤ dp-rk M . (cid:3) In Section 5.1 we show that finiteness of op-rank is also preserved.4.2.1.
Strong Erd¨os-Hajnal.
Distality is not preserved under trace definability asit is not preserved under interpretations. We discuss a combinatorial consequenceof distality which is preserved. A bipartite graph (
V, W ; E ) satisfies the StrongErd¨os-Hajnal property if there is a real number δ > A ⊆ V, B ⊆ W there are A ′ ⊆ A , B ′ ⊆ B such that | A ′ | ≥ δ | A | , | B ′ | ≥ δB ,and A ′ × B ′ is either contained in or disjoint from E . A structure O satisfiesthe strong Erd¨os-Hajnal property if every O -definable bipartite graph has thestrong Erd¨os-Hajnal property and a theory has the strong Erd¨os-Hajnal propertywhen its models do.There are countable bipartite graphs that do not have the strong Erd¨os-Hajnalproperty, so the generic countable bipartite graph does not have the strong Erd¨os-Hajnal property. Thus by Proposition 4.7 a theory with the strong Erd¨os-Hajnalproperty is NIP. Chernikov and Starchenko show that distal structures have thestrong Erd¨os-Hajnal property [6, Theorem 1.9]. Proposition 4.9 is clear from thedefinitions. Proposition 4.9. If T has the strong Erd¨os-Hajnal property than any theory tracedefinable in T has the strong Erd¨os-Hajnal property. Fact 4.10 is due to Chernikov and Starchenko [6, Section 6].
Fact 4.10.
An infinite field of positive characteristic does not have the Erd¨os-Hajnal property.
Proposition 4.11 follows from Fact 4.10 and Proposition 4.9.
Proposition 4.11.
A distal structure cannot trace define an infinite field of positivecharacteristic.
Near linear Zarankiewicz bounds.
Let (
V, W ; E ) be a bipartite graph. We saythat ( V, W ; E ) is not K kk -free if there are V ′ ⊆ V , W ′ ⊆ W with | V ′ | = k = | W ′ | and V ′ × W ′ ⊆ E . We say that ( V, W ; E ) has near linear Zarankiewicz bounds if for any k and ε ∈ R > there is c ∈ R > such that if V ′ ⊆ V , W ′ ⊆ W are finiteand ( V ′ , W ′ ; E ∩ [ V ′ × W ′ ]) is K kk -free then | E ∩ ( V ′ × W ′ ) | ≤ c | V ′ + W ′ | ε . We saythat M has near linear Zarankiewicz bounds if every M -definable bipartite graphhas near linear Zarankiewicz bounds and T has near linear Zarankiewicz bounds ifevery (equivalently: some) M | = T has near linear Zarankiewicz bounds. The useof “Zarankiewicz” arises from the connection to Zarankiewicz’s problem, see [1].Lemma 4.12 is clear. Lemma 4.12.
Suppose that T has near linear Zarankiewicz bounds and T ′ is tracedefinable in T . Then T ′ has near linear Zarankiewicz bounds. The first claim of Fact 4.13 shows that this notion is non-trivial.
Fact 4.13.
The generic countable bipartite graph does not have near linear Zarankiewiczbounds. A theory with near linear Zarankiewicz bounds is
NIP . The first claim follows from a probabilistic construction given in [11] as every finitebipartite graph is a substructure of the generic countable bipartite graph. Thesecond claim follows from the first claim and Proposition 4.7.The real ordered vector space R Vec is shown to have near linear Zarankiewicz boundsin [1, Theorem C]. Fact 4.14 is not explicitly stated in [1], but follows directly fromthe proof of Theorem C in that paper and quantifier elimination for ordered vectorspaces.
Fact 4.14.
An ordered vector space has near linear Zarankiewicz bounds.
We now prove Proposition 4.15.
Proposition 4.15.
An infinite field does not have near linear Zarankiewicz bounds.
Proposition 4.15 follows from Lemma 4.16 and some standard incidence geometry.
Lemma 4.16.
Suppose that K is an infinite NIP field of characteristic p > .Then K trace defines the algebraic closure of the field with p elements.Proof. Let F alg p be the algebraic closure of the field with p elements. As K isNIP it follows from Kaplan, Scanlon, and Wagner [27, Corollary 4.5] that thereis a field embedding τ : F alg p → K . Proposition 3.3 and quantifier elimination foralgebraically closed fields shows that K trace defines F alg p via τ . (cid:3) We now prove Proposition 4.15.
Proof.
We let K be an infinite field. Let V = K , W be the set of lines in K , and E be the set of ( p, ℓ ) ∈ V × W such that p is on ℓ . Note that ( V, W ; E ) is K -free.We first treat the case when K is of characteristic zero. We just need to recall theusual witness for sharpness in the lower bounds of Szemeredi-Trotter. Fix n ≥ V ′ = { , . . . , n } × { , . . . , n } and W ′ be the set of lines with slope in { , . . . , n } and y -intercept in { , . . . , n } . Then | V ′ + W ′ | = 3 n and | E ∩ ( V ′ × W ′ ) | = n .So ( V, W ; E ) does not have near linear Zarankiewicz bounds.Now suppose that K has characteristic p >
0. By Fact 4.13 we may suppose that K is NIP. By Lemma 4.16 and Lemma 4.12 we may suppose that K is the algebraicclosure of the field with p elements. Fix n ≥ L be the subfield of K with q = p n elements. Let V ′ = L and W ′ be the set of non-vertical lines betweenelements of L . Let | V ′ | = q = | W ′ | and as every ℓ ∈ W ′ contains q points in L we have | E ∩ ( V ′ × W ′ ) | = q . So ( V, W ; E ) does not have near linear Zarankiewiczbounds. (cid:3) Theorem 4.17 follows from what is above.
Theorem 4.17.
An ordered vector space cannot trace define an infinite field. Indiscernible collapse
Stability and NIP can both be defined in terms of indiscernible collapse: M isunstable if and only if M admits an uncollapsed indiscernible picture of ( Q ; < )and M has IP if and only if M admits an uncollapsed indiscernible picture of thegeneric countable ordered graph. We explore the more general connection betweentrace definability and indiscernible collapse.We first recall some background definitions on indiscernible collapse. Some of ourconventions are a bit different then other authors, so some care is warranted. Wesuppose that M is a monster model of an L -theory T and A is a small set of param-eters from M . Let L indis be a finite relational language and I be a homogeneous L indis -structure such that Age( I ) has the Ramsey property. A picture of I in M is an injection γ : I → M n for some n . Given an injection γ : I → M n and a = ( a , . . . , a k ) ∈ I k we let γ ( a ) = ( γ ( a ) , . . . , γ ( a k )). A picture γ of I in M is A -indiscernible iftp I ( a ) = tp I ( b ) = ⇒ tp M ( γ ( a ) | A ) = tp M ( γ ( b ) | A )for any k and a, b ∈ I k . An A -indiscernible picture γ of I in M is uncollapsed iftp I ( a ) = tp I ( b ) ⇐⇒ tp M ( γ ( a ) | A ) = tp M ( γ ( b ) | A )for any k ∈ N and a, b ∈ I k . Recall that I has quantifier elimination so we coulduse qftp I ( a ) in place of tp I ( a ). Let γ, γ ′ be pictures of I in M and suppose that γ ′ is indiscernible. Then γ ′ is based on γ if for any finite L ⊆ L and a ∈ I k there is b ∈ I k such that qftp I ( a ) = qftp I ( b ) and tp M ( γ ′ ( a ) | A ) ↾ L = tp M ( γ ( b ) | A ) ↾ L . Itis shown in [15, Theorem 2.13] that if γ is a picture of of I in M then there is an A -indiscernible picture γ ′ of I in M such that γ ′ is based on γ . (This requires theRamsey property for Age( I ).) Proposition 5.1.
Suppose as above that I is a finitely homogeneous structure suchthat Age( I ) has the Ramsey property. Then the following are equivalent:(1) M trace defines I via an injection I → M m ,(2) there is a small set A of parameters and a non-collapsed A -indiscerniablepicture I → M m of I in M .Therefore the following are equivalent:(4) M trace defines I ,(5) any ℵ -saturated elementary substructure of M trace defines I ,(6) there is a small set A of parameters and a non-collapsed A -indiscerniblepicture of I in M . As I is ℵ -categorical we can and will apply Proposition 3.4. Proof.
Note that the equivalence of (4) − (6) follows from the equivalence of (1) − (2)and Proposition 3.8. Suppose that M trace defines I via τ : I → M m . So for every k ∈ N and p ∈ S k ( I ) there is an M -definable Y p ⊆ M mk such that or any a ∈ I k we have qftp I ( a ) = p ⇐⇒ τ ( a ) ∈ Y p . We reduce to the case when the Y p arepairwise disjoint. Note that the sets Y p ∩ τ ( I ) k are pairwise disjoint. For each p ∈ S k ( I ) we let Y ′ p = Y p \ S q ∈ S k ( I ) \{ p } Y q . Then the Y ′ p are pairwise disjoint and Y ′ p ∩ τ ( I ) k = Y p ∩ τ ( I ) k for all p ∈ S k ( I ). So after replacing each Y p with Y ′ p wemay suppose that the Y p are pairwise disjoint. Let A be a small set of parameters such that each Y p is A -definable and γ bean A -indiscernible picture of I in M which is based off of τ . We show that γ isuncollapsed. Suppose that a , a ∈ I k and p := qftp I ( a ) = qftp I ( a ) =: p . Let L ⊆ L be finite such that Y p is L ( A )-definable for all p ∈ S k ( I ). As γ is based on τ there are b , b ∈ I k such that for each i ∈ { , } we have(1) qftp I ( b i ) = p i , and(2) tp M ( γ ( b i ) | A ) ↾ L = tp M ( τ ( a i ) | A ) ↾ L .We have τ ( a ) ∈ Y p and τ ( a ) ∈ Y p . Therefore γ ( b ) ∈ Y p and γ ( b ) ∈ Y p . As Y p ∩ Y p = ∅ we have tp M ( γ ( b ) | A ) = tp M ( γ ( b ) | A ).Now suppose that γ : I → M m is a non-collapsed A -indiscernible picture of I in M .We show that M trace defines I via γ . We fix k and let S k ( I ) = { p , . . . , p n } wherethe p i are distinct. We produce A -definable Y , . . . , Y n such that for any a ∈ I k we have qftp I ( γ ( a )) = p i if and only if τ ( a ) ∈ Y i . Fix a , . . . , a n ∈ I k such thatqftp I ( a i ) = p i for each i . As γ is non-collapsed tp M ( γ ( a i ) | A ) = tp M ( γ ( a j ) | A ) when i = j . So for each i, j there is an A -definable Y ij ⊆ M mk such that γ ( a i ) ∈ Y ij and γ ( a j ) / ∈ Y ij . Let Y i = T nj =1 Y ij . So we have γ ( a i ) ∈ Y i and γ ( a j ) / ∈ Y i for all i = j .Fix b ∈ I k . We show that tp I ( b ) = p i ⇐⇒ γ ( b ) ∈ Y i . Suppose that tp I ( b ) = p i . As γ is indiscernible we have tp M ( γ ( b ) | A ) = tp M ( γ ( a i ) | A ), so b ∈ Y i . Now supposethat γ ( b ) ∈ Y i . Then γ ( b ) / ∈ Y j when j = i , so tp I ( b ) = p j for all j = i . Hencetp I ( b ) = p i . (cid:3) We continue to suppose that I is finitely homogeneous and Age( I ) has the Ram-sey property. We let C I be the class of complete first order theories T such thatthe monster model of T does not admit an uncollapsed indiscernible picture of I .Corollary 5.2 is immediate from Proposition 5.1. Corollary 5.2.
Let I be a finitely homogeneous structure such that Age( I ) has theRamsey property. If T ∈ C I and T trace defines T ′ then T ′ ∈ C I . Proposition 5.3.
Suppose that I and J are finitely homogeneous structures suchthat Age( I ) and Age( J ) have the Ramsey property. Then C J ⊆ C I if and only if Th( I ) trace defines Th( J ) . So C I = C J if and only if I and J are trace equivalent.Proof. It is enough to prove the first claim. Suppose that Th( I ) trace definesTh( J ). By Proposition 3.8 any ℵ -saturated elementary extension of I trace defines J . Suppose that T / ∈ C I and M | = T . By Proposition 5.1 M admits an uncollapsedindiscernible picture of I , so M trace defines I , so by Proposition 3.8 M trace definesan ℵ -saturated elementary extension of I , so M trace defines J , so M admits anuncollapsed indiscernible picture of J . So T / ∈ C J . Now suppose that C J ⊆ C I .We have Th( I ) / ∈ C I , hence Th( I ) / ∈ C J , so I admits an uncollapsed indiscerniblepicture of J , so Th( I ) trace defines Th( J ). (cid:3) It would be interesting to classify finitely homogeneous structures up to trace equiv-alence. We take a few steps towards this below.5.1.
Op-dimension.
Op-dimension was introduced by Guingona and Hill [14]. Welet opd M ( X ) be the op-dimension of an M -definable set X and opd( T ) be the op-dimension of T . Recall that opd( T ) = opd( M ) for any M | = T . A structure withfinite op-dimension has NIP, see [15, Section 3.1]. We will also apply subadditivityof op-dimension, see [14, Theorem 2.2]. A k -order is a structure ( P ; < , . . . , < k ) where each < k is a linear order on P . Finite k -orders form a Fra¨ıss´e class, we denotethe Fra¨ıss´e limit by P k . Fact 5.4 is [15, Theorem 3.4]. Fact 5.4.
Suppose that M is NIP and let X be an M -definable set. Then thefollowing are equivalent for any k ≥ :(1) opd M ( X ) ≥ k (2) there is a small set A of parameters and an uncollapsed A -indiscerniblepicture P → X of P k in M . Proposition 5.5.
Suppose that T has finite op-dimension and T ′ is trace definablein T . Then T ′ has finite op-dimension.Proof. By Proposition 3.8 we may suppose that M | = T and O | = T ′ are ℵ -saturated, O is a subset of M m , and M trace defines O via the identity O → M m .We prove opd( T ′ ) ≤ m opd( T ). Suppose that opd( T ′ ) ≥ k . As T has finite op-rank, T is NIP, so T ′ is NIP by Proposition 4.6. Hence by Fact 5.4 there is anuncollapsed indiscernible picture P → O of P k in O . By Proposition 3.8 O tracedefines P k via an injection τ : P → O . So M trace defines P k via τ : P → M m . ByProposition 5.1 there is an uncollapsed indiscernible picture P → M m of P k in M .Thus opd M ( M m ) ≥ k . By subadditivity of op-dimension m opd( T ) ≥ k . (cid:3) Pictures in a power.
We prove Proposition 5.7, which will be used below.Suppose that L is relational and fix k ≥
2. Let L [ k ] be the language with n -aryrelation symbols R , . . . , R k for each n -ary R ∈ L . Given an L [ k ]-structure I and1 ≤ i ≤ k we let I i be the L -structure with domain I such that for every n -ary R ∈ L and a ∈ I n we have I i | = R ( a ) ⇐⇒ I | = R i ( a ). Given a (possibly incomplete) L -theory S we let S [ k ] be the L k theory where I | = S [ k ] if and only if I i | = S forall 1 ≤ i ≤ k .Suppose that I is an L -structure. For each 1 ≤ i ≤ k let π i : I k → I be given by π i ( x , . . . , x k ) = x i . Let I k be the S [ k ]-structure with domain I k where for any n -ary R ∈ L , 1 ≤ i ≤ k , and a , . . . , a n ∈ I k we have I k | = R i ( a , . . . , a n ) ⇐⇒ I | = R ( π i ( a ) , . . . , π i ( a n )) . Lemma 5.6.
Suppose that I is a homogeneous L -structure and S = Th( I ) ∀ . Anycountable S [ k ] -model embeds into I k .Proof. Suppose that J | = S [ k ] is countable. For each 1 ≤ i ≤ k we have J i | = S . As I is a universal countable S -model, we have embeddings e i : J i → I . Let e : J → I k be e ( x ) = ( e ( x ) , . . . , e k ( x )). We show that e is an embedding J → I k . Supposethat R ∈ L is n -ary, a , . . . , a n ∈ J , and 1 ≤ i ≤ k . Then I k | = R i ( e ( a ) , . . . , e ( a n )) ⇐⇒ I | = R ( π i ( e ( a )) , . . . , π i ( e ( a n ))) ⇐⇒ I | = R ( e i ( a ) , . . . , e i ( a n )) ⇐⇒ J i | = R ( a , . . . , a n ) ⇐⇒ J | = R i ( a , . . . , a n ) . (cid:3) Proposition 5.7.
Suppose that I is a homogeneous L -structure and S = Th( I ) ∀ .Any countable J | = S [ k ] with quantifier elimination is trace definable in I .Proof. Note that I k is interpretable in I and hence trace definable in I . Lemma 5.6and Proposition 3.3 together show that J is trace definable in I k . (cid:3) k -dependence Suppose k ≥
2. We discuss k -dependence. We refer to [5] for the definition andbasic results. Proposition 6.1 is clear from the usual definition of k -dependence. Proposition 6.1. If T is k -dependent and T ′ is trace definable in T then T ′ is k -dependent. The generic countable k -hypergraph. The generic countable k -hypergraphis ( k − k -dependent. Thus the generic countable k -hypergraphdoes not trace define the generic countable ( k + 1)-hypergraph. We let Hyp k be thetheory of the generic countable k -hypergraph. The class of structures ( V ; E ) where V is finite and E is an arbitrary k -ary relation is a Fra¨ıss´e class, we refer to theFra¨ıss´e limit as the generic countable k -ary relation and let Rel k be its theory. Lemma 6.2.
Fix k ≥ . The generic countable k -hypergraph and the genericcountable k -ary relation are trace equivalent. We do not know if the generic countable k -hypergraph interprets the generic count-able k -ary relation, even when k = 2. Proof.
Suppose that ( W ; R ) is the generic countable k -ary relation. Define a k -ary relation E on W by declaring E ( a , . . . , a k ) when a i = a j for all i = j and R ( a σ (1) , . . . , a σ ( k ) ) holds for some permutation σ of { , . . . , k } . It is easy to seethat E satisfies the extension axioms of Hyp k , so ( W ; E ) is a copy of the randomcountable k -hypergraph.Let ( V ; E ) be the generic countable k -hypergraph, we show that ( V ; E ) trace defines( W ; R ). We will produce a ( V ; E )-definable k -ary relation R E with domain V k suchthat the generic countable k -ary relation embedds into ( V k ; R E ). This suffices byProposition 3.4 and quantifier elimination for Rel k .We first describe R E , which is defined more generally for any k -hypergraph. Let( X ; F ) be a k -hypergraph. Given elements a , . . . , a k of X k with a i = ( a i , . . . , a ik )we declare R F ( a , . . . , a k ) ⇐⇒ F ( a , a , . . . , a kk ). Claim.
Let X be a set and S be a k -ary relation on X . Then there is a k -hypergraph ( Y ; F ) such that ( X ; S ) embedds into ( Y k ; R F ) . If Y is infinite then we may take | Y | = | X | . We first explain why the claim is enough. Suppose that the claim holds. By theclaim we may suppose that ( W ; R ) is a substructure of ( Y k ; R F ) for some countable k -hypergraph ( Y ; F ). Every countable k -hypergraph embedds into ( V ; E ), so wemay suppose that ( Y ; F ) is a substructure of ( V ; E ). Finally, it is easy to see that( Y k ; R F ) is a substructure of ( V k ; R E ).We now prove the claim. Fix a k -ary relation S on a set X . Let Y = X ×{ , . . . , k } .We define a k -hypergraph F on Y . Let ( a , i ) , . . . , ( a k , i k ) ∈ Y . Then we declare F (( a , i ) , . . . , ( a k , i k )) when { i , . . . , i k } = { , . . . , k } and R ( a σ (1) , . . . , a σ ( k ) ) where σ is the unique permutation of { , . . . , k } such that i σ (1) = 1 , . . . , i σ ( k ) = k . Wenow describe an embedding e : ( X ; R ) → ( Y k , R F ). Let e : X → Y k be given by e ( a ) = (( a, , . . . , ( a, k )). Suppose that a , . . . , a k ∈ X . Then R ( a , . . . , a k ) ⇐⇒ F (( a , , ( a , , . . . , ( a k , k )) ⇐⇒ R F ( e ( a ) , . . . , e ( a k )) So e is an embedding. (cid:3) Lemma 6.3.
The generic countable k -ary relation trace defines any countablefinitely k -ary structure. The generic countable k -ary hypergraph trace defines anycountable finitely k -ary structure. We remind the reader that the definition of “finitely k -ary” is back in the conven-tions. Proof.
The second claim follows from the first by Lemma 6.2. Suppose that O is acountable finitely k -ary L -structure. We first reduce to the case when every R ∈ L has airity k . Morleyizing reduces to the case when L is a finite relational language,every R ∈ L has airity ≤ k , and O admits quantifier elimination. Let L ∗ be thelanguage which contains each k -ary R ∈ L and contains a k -ary relation symbol R ∗ for each R ∈ L of airity < k . We let O ∗ be the L ∗ -structure with domain O whereeach k -ary R ∈ L is given the same interpretation as in L and if R ∈ L has airity ℓ < k then for any a , . . . , a k ∈ O we declare O ∗ | = R ∗ ( a , . . . , a k ) ⇐⇒ O | = R ( a , . . . , a ℓ ) ∧ ( a ℓ = a ℓ +1 = . . . = a k ) . Note that O ∗ admits quantifier elimination and is interdefinable with O . So afterpossibly replacing O with O ∗ we may suppose that every R ∈ L is k -ary.Let ( V ; R ) be the generic countable k -ary relation. We may suppose that L = { R , . . . , R m } where each R i is k -ary. Now apply Proposition 5.7 in the case when L = { R } and I is ( V ; R ). In this case S is the empty L -theory so O is an S [ m ]-modelwith quantifier elimination. Thus ( V ; R ) trace defines O . (cid:3) Proposition 6.4.
Fix k ≥ . Suppose that M is ℵ -saturated. Then the followingare equivalent.(1) M is ( k − -independent,(2) M trace defines the generic countable k -hypergraph,(3) M trace defines the generic countable k -ary relation,(4) M trace defines any countable finitely k -ary structure.The following are also equivalent.(5) T is k -independent,(6) T trace defines Hyp k ,(7) T trace defines Rel k ,(8) T trace defines any finitely k -ary theory. An ordered k -hypergraph is a structure ( V ; E, < ) where ( V ; E ) is a k -hypergraphand ( V ; < ) is a linear order. Finite ordered k -hypergraphs form a Fra¨ıss´e classwith the Ramsey property, we refer to the Fra¨ıss´e limit of this class as the genericcountable ordered k -hypergraph. If ( V ; E, < ) is the generic countable ordered k -hypergraph then ( V ; E ) | = Hyp k . We now prove Proposition 6.4. Proof.
It is enough to show that (1) − (4) are equivalent. Lemma 6.2 shows that(2) and (3) are equivalent. The random k -ary relation is k -independent, so (3)implies (1) by Proposition 6.1. It is clear that (4) implies (3). By Lemma 6.3 (3)implies (4). We finish by showing that (1) implies (2). Suppose that M is ( k − M admits an uncollapsed generic countable ordered k -hypergraph indiscernible. By Proposition 5.1 M trace definesthe generic countable k -hypergraph. (cid:3) Corollary 6.5 follows from Proposition 6.4.
Corollary 6.5.
Any finitely binary IP structure is trace equivalent to the Erd¨os-Rado graph. We use the classification of homogeneous graphs to prove Corollary 6.6.
Corollary 6.6.
Any countable homogeneous graph is trace equivalent to either:(1) an infinite set with no additional structure, or(2) the Erd¨os-Rado graph.
We apply Fact 6.7 below, a theorem of Lachlan and Woodrow [31]. Recall that the n th Henson graph is the generic countable K n -free graph for n ≥
2. Given a graph( V ; E ) we let ( V ; E c ) be the graph where E c ( a, b ) ⇐⇒ ( a = b ) ∧ ¬ E ( a, b ). Fact 6.7.
Suppose that ( V ; E ) is a homogeneous graph. Then either ( V ; E ) or ( V ; E c ) is isomorphic to one of the following:(1) a countable union of copies of K n for some n ≤ ω ,(2) the Erd¨os-Rado graph, or(3) the n th Henson graph for some n ≥ . We now prove Corollary 6.6.
Proof.
Suppose that ( V ; E ) is a homogeneous graph. Note that ( V ; E ) and ( V ; E c )are interdefinable. By Fact 6.7 we may suppose that ( V ; E ) is one of the following:(1) a countable union of copies of K n for some n ≤ ω ,(2) the Erd¨os-Rado graph, or(3) the n th Henson graph for some n ≥ V ; E ) is interpretable in the theory of aninfinite set. It is also easy to see that each Henson graph is IP. Apply Corollary 6.5. (cid:3) Simplicity and trace maximality
We say that T is trace maximal if T trace defines every theory and that M istrace maximal if Th( M ) is trace maximal. We prove some general facts about tracemaximality and show that infinite boolean algebras and non separably closed PACfields are trace maximal. It follows that pseudofinite fields are trace maximal.7.1. Basic results.Lemma 7.1.
Suppose that M is ℵ -saturated. Then the following are equivalent:(1) M is trace maximal,(2) There is an infinite A ⊆ M m such that for any X ⊆ A k there is an M -definable Y ⊆ M mk such that X = A k ∩ Y ,(3) there is an infinite A ⊆ M m , a sequence ( ϕ k ( x k , y k ) : k < ω ) of formulaswith | x k | = mk , and elements ( a k,X ∈ M | y k | : k < ω, X ⊆ A k ) such thatfor every b ∈ A k we have M | = ϕ k ( b, a k,X ) ⇐⇒ b ∈ X , (4) there is an infinite A ⊆ M m , a sequence ( ϕ k ( x k , y k ) : k < ω ) with | x k | = mk , and ( a k,X,Y ∈ M | y k | : k < ω, X, Y ⊆ A k , | X ∪ Y | < ℵ , X ∩ Y = ∅ ) such that for any k < ω , disjoint finite X, Y ⊆ A k , and b ∈ A k we have b ∈ X = ⇒ M | = ϕ k ( b, a k,X,Y ) b ∈ Y = ⇒ M | = ¬ ϕ k ( b, a k,X,Y ) . Standard coding arguments shows that ( Z ; + , × ) satisfies (4) above with A = Z .Thus ( Z ; + , × ) is trace maximal. Proof.
We first show that (1) implies (2). Suppose that T is trace maximal. Let A be a countable set and A be a structure on A which defines every subset of every A k . Then Th( A ) is trace definable in T , by Proposition 3.8 M trace defines A . Itis easy to see that (3) and (4) are equivalent.We show that (3) implies (1). It is easy to see that for every infinite cardinal λ there is M ≺ N and A ⊆ N m such that | A | = λ and every subset of every A k is ofthe form Y ∩ A k for some N -definable Y ⊆ N mk . This yields trace maximality. Wefinish the proof by showing that (2) implies (4). Trace maximality follows directly.Suppose that A ⊆ M m satisfies the condition of (2). The collection of disjoint pairsof nonempty subsets of A has cardinality | A | . So we can fix a subset Z of A k +1 suchthat for any disjoint finite X, Y ⊆ A k there is b X,Y ∈ A such that for all a ∈ A k wehave a ∈ X = ⇒ ( a, b X,Y ) ∈ Za ∈ Y = ⇒ ( a, b X,Y ) / ∈ Z Let ϕ ( x, y ) be a formula, possibly with parameters, such that for any ( a, b ) ∈ A k +1 we have M | = ϕ ( a, b ) ⇐⇒ ( a, b ) ∈ Z . (4) follows. (cid:3) Lemma 7.2.
Suppose that M is ℵ -saturated. Then the following are equivalent:(1) M is trace maximal,(2) there is an infinite A ⊆ M m such that for every k -hypergraph E on A thereis an M -definable X ⊆ M mk such that E ( a , . . . , a k ) ⇐⇒ ( a , . . . , a k ) ∈ X for all a , . . . , a k ∈ A ,(3) there is a sequence ( a i : i < ω ) of elements of some M m such that forany k -hypergraph E on ω there is an M -definable Y ⊆ M mk such that E ( i , . . . , i k ) ⇐⇒ ( a i , . . . , a i k ) ∈ Y for all i , . . . , i k < ω . We suppose that M is an L -structure. Proof.
It is clear that (2) and (3) are equivalent. Lemma 7.1 shows that (1) implies(2). We show that (2) implies (1). Suppose (2). Let E be a graph on A and( a i : i < ω ), ( b j : j < ω ) be sequences of distinct elements of A such that for all i, j we have E ( a i , b j ) ⇐⇒ i < j . Let δ ( x, y ) be an L ( M )-formula such that for all a, a ′ ∈ A we have E ( a, a ′ ) ⇐⇒ M | = δ ( a, a ′ ). For each i let c i = ( a i , b i ) and let φ ( x , y , x , y ) be δ ( x , y ). So for any i, j we have M | = φ ( c i , c j ) ⇐⇒ M | = φ ( a i , b i , a j , b j ) ⇐⇒ M | = δ ( a i , b j ) ⇐⇒ i < j. We show that for any X ⊆ ω k there is an M -definable Y ⊆ M k such that( i , . . . , i k ) ∈ X ⇐⇒ ( c i , . . . , c i k ) ∈ Y for all ( i , . . . , i k ) ∈ ω k . Trace maximalityof M follows by Lemma 7.1. We apply induction on k ≥ k = 1 and X ⊆ ω . Let F be a graph on A and d ∈ A be such that E ( a i , d ) ⇐⇒ i ∈ X for all i < ω . Let θ ( x, y ) be an L ( M )-formula such that F ( a, a ′ ) ⇐⇒ M | = θ ( a, a ′ ) for all a, a ′ ∈ A . Let Y be the set of ( a, b ) ∈ M suchthat M | = θ ( a, d ). So for any i < ω we have i ∈ X ⇐⇒ c i ∈ Y .We now suppose that k ≥ X ⊆ ω k . Abusing notation we let ω denotethe structure ( ω ; < ). We let qftp ω ( i , . . . , i k ) be the quantifer free type (equiva-lently: order type) of ( i , . . . , i k ) ∈ ω k and let S k ( ω ) be the set of quantifier free k -types. Note that qftp ω ( a ) is just the order type of a = ( a , . . . , a k ). Note that S k ( ω ) is finite. For each p ∈ S k ( ω ) we fix an M -definable Y p ⊆ M k such thatqftp ω ( i , . . . , i k ) = p ⇐⇒ ( c i , . . . , c i k ) ∈ Y p for all ( i , . . . , i k ) ∈ ω k . We showthat for every p ∈ S k ( ω ) there is an M -definable X p ⊆ M k such that for any( i , . . . , i k ) ∈ ω k satisfying qftp ω ( i , . . . , i k ) = p we have ( c i , . . . , c i k ) ∈ X ⇐⇒ ( c i , . . . , c i k ) ∈ X p . It then follows that for any ( i , . . . , i k ) ∈ ω k we have( c i , . . . , c i k ) ∈ X ⇐⇒ ( c i , . . . , c i k ) ∈ [ p ∈ S k ( ω ) ( Y p ∩ X p ) . We fix p ( x , . . . , x k ) ∈ S k ( ω ) and produce X p . We first treat the case when p | =( x i = x j ) for some i = j . To simplify notation we suppose that p | = ( x = x ).Let X ′ be the set of ( i , . . . , i k − ) ∈ ω k − such that ( i , i , i , . . . , i k − ) ∈ X . Byinduction there is an M -definable Y ′ ⊆ M k − such that for all ( i , . . . , i k − ) ∈ ω k − we have ( i , . . . , i k − ) ∈ X ′ ⇐⇒ ( c i , . . . , c i k − ) ∈ Y ′ . Let X p be the setof ( d , . . . , d k ) ∈ M k such that d = d and ( d , . . . , d k ) ∈ Y ′ , note that X p isdefinable in M .Now suppose that p | = ( x i = x j ) when i = j . So if ( i , . . . , i k ) ∈ ω k and |{ i , . . . , i k }| = k then there is a unique permutation σ p of { , . . . , k } such thatqftp ω ( i σ p (1) , . . . , i σ p ( k ) ) = p . Let H ′ be the k -hypergraph on { c i : i < ω } where H ′ ( c i , . . . , c i k ) if and only if |{ i , . . . , i k }| = k and ( i σ p (1) , . . . , i σ p ( k ) ) ∈ X . Let H be the k -hypergraph on { a i : i < ω } where H ( a i , . . . , a i k ) ⇐⇒ H ′ ( c i , . . . , c i k ) forall ( i , . . . , i k ) ∈ ω k . By assumption there is an L ( M )-formula ϕ ( x , . . . , x k ) suchthat H ( a i , . . . , a i k ) ⇐⇒ M | = ϕ ( a i , . . . , a i k ) for any ( i , . . . , i k ) ∈ ω k . Let X p bethe set of (( d , e ) , . . . , ( d k , e k )) ∈ M k such that M | = ϕ ( d , . . . , d k ). (cid:3) Examples.
We first describe two simple trace maximal theories. First, sup-pose that L is a relational language and L contains an n -ary relation symbol for all n ≥
2. Let ∅ L be the empty L -theory and ∅ ∗ L be the model companion of ∅ L , whichexists by a theorem of Winkler [47]. By [29] ∅ ∗ L is simple. We show that ∅ ∗ L is tracemaximal. Suppose that M | = ∅ ∗ L is ℵ -saturated and let A ⊆ M be countably infi-nite. It is easy to see that for each X ⊆ A k and ( k +1)-ary R ∈ L there is c ∈ M suchthat for all a , . . . , a k ∈ A we have ( a , . . . , a k ) ∈ X ⇐⇒ M | = R k +1 ( a , . . . , a k , c ).Apply Lemma 7.1.We now describe a symmetric analogue of ∅ ∗ L . For each k ≥ E k be a k -aryrelation symbol and let L = { E k : k < ω } . Let T be the L -theory such that( M ; ( E k ) k ≥ ) if and only if each E k is a k -hypergraph on M . Then T has a modelcompanion T ∗ and T ∗ is simple. Suppose that M | = T is ℵ -saturated and A ⊆ M is countably infinite. We show that M is trace maximal. Towards an applicationof Lemma 7.2 we suppose that E is a k -hypergraph on M . It is easy to see thatthere is c ∈ M such that we have E ( a , . . . , a k ) ⇐⇒ M | = E k +1 ( a , . . . , a k , c ) forall a , . . . , a k ∈ A . Proposition 7.3.
Every infinite boolean algebra is trace maximal.
A subset A of a boolean algebra is independent if the subalgebra generated by A is free, equivalently for any a , . . . , a k , b , . . . , b ℓ ∈ A with a i = b j for all i, j wehave ( a ∧ . . . ∧ a k ) ∧ ( ¬ b ∧ . . . ∧ ¬ b ℓ ) = 0. Lemma 7.4.
Suppose that B is a boolean algebra, A is an independent subset of B , and ( b i : i ≤ k ) , ( a ij : 1 ≤ i ≤ n, ≤ j ≤ k ) are elements of A such that for all ≤ i ≤ n we have(1) |{ b , . . . , b k }| = k = |{ a i , . . . , a ik }| , and(2) { b , . . . , b k } 6 = { a i , . . . , a ik } .Then ( b ∧ . . . ∧ b k ) (cid:10) W ni =1 ( a i ∧ . . . ∧ a ik ) .Proof. Note that for each 1 ≤ i ≤ n there is j ( i ) such that a ij ( i ) / ∈ { b , . . . , b k } . Byindependence we have c := ( b ∧ . . . ∧ b k ) ∧ ( ¬ a j (1) ∧ . . . ∧ ¬ a nj ( n ) ) = 0 . Then c ≤ ( b ∧ . . . ∧ b k ) and c ∧ W ni =1 ( a i ∧ . . . ∧ a ik ) = 0. (cid:3) We now prove Proposition 7.3.
Proof.
We first make some basic remarks on boolean algebras which can probablybe found in any standard reference. Any finite boolean algebra is isomorphic tothe boolean algebra of subsets of { , . . . , n } for some n . It follows that if B ′ , B ′′ are finite boolean algebras with | B ′ | ≤ | B ′′ | then there is an embedding B ′ → B ′′ .The free boolean algebra on n generators has cardinality 2 n and clearly has anindependent subset of cardinality n . It follows that any finite boolean algebra with ≥ n elements contains an independent subset of cardinality n . Recall also thata finitely generated boolean algebra is finite. It follows that any infinite booleanalgebra contains an independent subset of cardinality n for all n .Let B = ( B ; ∧ , ∨ , <, ,
1) be an infinite boolean algebra. We may suppose that B is ℵ -saturated. Applying the previous paragraph and saturation we fix a countablyinfinite independent subset A of B . By Lemma 7.2 it suffices to suppose that E is a k -hypergraph on A and produce a definable Y ⊆ B k such that for all a , . . . , a k ∈ A we have E ( a , . . . , a k ) ⇐⇒ ( a , . . . , a k ) ∈ Y . Let f : A k → B begiven by f ( a , . . . , a k ) = a ∧ . . . ∧ a k . Let D be the set of ( a , . . . , a k ) ∈ A k suchthat |{ a , . . . , a k }| = k . Let ( a i : i < ω ) be an enumeration of E . Lemma 7.4 showsthat if b ∈ D and ¬ E ( b ) then f ( b ) (cid:10) W ni =1 f ( a i ) for all n . So for any n there is a c ∈ B such that f ( a i ) ≤ c for all i n and f ( b ) (cid:10) c for all b ∈ D such that ¬ E ( b ).By saturation there is c ∈ B such that f ( a i ) c for all i < ω and f ( b ) (cid:10) c for all b ∈ D with ¬ E ( b ). Therefore we let Y be the set of a ∈ B k such that a ∈ D and f ( a ) c . (cid:3) We now show that some of the main examples of model-theoretically tame IP fieldsare trace maximal. We do not know of an IP field which is not trace maximal. It isa conjecture of Chernikov and Hempel that a field which is k -independent for some k ≥ k -independent for all k [17, Conjecture 1]. Proposition 7.5.
Let K be a field and suppose that one of the following holds:(1) K is PAC and not separably closed, or(2) K is pseudo real closed and not real closed.Then K is trace maximal. Furthermore a pseudofinite field is trace maximal. Separably closed fields are stable and hence not trace maximal. Psuedofinite fieldsare simple, so any theory is trace definable in a simple theory. We largely follow theproof of Duret’s theorem [8] that a non-separably closed PAC field is IP. Fact 7.6was essentially proven by Macintryre [34].
Fact 7.6.
Suppose that K is not separably closed. Then there is a finite extension L of K such that one of the following holds:(1) the Artin-Schrier map ℘ : L → L is not surjective, or(2) there is a prime p = Char( K ) such that L contains a primitive p th root ofunity and the p th power map L × → L × is not surjective. Fact 7.7 is [8, Lemma 6.2].
Fact 7.7.
Suppose that K is a PAC field, p = Char( K ) is a prime, K containsa primitive p th root of unity, and the p th power map K × → K × is not surjective.Let A, B be disjoint finite subsets of K . Then there is c ∈ K such that c + a is a p th power for every a ∈ A and c + b is not a p th power for every b ∈ B . Given a field K of characteristic p = 0 we let ℘ : K → K be the Artin-Schrier map ℘ ( x ) = x p − x . Fact 7.8 is [8, Lemma 6.2]. Fact 7.8.
Suppose that K is a PAC field of characteristic p = 0 . Suppose that ℘ : K → K is not surjective and A, B are disjoint finite subsets of K such that A ∪ B is linearly independent over F p . Then there is c ∈ K such that ac is in theimage of ℘ : K → K for any a ∈ A and bc is not in the image of ℘ : K → K forany b ∈ B . We now prove Proposition 7.5.
Proof.
The second claim follows from the first claim as pseudofinite fields are PACand not separably closed by a theorem of Ax. If K is pseudo real closed and not realclosed then K ( √−
1) is PAC and not separably closed, so the pseudo real closedcase follows from the PAC case. Therefore we suppose that K is PAC and notseparably closed. We may also suppose that K is ℵ -saturated.By Fact 7.6 there is a finite L/K such that either:(1) Char( K ) = 0 and the Artin-Schrier map ℘ : L → L is not surjective, or(2) there is a prime q = Char( K ) such that L contains a primitive q th root ofunity and the q th power map L × → L × is not surjective.A finite extension of a PAC field is PAC, so L is PAC. If L is separably closedthen K is either real closed or separably closed. A real closed field is not PAC, so L is not separably closed. Recall that L is interpretable in K so L is ℵ -saturated.It is enough to show that L is trace maximal. Suppose (2) and fix a relevantprime q . Let A be an infinite subset of K and t be an element of K which is notin the algebraic closure of Q ( A ). Let f : A k → K be given by f ( a , . . . , a k − ) = t k + a k − t k − + . . . + a t + a . Let X be a subset of A k . By injectivity of f , Fact 7.7,and saturation there is c ∈ K such that a ∈ X ⇐⇒ L | = ∃ x ( x q = f ( a ) + c ) for all a ∈ A k . By Lemma 7.1 L is trace maximal. We now suppose (1). Let A be an infinite subset of L which is algebraically indepen-dent over F p . By Lemma 7.2 it is enough to suppose that E is a k -hypergraph on A and produce definable Y ⊆ L k such that E ( a , . . . , a k ) ⇐⇒ ( a , . . . , a k ) ∈ Y for all a , . . . , a k ∈ A . Let D be the set of ( a , . . . , a k ) ∈ A k such that |{ a , . . . , a k }| = k and f : D → A be given by f ( a , . . . , a k ) = a a . . . a k . By algebraic indepen-dence f ( a ) = f ( b ) ⇐⇒ { a , . . . , a k } = { b , . . . , b k } for all a = ( a , . . . , a k ) , b =( b , . . . , b k ) ∈ D and { f ( a ) : a ∈ D } is linearly independent over F p . By Fact 7.8and saturation there is c ∈ L such that E ( a ) ⇐⇒ L | = ∃ x ( x p − x = f ( a ) c ) for all a ∈ D . (cid:3) Basic Examples
We first give a few basic examples.
Proposition 8.1. ( N ; < ) is trace definable in ( R ; + , < ) . Note that ( N ; < ) is not interpretable in ( R ; + , < ) as ( R ; + , < ) eliminates both imag-inaries and ∃ ∞ . Proposition 8.1 shows that elimination of ∃ ∞ is not preserved undertrace definability. Proof.
Let s ( x ) = x + 1. Recall that ( N ; <, s,
0) admits quantifier elimination andnote that ( N ; <, s,
0) is a substructure of ( R ; <, s, (cid:3) It follows from Lemma 4.2 that any ℵ -saturated linear order trace defines DLO.The next three results are similar. Proposition 8.2.
Suppose that G is a group of infinite exponent. Then Th( G ) trace defines the theory of torsion free divisible abelian groups, i.e. Th( Q ; +) . In particular Th( Z ; +) trace defines Th( Q ; +). Note that Th( Q ; +) does not tracedefine Th( Z ; +) as ( Q ; +) is ℵ -stable and ( Z ; +) is not. We do not know if a vectorspace over a finite field can trace define ( Q ; +). Proof.
After possibly passing to an elementary extension we suppose that there isa non-torsion a ∈ G . Let Z = { a m : m ∈ Z } and ( G ′ , Z ′ ) be an ℵ -saturatedelementary extension of ( G, Z ). So Z ′ is an ℵ -saturated model of Th( Z ; +) andis hence isomorphic to ( Q ; +) κ × Z where κ is an infinite cardinal and Z is theprofinite completion of ( Z ; +). So there is an embedding τ : ( Q ; +) → Z ′ . ApplyProposition 3.3. (cid:3) Proposition 8.3. If ( G ; + , < ) is an ordered abelian group then Th( G ; + , < ) tracedefines the theory of divisible ordered abelian groups, i.e. Th( Q ; + , < ) . Proposition 8.3 follows in the same way as Proposition 8.2 and an application ofquantifier elimination for divisible ordered abelian groups.
Proposition 8.4.
Suppose that R is a (2 ℵ ) + -saturated ordered field. Then R tracedefines ( R ; + , × ) . Thus if S is an ordered field then Th( S ) trace defines RCF .Proof.
Let V be the convex hull of Z in R and m be the set of a ∈ V such that | a | < /n for all n ≥
1. Let st : V → R be given by st( a ) = sup { q ∈ Q : q < a } .By saturation st is surjective by saturation, so we identify V / m with R . Then V isa valuation ring with maximal ideal m and st is the residue map. Let τ : R → V be a section of st. We show that R trace defines ( R ; + , × ) via τ . Suppose that X is an ( R ; + , × )-definable subset of R n . By Proposition 3.6 we may suppose that X is definable without parameters. By quantifier elimination for real closed fieldswe may suppose that X = { a ∈ R n : f ( a ) ≥ } for some f ∈ Z [ x , . . . , x n ]. Sofor any a ∈ R n we have a ∈ X if and only if f ( a ) = f (st( τ ( a )) ≥
0. We have f (st( b )) = st( f ( b )) for all b ∈ V n , so a ∈ X ⇐⇒ st( f ( τ ( a ))) ≥ ⇐⇒ f ( τ ( a )) ≥ c for some c ∈ m . By saturation the downwards cofinality of m is at least (2 ℵ ) + , so there is c ∈ m such that for any a ∈ R n we have st( f ( τ ( a ))) ≥ f ( τ ( a )) ≥ c . Let Y be the set of a ∈ R n such that f ( a ) ≥ c . So Y is R -definable and X = τ − ( Y ). (cid:3) Algebraically closed fields
Many interesting NIP structures have finite dp-rank, finiteness of dp-rank is pre-served under trace definability and there is now a reasonable classification of finitedp-rank fields [25]. So it should be both possible and interesting to classify finitedp-rank fields up to trace definability. We make a first step in this direction andrecord some facts on trace definability in real and algebraically closed fields.We let K alg be the algebraic closure of a field K . Theorem 9.1.
Suppose that K is an algebraically closed field. Then any infinitefield trace definable in K is algebraically closed. If K is in addition characteristiczero then any field trace definable in K is characteristic zero. It easily follows that if K is algebraically closed of characteristic zero, the transendencedegree of K/ Q is infinite, and L is an infinite field trace definable in K , then there isan elementary embedding L → K . This is sharp as elementary substructures are al-ways trace definable. We don’t know if an algebraically closed field of characteristic p > = p . Proof.
The first claim follows from Corollary 4.4 and Macintyre’s theorem [34] thatan infinite ℵ -stable field is algebraically closed. Suppose K is characteristic zero.Fix a real closed subfield R of K such that K = R ( √− K ; R ) is distal.Apply Proposition 4.11. (cid:3) We now make some remarks concerning trace definability in RCF. It is natural toconjecture that an infinite field trace definable in a real closed field is either real oralgebraically closed of characteristic zero. By Proposition 4.11 a real closed fieldcannot trace define an infinite field of positive characteristic.
Proposition 9.2.
Suppose that M expands a linear order, Th( M ) is weakly o-minimal, and K is an infinite field which is trace definable in M via an injection K → M . Then K is real closed or algebraically closed of characteristic zero. Proposition 9.2 shows that if R is a real closed field and K is an infinite field tracedefinable in R via an injection K → R , then K is either real or algebraically closedof characteristic zero.Proposition 9.2 follows easily from some known results. We recall a notion ofFlenner and Guingona [12]. We say that M is convexly orderable if there is alinear order ⊳ on M such that for every formula ϕ ( x, y ) with | x | = 1 there is n suchthat { a ∈ M : M | = ϕ ( a, b ) } is a union of at most n ⊳ -convex sets for all b ∈ M | y | . A convexly orderable structure M need not define a linear order on M , for examplea strongly minimal structure is convexly orderable. It is easy to see that convexorderability is preserved under elementary equivalence, so we say that T is convexlyorderable if some (equivalently: every) T -model is convexly orderable.Lemma 9.3 is clear from the definitions. Lemma 9.3. If M is convexly orderable and O is trace definable in M via aninjection O → M then O is convexly orderable. A weakly o-minimal strucutre is distal so by Proposition 4.11 a weakly o-minimalstructure cannot trace definable an infinite positive characteristic field. Proposi-tion 9.2 follows from this observation, Lemma 9.3, and the theorem of Johnson [24]that an infinite convexly orderable field is either real closed or algebraically closed.10.
Shelah Completions
Lemma 10.1 is immediate from the definitions.
Lemma 10.1.
Suppose that A ⊆ M m , A is the structure induced on A by M , and A admits quantifier elimination. Then M trace defines A . Lemma 10.2.
Let λ be a cardinal. Suppose that M is NIP and λ -saturated and O is an elementary submodel of M of cardinality < λ . Then M trace defines O Sh .Proof. Suppose that X ⊆ O n is O Sh -definable. By Fact 2.5 X is externally definablein O . By Lemma 2.3 there is an M -definable Y ⊆ M n such that X = Y ∩ O n . (cid:3) It is clear that the Shelah completion of M trace defines M , the other direction ofProposition 10.3 follows from Lemma 10.2. Proposition 10.3.
Suppose that M is NIP . Then M and M Sh are trace equivalent. Proposition 10.4.
Let λ be an uncountable cardinal. Suppose that M is λ -saturated and NIP . Suppose M Sh trace defines O and | O | < λ . Then M tracedefines O . If M Sh interprets P and | P | < λ then M trace defines P .Proof. The second claim follows from the first claim and Proposition 3.2. We maysuppose that O ⊆ M m and that M trace defines O via the identity O → M m . Fixan O -definable subset X of O n . Let X be an M Sh -definable subset of M mn suchthat O n ∩ Y = X . By Fact 2.5 X is externally definable so by Lemma 2.3 there isan M -definable Y ⊆ M mn such that X ∩ O n = Y ∩ O n . (cid:3) Corollary 10.5 follows from Proposition 10.3 and Fact 2.2.
Corollary 10.5.
Suppose that M is a NIP expansion of a linear order ( M ; < ) and C is a collection of convex subsets of M . Then ( M , C ) is trace equivalent to M . Thus the theory of real closed valued fields is trace equivalent to the theory ofreal closed fields. Recall that RCF is rosy and RCVF is not. Thus rosiness is notpreserved under trace definability.
Corollary 10.6.
Suppose that K is a NIP field and v is a Henselian valuation on K such that the residue field of v is not separably closed. Then K and ( K, v ) aretrace equivalent. Corollary 10.6 follows from Proposition 10.3 and the fact that if K and v satisfythe conditions above then v is externally definable in K , see [23].We now assume that the reader is somewhat familiar with the theory of definablegroups in NIP structures. Suppose that M is NIP and G is a definable group. Let π be the quotient map G → G/G . The structure induced on G/G by M is theexpansion of the pure group G/G by all sets { ( π ( a ) , . . . , π ( a n )) : a , . . . , a n ∈ X } for definable X ⊆ G n . Proposition 10.7.
Suppose that M is NIP , G is a definably amenable definablegroup, and G/G is a Lie group. Then the structure induced on G/G by M istrace definable in M . If M is o-minimal and G is a definably compact definablegroup then the structure induced on G/G by M is trace definable in M .Proof. By Proposition 3.8 and Proposition 10.3 it is enough to show that M Sh interprets G . This follows from work of Pillay and Hrushovski who showed that G is externally definable in M . They proved this in the case when M is o-minimal and G is definably compact [21, Lemma 8.2] and pointed out that theproof goes through when M is NIP, G is definably amenable, and G/G is a Liegroup [21, Remark 8.3]. (cid:3) DLO
Let DLO be the theory of dense linear orders without endpoints, i.e. Th( Q ; < ). ByProposition 4.1 T is unstable if and only if T trace defines DLO, so the minimalunstable trace equivalence class is that of DLO. We describe some theories in thisclass and show that DLO does not trace define an infinite group.Given λ ≤ ω we let DLO λ eq be the theory of ( L ; <, E ) where ( L ; < ) | = DLO and E isan equivalence relation on L with λ classes each of which is dense and codense in L .It is easy to see that each DLO λ eq is ℵ -categorical and admits quantifier elimination.Note also that DLO is just the theory of ( L ; <, X ) when ( L ; < ) | = DLO and X isa dense and co-dense subset of L . Proposition 11.1.
Suppose λ ≤ ω . Then DLO λ eq is trace equivalent to DLO .Furthermore
DLO does not interpret
DLO λ eq .Proof. The second claim follows from Corollary 2.10. We suppose ( L ; <, E ) | =DLO λ eq and show that ( L ; < ) trace defines ( L ; <, E ). Let [ a ] be the E -class of a ∈ L , ι : L/E → L be an injection, τ : L → L be given by τ ( a ) = ( a, ι ([ a ])), and O = τ ( L ). Let ⊳ be the order on L where ( a, b ) ⊳ ( a ′ , b ′ ) when a < a ′ and F bethe equivalence relation on L where F (( a, b ) , ( a ′ , b ′ )) ⇐⇒ b = b ′ . Note that ⊳ and F are both ( L ; < )-definable and τ induces an isomorphism ( L ; <, E ) → ( O ; ⊳, F ).Apply Proposition 3.4 and quantifier elimination for DLO λ eq . (cid:3) We let DLO k be the theory of the generic countable k -order, so DLO = DLO. Proposition 11.2.
All countable homogeneous k -orders are trace equivalent to ( Q ; < ) . So DLO k and DLO ℓ are trace equivalent for any k, ℓ . Suppose that k ≥ O ; < , . . . , < k ) | = DLO k . It is easy to see that anynonempty open < -interval is dense and codense in the < -topology. ThereforeDLO does not interpret DLO k . Proof.
It is enough to prove the first claim. Suppose that O = ( O ; < , . . . , < k ) isa countable homogeneous k -order. An application of Proposition 5.7 in the casewhen L = { < } and I = ( Q ; < ) shows that O is trace definable in ( Q ; < ). (cid:3) We now show that DLO does not trace define an infinite group. Equivalently: thereis an unstable structure that does not trace define an infinite group. A coloredlinear order is an expansion of a linear order ( O ; < ) by a family of unary predicatesdefining subsets of O and constant symbols defining elements of O . Proposition 11.3.
Suppose that O is a colored linear order. Then O does not tracedefine an infinite group. Poizat showed that a colored linear order cannot interpret an infinite group [38].We follow the proof of this result as presented by Hodges [18, A.6]. Fact 11.4 is[18, Lemma A.6.8].
Fact 11.4.
Suppose that O is a colored linear order. Suppose that a = ( a , . . . , a m ) ∈ O m and b = ( b , . . . , b n ) ∈ O n . Then there is a subset I of { , . . . , m } of car-dinality at most n such that if c = ( c , . . . , c m ) ∈ O m , tp O ( c ) = tp O ( a ) and tp O ( b, ( a i ) i ∈ I ) = tp O ( b, ( c i ) i ∈ I ) then tp O ( b, a ) = tp O ( b, c ) . We now prove Proposition 11.3.
Proof.
Let L be the language of O . Suppose that O trace defines an infinite group G . We suppose that G is a subset O m and that G is trace definable via the identity G → O m . By Proposition 3.8 we may suppose that O and G are both highlysaturated. For each k there is an L ( O )-formula ϕ ( x , . . . , x k , y ) such that for any a , . . . , a k , b ∈ G we have O | = ϕ ( a , . . . , a k , b ) ⇐⇒ a a . . . a k = b . After addingnew constant symbols to L we may suppose that each ϕ k is parameter-free.Fix a sequence ( a q : q ∈ Q ) of distinct elements of G which is indiscernible in O . Let a a . . . a k = c k for all k ≥
1. By Fact 11.4 there is k and i ≤ k suchthat tp O ( a , a , . . . , a k , c k ) is determined by tp O ( a , . . . , a i − , a i +1 , . . . , a k , c k ) andtp O ( a , . . . , a k ). So by indiscernibility we havetp O ( a , . . . , a i − , a i + , a i +1 , . . . , a k , c k ) = tp O ( a , . . . , a i − , a i , a i +1 , . . . , a k , c k ) . Then O | = ϕ k ( a , . . . , a i − , a i + , a i +1 , . . . , a k , c k ), so a . . . a i − a i + a i +1 . . . a k = c k . But this implies a i = a i + , contradiction. (cid:3) Corollary 11.5.
Suppose that M is ℵ -categorical and ℵ -stable. Then M tracedefines an infinite group if and only if M interprets an infinite group. The argument below is probably only comprehensible to the reader who is somewhatfamiliar with the structure theory of ℵ -categorical and ℵ -stable structures. Fact 11.6.
Suppose that M is countable, ℵ -categorical, ℵ -stable, and the combi-natorial geometry on each strongly minimal set interpretable in M is trivial. Then M is interpretable in ( Q ; < ) . Lachlan [30] proved Fact 11.6 under the additional assumption that the languageof M is finite. Hrushovksi [19, Theorem 2.1(a)] showed that if M is ℵ -categoricaland ℵ -stable then M is interdefinable with a structure in a finite language. Wenow prove Corollary 11.5. Proof.
Recall that M is coordinatized by finitely many strongly minimal sets in-terpretable in M and that the combinatorial geometry associated to each stronglyminimal set is either trivial or an affine or projective geometry over a finite field [4].If the geometry on some strongly minimal interpretable set is affine or projectiveover a finite field then M interprets an infinite group, see for example [37, Chapter5]. Apply Fact 11.6 and Proposition 11.3. (cid:3) Simon [42] has shown that if M is ℵ -categorical, NIP, and unstable, then M interprets DLO. Equivalently: If M is ℵ -categorical and NIP then M trace definesDLO if and only if M interprets DLO. This seems analogous to Corollary 11.5, soone wonders if there is more general trace rigidity phenomenon for (certain) ℵ -categorical theories. Note that by Proposition 7.3 the ℵ -categorical theory of aninfinite atomless boolean algebra trace defines many structures which it does notinterpret. 12. Proof of Theorem 1.3 and two examples
We first prove a general lemma.
Lemma 12.1.
Suppose that R is an o-minimal expansion of ( R ; + , < ) , A is a densesubset of R , A is the structure induced on A by R , and A is NIP . Then
Th( A ) trace defines R . Lemma 12.1 probably holds without the assumption that A is NIP, but this as-sumption is convenient as it allows us to use Shelah completions. In the proof belowwe let k a k be the ℓ ∞ -norm of a and let Cl( X ) be the closure of X ⊆ R m . We usethe trivial identity k ( a, b ) k = max {k a k , k b k} to simplify a metric argument. Proof.
Let A ≺ B be ℵ -saturated. By Proposition 10.3 it is enough to showthat B Sh interprets R . We first realize R as a B Sh -definable set of imaginaries.Let D be the set of ( c, a, b ) ∈ A such that c > | a − b | < c . Note that D is A -definable. Let D ∗ be the subset of B defined by the same formula as D . Note that ( D ∗ c ) c> is a chain under inclusion. Let E = T c ∈ A,c> D ∗ c and V = S c ∈ A,c> { a ∈ B : (0 , a, c ) ∈ D ∗ } . By Lemma 2.4 V and E are both externallydefinable in B . Note that V is the set of b ∈ B such that | b | < n for some n and E is the set of ( a, b ) ∈ B such that | a − b | < /n for all n ≥
1. So E is an equivalencerelation on V . Let st : V → R be the usual standard part map. By saturationst is surjective and for any a, a ′ ∈ V we have st( a ) = st( a ′ ) ⇐⇒ ( a, a ′ ) ∈ E . Sowe identify V /E with R . We show that R is a reduct of the structure induced on R by B Sh . Suppose that X ⊆ A m is A -definable. Let X ∗ be the subset of B m defined by the same formula as X . An easy application of saturation shows thatst( X ∗ ∩ V m ) = Cl( X ). So Cl( X ) is B Sh -definable.Suppose Y is a nonempty R -definable subset of R m . We show that Y is B Sh -definable. By o-minimal cell decomposition Y is a boolean combination of closed R -definable subsets of R m , so we may suppose that Y is closed. Let W be theset of ( ε, c ) ∈ R > × R m for which there is c ′ ∈ Y satisfying k c − c ′ k < ε . So W ∩ ( A × A m ) is A -definable and Z := Cl( W ∩ ( A × A m )) is B Sh -definable. Let Y ′ be T t ∈ I,t> Z t . So Y ′ is B Sh -definable. We show that Y = Y ′ . Each Z t is closed, so Y ′ is closed. Suppose ε ∈ A > . Then W ε is open, so ( W ∩ ( A × A n )) ε is dense in W ε ,so Z ε contains W ε , hence Z ε contains Y . Thus Y ⊆ Y ′ . We now prove the otherinclusion. Suppose that p ′ ∈ Y ′ . We show that p ′ ∈ Y . As Y is closed it suffices to fix ε > p ∈ Y such that k p − p ′ k < ε . We may suppose that ε ∈ A . Wehave ( ε, p ′ ) ∈ Z , so there is ( δ, q ) ∈ W ∩ ( A × A n ) such that k ( ε, p ′ ) − ( δ, q ) k < ε .By definition of W we obtain p ∈ Y such that k p − q k < δ . We have | ε − δ | < ε , so δ < ε , hence k p − q k < ε . We also have k p ′ − q k < ε , so k p ′ − p k < ε . (cid:3) Proposition 12.2.
Let R , I , and A be as in Lemma 12.1. Suppose that the inducedstructure A eliminates quantifiers. Then A is trace equivalent to R . Proof of Theorem 1.3.
We first state Corollary 12.3, which follows fromFact 1.2 and Theorem 4.17.
Corollary 12.3.
Suppose that R is an o-minimal expansion of ( R ; + , < ) . Thenthe following are equivalent:(1) R defines ( R ; + , × ) ,(2) R defines an infinite field,(3) R trace defines an infinite field. Suppose that Q is a divisible subgroup of ( R ; +) and Q is a dp-rank one expansionof ( Q ; + , < ). We first describe the o-minimal completion Q (cid:3) of Q . This completionis closely associated to the o-minimal completion of a non-valuational weakly o-minimal structure constructed by Wencel [46], see also Keren [28]. Indeed Q (cid:3) is interdefinable with the Wencel completion of Q Sh . Let N be an ℵ -saturatedelementary extension of Q . Let V = { a ∈ N : | a | < t, for some t ∈ Q, t > } m = { a ∈ N : | a | < t, for all t ∈ Q, t > } . So V and m are convex subgroups of ( N, + , < ) and we may identify V / m with R and the quotient map V → R with the usual standard part. Note that V and m are externally definable, so we consider R to be an N Sh -definable set of imaginaries.We define Q (cid:3) to be the structure induced on R by N Sh . Lemma 12.4. Q (cid:3) is o-minimal.Proof. By Fact 2.7 Th( Q ) is weakly o-minimal, so N is weakly o-minimal. ByFact 2.5 N Sh is weakly o-minimal. The quotient map O → R is monotone, and animage of a convex set under a monotone map is convex, so Q (cid:3) is weakly o-minimal.Note that every convex subset of R is an interval, so Q (cid:3) is o-minimal. (cid:3) Fact 12.5 is a special case of the results of [45].
Fact 12.5.
The structure induced on Q by Q (cid:3) admits quantifier elimination andis interdefinable with Q Sh . Corollary 12.6 follows from Fact 12.5, Proposition 12.2, and Proposition 10.3.
Corollary 12.6. Q and Q (cid:3) are trace equivalent. Laskowski and Steinhorn [33] show that if Q is o-minimal then Q is an elementarysubmodel of a unique o-minimal expansion of ( R ; + , < ). In fact, if Q is o-minimalthen Q (cid:3) is interdefinable with this elementary extension, see [45]. If Q is not o-minimal then an application of Corollary 2.11 shows that Q is not interpretable in ano-minimal expansion of an ordered group, so in particular Th( Q ) is not interpretablein Th( Q (cid:3) ). We now prove Theorem 1.3. We first restate the theorem in a slightlydifferent form. A semilinear set is an R Vec -definable set.
Theorem.
Suppose that Q is a divisible subgroup of ( R ; +) and Q is a dp-rank oneexpansion of ( Q ; + , < ) . Then the following are equivalent:(1) Th( Q ) does not trace define RCF ,(2)
Th( Q ) does not trace define an infinite field,(3) Q has near linear Zarankiewicz bounds,(4) any Q -definable X ⊆ Q n is of the form Y ∩ Q n for semilinear Y ⊆ R n .(5) Q is trace equivalent to a reduct of R Vec .(6) Q (cid:3) is a reduct of R Vec .Proof.
Corollary 12.6 shows that (6) implies (5) and Fact 12.5 shows that (6) implies(4). Fact 4.14 and Lemma 4.12 together show that (5) implies (3) and Fact 4.14shows that (4) implies (3). Proposition 4.15 shows that (3) implies (2) and (2)clearly implies (1). Suppose (1). By Corollary 12.6 Q (cid:3) cannot trace define ( R ; + , × ),so Lemma 12.4 and Fact 1.2 together yield (6). (cid:3) Corollary 12.7.
Let Q Vec be the structure induced on Q by R Vec and suppose that Q is a dp-rank one expansion of ( Q ; + , < ) . Then Th( Q ) does not trace define RCF if and only if Q is a reduct of Q Vec .Proof.
It is shown in [13] that ( R Vec , Q ) is NIP and that every ( R Vec , Q )-definablesubset of Q n is of the form X ∩ Q n for semilinear X ⊆ R n . So Q Vec is trace definablein R Vec and hence has near linear Zarankiewicz bounds. So a reduct of Q Vec cannottrace define an infinite field. Suppose Th( Q ) does not trace define RCF. Then Q (cid:3) is a reduct of R Vec , so by Fact 12.5 Q is a reduct of Q Vec . (cid:3) A Mann pair.
We describe a dp-rank one expansion Q λ of ( Q ; + , < ) whichis trace equivalent to ( R ; + , × ) and does not interpret an infinite field. Fix a realnumber λ >
0, let λ Q = { λ q : q ∈ Q } , and Q λ be the expansion of ( Q ; + , < ) byall sets { ( q , . . . , q m ) ∈ Q m : ( λ q , . . . , λ q m ) ∈ X } for semialgebraic X ⊆ R m . Notethat q λ q gives an isomorphism between Q λ and the structure induced on λ Q by( R ; + , × ). Fact 12.8.
Every ( R ; + , × , λ Q ) -definable subset of ( λ Q ) n is of the form ( λ Q ) n ∩ X for semialgebraic X ⊆ R n . So the structure induced on λ Q by ( R ; + , × ) ad-mits quantifier elimination, hence Q λ is weakly o-minimal and trace definable in ( R ; + , × ) . The first claim of Fact 12.8 is a special case of a result of van den Dries andG¨unaydin [44, Theorem 7.2], the others follow from the first.
Corollary 12.9. Q λ is trace equivalent to ( R ; + , × ) . Corollary 12.9 follows from Fact 12.8 and Proposition 12.2.
Proposition 12.10. Q λ does not interpret an infinite field. It follows from [44] that algebraic closure in Q λ agrees with algebraic closure in( Q ; +), from this one can deduce that Q λ does not define an infinite field. However,Berenstein and Vassiliev have essentially already proven Proposition 12.10, so wejust apply their results. Proof.
By Eleftheriou [9] Q λ eliminates imaginaries. So it suffices to show that Q λ does not define an infinite field. It is a special case of a theorem of Berenstein andVassiliev [2, Proposition 3.16] that Q λ is weakly one-based and a weakly one-basedtheory cannot define an infinite field by [2, Proposition 2.11]. (cid:3) The induced structure on an independent set.
We describe a weaklyo-minimal structure that does not interpret an infinite group but is trace equivalentto ( R ; + , × ). Given an o-minimal structure M we say that A ⊆ M is independent if it is independent with respect to algebraic closure in M . Fact 12.11 is due toDolich, Miller, and Steinhorn [7]. Fact 12.11.
Suppose that M is o-minimal and H is a dense independent subsetof M . Any subset of H n definable in ( M , H ) is of the form X ∩ H n for some M -definable X ⊆ M n . Corollary 12.12 follows from Fact 12.11 and Proposition 12.2.
Corollary 12.12.
Suppose that R is an o-minimal expansion of ( R ; + , < ) , H is adense independent subset of R , and H is the structure induced on H by R . Then H is trace equivalent to R . In fact if H ≺ N is ℵ -saturated then N Sh interprets R . We prove Proposition 12.13.
Proposition 12.13.
Suppose that R is an o-minimal expansion of ( R ; + , < ) , H isa dense independent subset of R , and H is the structure induced on H by R . Then H does not interpret an infinite group.Proof. A theorem of Eleftheriou [10, Theorem C] shows that H eliminates imagi-naries, so it is enough to show that H does not define an infinite group. This followsby another result of Berenstein and Vassiliev [3, Corollary 6.3] (cid:3) A p -adic example Throughout this section we fix a prime p . In this section we give an example of astructure interpretable in Q p which is trace equivalent to Q p but does not, provideda natural conjecture holds, interpret an infinite field.We let Val p : Q × p → Z be the p -adic valuation on Q p . Given a ∈ Q p and r ∈ Z we let B ( a, r ) be the ball with center a and radius r , i.e. the set of b ∈ Q p suchthat Val p ( a − b ) > r . We let B be the set of balls in Z p . We consider B to bea Q p -definable set of imaginaries. Let ≈ be the equivalence relation on Z p × N ≥ where ( a, r ) ≈ ( a ′ , r ′ ) if and only if B ( a, r ) = B ( a, r ′ ), equivalently Val p ( a − a ′ ) > r .We identify B with ( Z × N ≥ ) / ≈ and let B be the structure induced on B by Q p . Proposition 13.1. B is trace equivalent to Q p . Before proving Proposition 13.1 we show that modulo a reasonable conjecture, B does not interpret an infinite field. Conjecture 1 is a well-known and well-believed conjecture. It strengthens Pillay’s theorem [36] that if K is a p -adicallyclosed field then any infinite field definable in K is definably isomorphic to a finiteextension of K . The analogue of Conjecture 1 for ACVF is a result of Hrushovskiand Rideau [22]. Conjecture 1.
Suppose that K is a p -adically closed field. Then any infinite fieldinterpretable in K is definably isomorphic to a finite extension of K . We first prove a lemma. Given a p -adically closed field K we let B K be the set ofballs in K and B K be the structure induced on B K by K . Lemma 13.2.
Suppose that K is a p -adically closed field. Any definable function B mK → K n has finite image. Suppose that K is p -adically closed and B K interprets an infinite field L . AssumingConjecture 1 we may suppose that L = K n , so we would obtain a K -definablesurjection B mK → K n for some m , this contradicts Lemma 13.2. Proof. If f : B mK → K n has infinite image then there is a coordinate projection e : K m → K such that e ◦ f has infinite image. So we suppose m = 1. Recall thatif X is a definable subset of K then X is either finite or has interior. So it sufficesto show that the image of any definable function B mK → K has empty interior. It isenough to show that the image of any function B m → Q p has empty interior. Thisholds as B is countable and every nonempty open subset of Z p is uncountable. (cid:3) The remainder of this section is devoted to the proof of Proposition 13.1. We needto show that Th( B ) trace defines Q p . By Proposition 10.3 it suffices to produce B ≺ D such that D Sh interprets Q p . Let K be an ℵ -saturated elementary extensionof Q p of cardinality 2 ℵ . Let D ′ be the structure induced on B K by K . We let L be the language of B , recall that L contains an n -ary relation symbol R X for each Q p -definable X ⊆ B n . We naturally consider L to be a sublanguage of the languageof D ′ . We define D = D ′ ↾ L . Note that D is an ℵ -saturated elementary extensionof B .We first realize Q p as an K Sh -definable set of imaginaries. We let Val p : K × → Γ bethe p -adic valuation on K . Note that Z is the minimal non-trivial convex subgroupof Γ. By Fact 2.2 Z is externally definable. Let v : K × → Γ / Z be the compositionof Val p with the quotient Γ → Γ / Z . We equip Γ / Z with a group order by declaring a + Z ≤ b + Z when a ≤ b , so v is an externally definable valuation on K . Let W be the valuation ring of v and m W be the maximal ideal of W . So W is the set of a ∈ K such that Val p ( a ) ≥ m for some m ∈ Z and m W is the set of a ∈ K such thatVal p ( a ) > m for all m ∈ Z . It is easy to see that for every a ∈ W there is a unique a ′ ∈ Q p such that a − a ′ ∈ m W . We identify W/ m W with Q p so that the residuemap st : W → Q p is the usual standard part map. So Q p is a K Sh -definable set ofimaginaries.Lemma 13.3 is a saturation exercise. Lemma 13.3.
Suppose that X is a closed definable subset of Z np and X ′ is thesubset of W n defined by any formula defining X . Then st( X ′ ) agrees with X . Given B ∈ B K such that B = B ( a, t ) we let rad( B ) = t . As rad : B → Γ ≥ issurjective and K -definable we consider Γ ≥ to be an imaginary sort of B and rad tobe a B -definable function. Fix γ ∈ Γ such that γ > N . Let E be the set of B ∈ B K such that rad( B ) = γ , so E is D -definable.We finally show that D Sh interprets ( Z p ; + , × ). Note that for any a ∈ V and b ∈ B ( a, γ ) we have st( a ) = st( b ). So we define a surjection f : E → Z p bydeclaring f ( B ( a, γ )) = st( a ) for all a ∈ V . Let ≈ be the equivalence relation on E where B ≈ B if and only if f ( B ) = f ( B ). Note that for any B , B ∈ B K wehave B ≈ B if and only if { B ′ ∈ B K : rad( B ′ ) ∈ N , B ⊆ B ′ } = { B ′ ∈ B : rad( B ′ ) ∈ N , B ⊆ B ′ } . So ≈ is D Sh -definable. Let f : E n → Z np be given by f ( B , . . . , B n ) = ( f ( B ) , . . . , f ( B n )) for all B , . . . , B n ∈ E. Suppose that X is a Q p -definable subset of Z np . We show that f − ( X ) is D Sh -definable. As X is Q p -definable it is a boolean combination of closed Q p -definablesubsets of Z np , so we may suppose that X is closed. Let X ′ be the subset of V n defined by any formula defining X . Let Y be the set of B ∈ E such that B ∩ X ′ = ∅ and Y be the set of B ∈ E such that B ≈ B for some B ∈ Y . Observe that Y is D -definable and Y is D Sh -definable. Note that Y is the set of balls of the form B ( a, γ ) for a ∈ X ′ .We show that Y = f − ( X ). Suppose B ( a, γ ) ∈ f − ( X ). So st( a ) ∈ X . Wehave B (st( a ) , γ ) ∈ Y and B (st( a ) , γ ) ≈ B ( a, γ ), so B ( a, γ ) ∈ Y . Now supposethat B ( a, γ ) ∈ Y and fix B ( b, γ ) ∈ Y such that B ≈ B . We may suppose that b ∈ X ′ . As X is closed an application of Lemma 13.3 shows that st( b ) ∈ X . Sost( a ) = st( b ) ∈ X . So B ( a, γ ) ∈ f − ( X ).14. Geometric sorts in Q p It is shown in [20] that Q p eliminates imaginaries down to certain “geometric sorts”.We discuss one of the two families of geometric sorts. We first recall some well-known facts about the “geometric sorts” introduced in [16]. Let L be a valued fieldwith valuation ring V . A k -lattice is a free V -submodule of ( L k ; +) of rank k .Let S k ( L ) be the set of k -lattices for each k ≥
1. There is a canonical bijection S k ( L ) → Gl k ( L ) / Gl k ( V ) so we take S k ( L ) to be an L -definable set of imaginaries.We let S k be the structure induced on S k ( Q p ) by Q p .We describe the canonical injection B → S ( Q p ). Fix B ∈ B . Let R be the Z p -submodule of ( Q p ; +) generated by { } × B . It is easy to see that R is a 2-latticeand R ∩ [ { } × Q p ] = B . The inclusion Gl ( Q p ) → Gl k ( Q p ) induces an injection S ( Q p ) → S k ( Q p ) for all k ≥
2. So Proposition 14.1 follows from Proposition 13.1.
Proposition 14.1. If k ≥ then S k ( Q p ) is trace equivalent to Q p . As Gl k ( Q p ) / Gl k ( Z p ) is countable any definable function S k ( Q p ) → Q p has finiteimage. So Conjecture 1 implies that S k ( Q p ) does not interpret an infinite field.The case when k = 1 remains. It is easy to see that Val p : Q p → Z induces abijection S ( Q p ) → Z , so we identify S ( Q p ) with the Z . Recall that the structureinduced on Z by Q p is interdefinable with ( Z ; + , < ). Proposition 14.2. ( Z ; + , < ) has near linear Zarankiewicz bounds. So Th( Z ; + , < ) does not trace define an infinite field. The second claim of Proposition 14.2 follows from the first and Proposition 4.15.Fact 4.14 shows that ( R ; + , < ) has near linear Zarankiewicz bounds. Thus the firstclaim of Proposition 14.2 follows by Proposition 14.3. Proposition 14.3.
Any bipartite graph which is definable in ( Z ; + , < ) embeds intoa bipartite graph which is definable in ( R ; + , < ) . We do not know if ( Z ; +) is trace definable in ( R ; + , < ). Proof.
It is enough to consider bipartite graphs of the form ( Z m , Z n ; E ) where E is a ( Z ; + , < )-definable subset of Z m + n . We construct an ( R ; + , < )-definablebipartite graph ( V , V ; F ) and an embedding e : ( Z m , Z n ; E ) → ( V , V ; F ). Theusual quantifier elimination for Presburger arithmetic shows that there is ℓ ∈ N such that E ⊆ Z m + n is a finite union of finite intersections of sets of the followingtwo forms: (1) X ∩ Z m + n for ( R ; + , < )-definable X ⊆ R m + n , and(2) { ( a , . . . , a m + n ) ∈ Z m + n : a i ≡ r (mod ℓ ) } for r ∈ { , . . . , ℓ − } , and1 ≤ i ≤ m + n .Let A be { , . . . , ℓ − } m + n . For each a = ( a , . . . , a m + n ) ∈ A we let C a be the set of( b , . . . , b m + n ) ∈ Z m + n such that b i ≡ a i (mod ℓ ) for all i . It easily follows that foreach a ∈ A there is an ( R ; + , < )-definable E a ⊆ R m + n such that E ∩ C a = E a ∩ C a .We let A be { , . . . , ℓ − } m , A be { , . . . , ℓ − } n , and a a ∈ A be the concate-nation of a ∈ A and a ∈ A . Let V = R m × A and V = R n × A . We declare F (( b, a ) , ( b ′ , a ′ )) ⇐⇒ ( b, b ′ ) ∈ E aa ′ . So ( V , V ; F ) is ( R ; + , < )-definable. We let e : Z m → V be given by e ( b , . . . , b m ) = ( b , . . . , b m , r , . . . , r m ) where each r i isthe remainder mod ℓ of b i . We define e : Z n → V in the same way. It is easy tosee that e is an embedding. (cid:3) Conjecture 2.
An ordered abelian group cannot trace define an infinite field.
References [1] A. Basit, A. Chernikov, S. Starchenko, T. Tao, and C.-M. Tran. Zarankiewicz’s problem forsemilinear hypergraphs, 2020,arXiv:2009.02922.[2] A. Berenstein and E. Vassiliev. Weakly one-based geometric theories.
J. Symbolic Logic ,77(2):392–422, 2012.[3] A. Berenstein and E. Vassiliev. Geometric structures with a dense independent subset.
SelectaMath. (N.S.) , 22(1):191–225, 2016.[4] G. Cherlin, L. Harrington, and A. H. Lachlan. ℵ -categorical, ℵ -stable structures. Ann. PureAppl. Logic , 28(2):103–135, 1985.[5] A. Chernikov, D. Palacin, and K. Takeuchi. On n -dependence. Notre Dame J. Form. Log. ,60(2):195–214, 2019.[6] A. Chernikov and S. Starchenko. Regularity lemma for distal structures.
Journal of the Eu-ropean Mathematical Society , July 2015.[7] A. Dolich, C. Miller, and C. Steinhorn. Expansions of o-minimal structures by dense inde-pendent sets.
Ann. Pure Appl. Logic , 167(8):684–706, 2016.[8] J.-L. Duret. Les corps faiblement alg´ebriquement clos non s´eparablement clos ont la propri´et´ed’ind´ependence. In
Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979) ,volume 834 of
Lecture Notes in Math. , pages 136–162. Springer, Berlin-New York, 1980.[9] P. Eleftheriou. Small sets in mann pairs. arXiv:1812.07970 , 2018.[10] P. E. Eleftheriou. Small sets in dense pairs.
Israel J. Math. , 233(1):1–27, 2019.[11] P. Erd˝os. On extremal problems of graphs and generalized graphs.
Israel J. Math. , 2:183–190,1964.[12] J. Flenner and V. Guingona. Convexly orderable groups and valued fields.
J. Symb. Log. ,79(1):154–170, 2014.[13] A. B. Gorman, P. Hieronymi, and E. Kaplan. Pairs of theories satisfying a mordell-langcondition, 2018.[14] V. Guingona and C. D. Hill. On a common generalization of Shelah’s 2-rank, dp-rank, ando-minimal dimension.
Ann. Pure Appl. Logic , 166(4):502–525, 2015.[15] V. Guingona, C. D. Hill, and L. Scow. Characterizing model-theoretic dividing lines viacollapse of generalized indiscernibles.
Ann. Pure Appl. Logic , 168(5):1091–1111, 2017.[16] D. Haskell, E. Hrushovski, and D. Macpherson. Definable sets in algebraically closed valuedfields: elimination of imaginaries.
Journal fur die reine und angewandte Mathematik (CrellesJournal) , 2006(597), Jan. 2006.[17] N. Hempel and A. Chernikov. On n-dependent groups and fields ii, 2020, arxiv:1912.02385.[18] W. Hodges.
Model theory , volume 42 of
Encyclopedia of mathematics and its applications .Cambridge University Press, 1993.[19] E. Hrushovski. Totally categorical structures.
Trans. Amer. Math. Soc. , 313(1):131–159, 1989.[20] E. Hrushovski, B. Martin, and S. Rideau. Definable equivalence relations and zeta functionsof groups (with an appendix by raf cluckers).
Journal of the European Mathematical Society ,20(10):2467–2537, July 2018. [21] E. Hrushovski and A. Pillay. On NIP and invariant measures. J. Eur. Math. Soc. (JEMS) ,13(4):1005–1061, 2011.[22] E. Hrushovski and S. Rideau-Kikuchi. Valued fields, metastable groups.
Selecta Mathematica ,25(3), July 2019.[23] F. Jahnke. When does nip transfer from fields to henselian expansions? arXiv:1607.02953: ,2019.[24] W. Johnson. On dp-minimal fields. arXiv:1507.02745 , 2015.[25] W. Johnson. Dp-finite fields vi: the dp-finite shelah conjecture. arXiv:2005.13989 , 2020.[26] I. Kaplan, A. Onshuus, and A. Usvyatsov. Additivity of the dp-rank.
Trans. Amer. Math.Soc. , 365(11):5783–5804, 2013.[27] I. Kaplan, T. Scanlon, and F. O. Wagner. Artin-schreier extensions in NIP and simple fields.
Israel Journal of Mathematics , 185(1):141–153, Sept. 2011.[28] G. Keren. Definable compactness in weakly o-minimal structures. Master’s thesis, Ben GurionUniversity of the Negev, 2014.[29] A. Kruckman and N. Ramsey. Generic expansion and Skolemization in NSOP theories. Ann.Pure Appl. Logic , 169(8):755–774, 2018.[30] A. H. Lachlan. Structures coordinatized by indiscernible sets. volume 34, pages 245–273.1987. Stability in model theory (Trento, 1984).[31] A. H. Lachlan and R. E. Woodrow. Countable ultrahomogeneous undirected graphs.
Trans.Amer. Math. Soc. , 262(1):51–94, 1980.[32] M. Laskowski and S. Shelah. Karp complexity and classes with the independence property.
Annals of Pure and Applied Logic , 120(1-3):263–283, Apr. 2003.[33] M. C. Laskowski and C. Steinhorn. On o-minimal expansions of Archimedean ordered groups.
J. Symbolic Logic , 60(3):817–831, 1995.[34] A. Macintyre. On ω -categorical theories of fields. Fund. Math. , 71(1):1–25. (errata insert),1971.[35] Y. Peterzil and S. Starchenko. A trichotomy theorem for o-minimal structures.
Proc. LondonMath. Soc. (3) , 77(3):481–523, 1998.[36] A. Pillay. On fields definable in Q p . Arch. Math. Logic , 29(1):1–7, 1989.[37] A. Pillay.
Geometric stability theory , volume 32 of
Oxford Logic Guides . The Clarendon Press,Oxford University Press, New York, 1996. Oxford Science Publications.[38] B. Poizat. `a propos de groupes stables. In
Logic colloquium ’85 (Orsay, 1985) , volume 122of
Stud. Logic Found. Math. , pages 245–265. North-Holland, Amsterdam, 1987.[39] J. Ramakrishnan. Definable linear orders definably embed into lexicographic orders in o-minimal structures.
Proc. Amer. Math. Soc. , 141(5):1809–1819, 2013.[40] S. Shelah. Dependent first order theories, continued.
Israel J. Math. , 173:1–60, 2009.[41] P. Simon. On dp-minimal ordered structures.
J. Symbolic Logic , 76(2):448–460, 2011.[42] P. Simon. Linear orders in NIP theories. arXiv:1807.07949 , 2018.[43] P. Simon and E. Walsberg. Dp and other minimalities.
Preprint , arXiv:1909.05399, 2019.[44] L. van den Dries and A. G¨unaydı n. The fields of real and complex numbers with a smallmultiplicative group.
Proc. London Math. Soc. (3) , 93(1):43–81, 2006.[45] E. Walsberg. Externally definable quotients and nip expansions of the real ordered additivegroup, 2019, arXiv:1910.10572.[46] R. Wencel. Weakly o-minimal nonvaluational structures.
Ann. Pure Appl. Logic , 154(3):139–162, 2008.[47] P. M. Winkler. Model-completeness and Skolem expansions. pages 408–463. Lecture Notes inMath., Vol. 498, 1975.
Department of Mathematics, Statistics, and Computer Science, Department of Math-ematics, University of California, Irvine, 340 Rowland Hall (Bldg.
Email address : [email protected] URL ::