Pathological examples of structures with o-minimal open core
aa r X i v : . [ m a t h . L O ] J a n PATHOLOGICAL EXAMPLES OF STRUCTURES WITHO-MINIMAL OPEN CORE
ALEXI BLOCK GORMAN, ERIN CAULFIELD, AND PHILIPP HIERONYMI
Abstract.
This paper answers several open questions around structures witho-minimal open core. We construct an expansion of an o-minimal structure R by a unary predicate such that its open core is a proper o-minimal expansionof R . We give an example of a structure that has an o-minimal open core andthe exchange property, yet defines a function whose graph is dense. Finally,we produce an example of a structure that has an o-minimal open core anddefinable Skolem functions, but is not o-minimal. Introduction
Introduced by Miller and Speissegger [5] the notion of an open core has becomea mainstay of the model-theoretic study of ordered structures. However, there arestill many rather basic questions, in particular about structures with o-minimalopen cores, that have remained unanswered. In this paper, we are able to settlesome of the questions raised by Dolich, Miller and Steinhorn [2, 3].Throughout this paper, R denotes a fixed, but arbitrary expansion of a dense linearorder ( R, < ) without endpoints. We now recall several definitions from the afore-mentioned papers. We denote by R ◦ the structure ( R, ( U )), where U ranges overthe open sets of all arities definable in R , and call this structure the open coreof R .Given two structures S and S with the same universe S , we say S and S are interdefinable (short: S = df S ) if S and S define the same sets. For a giventheory T extending the theory of dense linear orders, we say that a theory T ′ is an open core of T if for every M | = T there exists M ′ | = T ′ such that M ◦ = df M ′ . Question 1 ([2, p. 1408]) . If S ⊆ R and ( R , S ) ◦ is o-minimal, is ( R , S ) ◦ = df R ◦ ? We give a negative answer to this question by constructing an expansion of thereal field by a single unary predicate whose open core is o-minimal, but defines anirrational power function. It is clear from the construction in Section 2 that thereare similar examples of expansions of the real ordered additive group that do notdefine an ordered field.We say R is definably complete (short: R | = DC) if every definable unary set hasboth a supremum and an infimum in R ∪ {±∞} . We denote by dcl R the definableclosure operator in R , and often drop the subscript R . We say that R has the Date : January 8, 2021.This is a preprint version. Later versions might contain significant changes. exchange property (short:
R | = EP) if b ∈ dcl( S ∪{ a } ) for all S ⊆ R and a, b ∈ R such that a ∈ dcl( S ∪ { b } ) \ dcl( S ). For a theory T , we say T has the exchangeproperty (short: T | = EP) if every model of T has the exchange property. We write R | = NIP if its theory does not have the independence property as introduced byShelah [6]. We refer the reader to Simon [8] for a modern treatment of NIP andrelated model-theoretic tameness notions.
Question 2 ([2, p. 1409]) . If R | = DC + EP + NIP and expands an ordered group,is R o-minimal? By [2, p. 1374] we know that R has o-minimal open core if R | = DC + EP andexpands an ordered group. Thus Question 2 asks whether there is a combinator-ical model-theoretic tameness condition that can be added to force o-minimality.Again, we give a negative answer to this question. We construct a counterexampleas follows: Let Q ( t ) be the field of rational functions in a single variable t . Weconsider an expansion R t of the ordered real additive group ( R , <, + , ,
1) into a Q ( t )-vector space such that for all c ∈ Q ( t ) \ Q , the graph of the function x cx is dense in R . We show that R t | = DC + EP + NIP, but is not o-minimal.The structure R t has infinite dp-rank. By Simon [7], if R | = DC, expands an or-dered group and has dp-rank 1, then R is o-minimal. However, we do not knowwhether R is o-minimal if R | = DC + EC, expands an ordered group and has finitedp-rank.In addition to showing that R t | = DC + EP + NIP, we prove that its open core isinterdefinable with its o-minimal reduct ( R , <, + , , x cx isdense for c ∈ Q ( t ) \ Q , the theory of R t provides a negative answer to the followingquestion. Question 3 ([3, p. 705]) . Let T be a complete o-minimal extension of the theory ofdensely ordered groups. If e T is any theory (in any language) having T as an opencore, and some model of e T defines a somewhere dense graph, must EP fail for e T ? Our counterexample R t does not expand a field and we don’t know whether Ques-tion 2 (or Question 3) has a positive answer if we require R (or T ) to expand anordered field.We say that R has definable Skolem functions if for every definable set A ⊆ R m × R n there is a definable function f : R m → R n such that ( a, f ( a )) ∈ A whenever a ∈ A and there exists b ∈ R n with ( a, b ) ∈ A . Every o-minimal expansion of anordered group with a distinguished positive element has definable Skolem functions,but all documented examples of non-o-minimal structures with o-minimal core donot. Question 4 ([2, p. 1409]) . If R has definable Skolem functions and R ◦ is o-minimal, is R o-minimal? The answer is again negative. We say R satisfies uniform finitness (short: R | =UF) if for every m, n ∈ N and every A ⊆ R m + n definable in R there exists N ∈ N such that for every a ∈ R m the set { b ∈ R n : ( a, b ) ∈ A } is either infinite or containsat most N elements. By [2, Theorem A], if R | = DC + UF and expands an orderedgroup, then R ◦ is o-minimal. Using a construction due to Winkler [9] and following ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 3 a strategy of Kruckman and Ramsey [4], we establish that if
R | = DC + UF, then R has an expansion S such that S has definable Skolem functions and satisfies S ◦ = df R ◦ . Thus if R also expands an ordered group, then R ◦ is o-minimal and sois S ◦ . Acknowledgements.
A.B.G. was supported by the National Science FoundationGraduate Research Fellowship Program under Grant No. DGE – 1746047. P.H. waspartially supported by NSF grant DMS-1654725. We thank Pantelis Eleftheriou forhelpful conversations on the topic of this paper, in particular for bringing Question1 to the attention of P.H. at a conference at the Bedlewo Conference Center. Wethank Chris Miller for helpful comments on a draft of this paper.
Notation.
We will use m, n for natural numbers and κ for a cardinal. Let L bea language and T an L -theory. Let M | = T . We follow the usual convention todenote the universe of M by M . In this situation, L -definable means L -definablewith parameters. Let b be a tuple of elements in M , and let A ⊆ M . We writetp L ( b | A ) for the L -type of b over A . If N is another model of T and h is anembedding of a substructure of M containing A into N , then h tp( b | A ) is the typecontaining all formulas of the form ϕ ( x, h ( a )) where ϕ ( x, a ) ∈ tp( b | A ).2. Question 1
Let R be the real ordered field ( R , <, + , · ) and let R exp be the expansion of thereal field by the exponential function exp. Let I ⊆ R be a dense dcl R exp -independentset. Let τ ∈ I be such that τ >
1. Set J := [ a ∈ I \{ τ } {| a | , | a | τ , | a | + | a | τ } . By [3, 2.25] the open core of ( R exp , I ) ◦ is interdefinable with R exp . Since ( R , J ) isa reduct of ( R exp , I ), we have that ( R , J ) ◦ is a reduct of ( R exp , I ) ◦ . As the latterstructure is o-minimal, we have that ( R , J ) ◦ is o-minimal as well. In order to showthat Question 1 has a negative answer, it is left to show that ( R , J ) ◦ defines a setnot definable in R . Since R only defines raising to rational powers, it suffices toprove the definability of x x τ on an unbounded interval. Lemma 2.1.
Let u , u , u ∈ J such that < u < u and u + u = u . Thenthere is a ∈ I \ { τ } such that u = | a | and u = | a | τ .Proof. For a ∈ I \ { τ } observe that | a | , | a | τ and | a | + | a | τ are interdefinable in R exp over τ . Since u + u = u , we have u , u , u are dcl R exp -dependent. Because I isdcl R exp -independent, there are a ∈ I \ { τ } and i, j ∈ { , , } such that u i , u j ∈ {| a | , | a | τ , | a | + | a | τ } . Let ℓ ∈ { , , } such that ℓ = i and ℓ = j . Note u ℓ is dcl R exp -dependent over u i and u j . Thus u ℓ ∈ {| a | , | a | τ , | a | + | a | τ } . Since | a | >
0, we obtain from u + u = u that u , u ∈ {| a | , | a | τ } . Since 1 < u < u and τ >
0, we have that u = | a | and u = | a | τ . (cid:3) Proposition 2.2.
The graph of x x τ on R ≥ is definable in ( R , J ) ◦ . ALEXI BLOCK GORMAN, ERIN CAULFIELD, AND PHILIPP HIERONYMI
Proof.
By Lemma 2.1 the structure ( R , J ) defines { ( | a | , | a | τ ) : | a | > , a ∈ I \ { τ }} . Since I is dense in R , the closure of this set is the graph of x x τ on R ≥ , andhence definable in ( R , J ) ◦ . (cid:3) We conclude that ( R , J ) ◦ is a proper expansion of R .3. Questions 2 and 3
In this section we give negative answers to Questions 2 and 3. Let Q ( t ) bethe field of rational functions in the variable t . We expand ( R , <, + , ,
1) to a Q ( t )-vector space such that for each non-constant q ( t ) ∈ Q ( t ) the graph of multiplicationby q ( t ) is dense.We now construct such a Q ( t )-vector space structure on R . Let be the multiplica-tive identity of Q ( t ). We fix a dense basis B of R as a Q -vector space, and a basis I of Q ( t ) as a Q -vector space such that ∈ I . We choose a sequence of functions { f f γ : I → B} γ ∈ ℵ such that B = [ γ ∈ ℵ f f γ ( I )and for all γ ∈ ℵ : • f f γ is injective, • for all η ∈ ℵ with η = γ , e f η ( I ) ∩ f f γ ( I ) = ∅ , • for all open intervals J , . . . , J m ⊆ R open intervals and all pairwise distinct p ( t ) , . . . , p m ( t ) ∈ I there exists γ ∈ ℵ such that f f γ ( p ( t )) ∈ J , . . . , f f γ ( p m ( t )) ∈ J m . Since the order topology on the real line has a countable base, it is easy to see thatsuch a sequence of functions exists. For each γ ∈ ℵ , f f γ is defined on the basis I of Q ( t ). Therefore, we can extend each f f γ : I → B to a Q -linear map f γ : Q ( t ) → R . Lemma 3.1.
Let a ∈ R . Then there are unique γ , . . . , γ n ∈ ℵ and p ( t ) , . . . , p n ( t ) ∈ Q ( t ) such that a = f γ ( p ( t )) + · · · + f γ n ( p n ( t )) . Proof.
Since B is a basis of R as a Q -vector space, there are unique b , . . . , b n ∈ B and u , . . . , u n ∈ Q such that a = u b + · · · + u n b n . By the above construction,there are unique γ , . . . , γ n ∈ ℵ and q ( t ) , . . . , q n ( t ) ∈ I such that b i = f γ i ( q i ( t ))for i = 1 , . . . , n . Then by Q -linearity of the f γ i ’s a = u b + · · · + u n b n = u f γ ( q ( t )) + · · · + u n f γ n ( q n ( t ))= f γ ( u q ( t )) + · · · + f γ n ( u n q n ( t )) . Set p i := u i q i ( t ) for i = 1 , . . . , n . (cid:3) We now introduce a Q -linear map λ : Q ( t ) × R → R . Let q ( t ) ∈ Q ( t ) and a ∈ R .By Lemma 3.1 there are unique γ , . . . , γ n ∈ ℵ and p ( t ) , . . . , p n ( t ) ∈ Q ( t ) suchthat a = f γ ( p ( t )) + · · · + f γ n ( p n ( t )) . ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 5
We define λ ( q ( t ) , a ) := f γ ( q ( t ) · p ( t )) + · · · + f γ n ( q ( t ) · p n ( t )) . By Lemma 3.1, the function λ is well-defined. For q ( t ) ∈ Q ( t ), we write λ q ( t ) forthe map taking a ∈ R to λ ( q ( t ) , a ) . Proposition 3.2.
The additive group ( R , +) with λ as scalar multiplication is an Q ( t ) -vector space.Proof. We only verify the following vector spaces axioms: for all a ∈ R and for all q ( t ) , q ( t ) ∈ Q ( t ). λ q ( t ) · q ( t ) ( a ) = λ q ( t ) (cid:0) λ q ( t ) ( a ) (cid:1) . The other axioms can be checked using similar arguments and we leave this to thereader.Let a ∈ R and let q ( t ) , q ( t ) ∈ Q ( t ). By Lemma 3.1 there are unique γ , . . . , γ n ∈ ℵ and p ( t ) , . . . , p n ( t ) ∈ Q ( t ) such that a = f γ ( p ( t )) + · · · + f γ n ( p n ( t )) . We obtain λ q ( t ) ( λ q ( t ) ( a )) = λ q ( t ) (cid:0) λ q ( t ) (cid:0) n X i =1 f γ i ( p i ( t )) (cid:1) = λ q ( t ) ( n X i =1 f γ i ( q ( t ) · p i ( t )))= n X i =1 f γ i ( q ( t ) · ( q ( t ) · p i ( t )))= λ q ( t ) · q ( t ) (cid:0) n X i =1 f γ i ( p i ( t )) (cid:1) = λ q ( t ) · q ( t ) ( a ) . (cid:3) Let L be the language of ( R , <, + , , T be its theory; that is the the-ory of ordered divisible abelian groups with a distinguished positive element. It iswell-known that T has quantifier-elimination and is o-minimal. We will use variousconsequences of this fact throughout this section. Most noteworthy: when M | = T , X ⊆ M n is L -definable over A and there is b = ( b , . . . , b n ) ∈ X such that b , . . . , b n are Q -linearly independent over A , then X has interior.Let R t = ( R , <, + , , , ( λ q ( t ) ) q ( t ) ∈ Q ( t ) ) be the expansion of ( R , <, + , ,
1) by func-tion symbols for λ q ( t ) where q ( t ) ∈ Q ( t ). We denote the language of R t by L t .3.1. Density.
Let p = ( p ( t ) , . . . , p n ( t )) ∈ Q ( t ) n . Let λ p : R → R n denote thefunction from R that maps a to ( λ p ( t ) ( a ) , . . . λ p n ( t ) ( a )). The main goal of thissubsection is to show the density of the image of λ p when the coordinates of p are Q -linearly independent. Lemma 3.3.
Let p = ( p ( t ) , . . . , p n ( t )) ∈ I be such that p i ( t ) = p j ( t ) for i = j .Then the image of λ p is dense in R n . ALEXI BLOCK GORMAN, ERIN CAULFIELD, AND PHILIPP HIERONYMI
Proof.
Let J , . . . , J n be open intervals in R . Since p ( t ) , . . . , p n ( t ) are distinctelements of I , there exists γ ∈ ℵ such that f γ ( p ( t )) ∈ J , . . . , f γ ( p n ( t )) ∈ J n . For each i ∈ { , . . . , n } , we have λ p i ( t ) ( f γ ( )) = f γ ( p i ( t ))by definition of λ p i ( t ) . Therefore, (cid:0) λ p ( t ) ( f γ ( )) , . . . , λ p n ( t ) ( f γ ( )) (cid:1) ∈ J × J × . . . × J n . (cid:3) Proposition 3.4.
Let q = ( q ( t ) , . . . , q m ( t )) ∈ Q ( t ) m be such that q ( t ) , . . . , q m ( t ) are Q -linearly independent. Then the image of λ q is dense in R n .Proof. Let n ∈ N , let p ( t ) , . . . , p n ( t ) ∈ I be distinct non-constant, and let A =( u i,j ) i =1 ,...,m,j =0 ,...,n be an m × ( n + 1) matrix with rational entries such that q ( t ) = u , + u , p ( t ) + . . . + u ,n p n ( t ) q ( t ) = u , + u , p ( t ) + . . . + u ,n p n ( t )... q m ( t ) = u m, + u m, p ( t ) + . . . + u m,n p n ( t ) . By definition of λ , λ p ( t ) , . . . , λ p n ( t ) , we have for each i ∈ { , . . . , m } λ q i ( t ) ( x ) = u i, λ ( x ) + u i, λ p ( t ) ( x ) + . . . + u i,n λ p n ( t ) ( x ) . Therefore, Aλ ( ,p ( t ) ,...,p n ( t )) = λ ( q ( t ) ,...,q m ( t )) . Since q ( t ) , . . . , q m ( t ) are Q -linearly independent, the matrix A has rank m . Thusmultiplication by A is a surjective map from R n to R m . Since matrix multi-plication is continuous and continuous surjections preserve density, the image of Aλ ( ,p ( t ) ,...,p n ( t )) is dense in R m by Lemma 3.3. (cid:3) Axiomatization and QE.
In this subsection, we will find an axiomatiza-tion of R t and show that this theory has quantifier elimination. Indeed, we willprove that the following subtheory of the L t -theory of R t already has quantifier-elimination. Definition 3.5.
Let T t be the L t -theory extending T by axiom schemata statingthat for every model M = ( M, <, + , , , ( λ q ( t ) ) q ( t ) ∈ Q ( t ) ) | = T t (T1) ( M, + , , ( λ q ( t ) ) q ( t ) ∈ Q ( t ) ) is a Q ( t ) -vector space. (T2) If q ( t ) , . . . , q m ( t ) ∈ Q ( t ) are Q -linearly independent, then the image of the λ ( q ( t ) ,...,q m ( t )) is dense in M m . By Proposition 3.2 and Proposition 3.4 we know that R t | = T t . Let M | = T t .We observe that by (T1) the L t -substructures of M are precisely the Q ( t )-linearsubspaces of M containing 1. Lemma 3.6.
Let
M | = T t and let A be an L t -substructure of M . Let b ∈ M \ A andlet p ( t ) , . . . , p n ( t ) ∈ Q ( t ) be Q -linearly independent. Then λ p ( t ) ( b ) , . . . , λ p n ( t ) ( b ) are Q -linearly independent over A . ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 7
Proof.
Since A is a Q ( t )-linear subspace of M , we know that λ q ( t ) ( b ) / ∈ A for allnon-zero q ( t ) ∈ Q ( t ). Let u , . . . , u n ∈ Q . Since M is a Q ( t )-vector space, u λ p ( t ) ( b ) + · · · + u n λ p n ( t ) ( b ) = λ u p ( t )+ ··· + u n p n ( t ) ( b ) . Because p ( t ) , . . . , p n ( t ) ∈ Q ( t ) are Q -linearly independent and b / ∈ A , we get that u λ p ( t ) ( b ) + · · · + u n λ p n ( t ) ( b ) ∈ A ⇒ u = · · · = u n = 0 . Thus λ p ( t ) ( b ) , . . . , λ p n ( t ) ( b ) are Q -linearly independent of A . (cid:3) Proposition 3.7.
The theory T t has quantifier elimination.Proof. Let M , N | = T t be such that N is |M| + -saturated. Let A ⊆ M be a sub-structure that embeds into N via the embedding h : A ֒ → N . Let b ∈ M \ A .To prove quantifier elimination of T t , it is enough to show that the embedding h extends to an embedding of the L t -substructure generated by A and b .Consider the tuple ( λ p ( t ) ( b )) p ( t ) ∈ I . We first find c ∈ N such that(1) h tp L (cid:0) ( λ p ( t ) ( b )) i ∈ I | A (cid:1) = tp L (cid:0) ( λ p ( t ) ( c )) i ∈ I | h ( A ) (cid:1) By saturation of N it is enough to show that for every • pairwise distinct p ( t ) , . . . , p n ( t ) ∈ I , • L -formula ψ ( x, y ) and a ∈ A | y | such that M | = ψ (cid:0) λ p ( t ) ( b ) , . . . , λ p n ( t ) ( b ) , a (cid:1) , there is c ∈ N \ h ( A ) such that N | = ψ (cid:0) λ p ( t ) ( c ) , . . . , λ p n ( t ) ( c ) , h ( a ) (cid:1) . Because I is a Q -linear basis of Q ( t ), the sequence ( λ p ( t ) ( b )) i ∈ I is Q -linear indepen-dent over A by Lemma 3.6. Thus the set { d ∈ N n : N | = ψ ( d, h ( a )) } has interior. The existence of c now follows from (T2) and saturation of N .Let c ∈ N be such that (1) holds. Let X be the Q -linear subspace of M generated by( λ p ( t ) ( b )) i ∈ I and A . Let Y be the Q -linear subspace of N generated by ( λ p ( t ) ( c )) i ∈ I and h ( A ). Observe that X is the Q ( t )-subspace of M generated by b and A , and Y isthe Q ( t )-subspace of N generated by c and h ( A ). Hence X and Y are L t -structuresof M and N respectively. Since c satisfies (1), there is an L -isomorphism h ′ : X → C extending h and mapping λ p ( t ) ( b ) to λ p ( t ) ( c ) for each p ( t ) ∈ I . It follows easily thatthis h ′ is Q ( t )-linear and hence an L t -isomorphism extending h . (cid:3) Corollary 3.8.
Let M , N | = T t , let A ⊆ M be an L t -substructure such that h : A ֒ → N is an L t -embedding. Let b ∈ M \ A and c ∈ N \ h ( A ) such that h tp L ( (cid:0) λ p ( t ) ( b ) (cid:1) p ( t ) ∈ I | A ) = tp L ( (cid:0) λ p ( t ) ( c ) (cid:1) p ( t ) ∈ I | h ( A )) . Then tp L t ( b | A ) = tp L t ( c | h ( A )) .Proof. Let X be the Q -linear subspace of M generated by ( λ p ( t ) ( b )) i ∈ I and A , andlet Y be the Q -linear subspace of N generated by ( λ p ( t ) ( c )) i ∈ I and h ( A ). It iseasy to check that X and Y are L t -substructures of M and N respectively. By ourassumption on b and c , the embedding h extends on an L -isomorphism h ′ between X and Y mapping λ p ( t ) ( b ) to λ p ( t ) ( c ) for each p ( t ) ∈ I . Since h is an L t -embedding, it ALEXI BLOCK GORMAN, ERIN CAULFIELD, AND PHILIPP HIERONYMI follows easily that h ′ is an L t -isomorphism between X to Y . Since T t has quantifierelimination, we get that tp L t ( b | A ) = tp L t ( c | h ( A )). (cid:3) Proposition 3.9.
The theory of R t is axiomatized by the theory T t in conjunctionwith the axiom scheme that specifies tp L ( (cid:0) λ p ( t ) (1) (cid:1) p ( t ) ∈ I ) .Proof. Let T ∗ t be the theory described in the statement. Since R t | = T t , we imme-diately get that R t | = T ∗ t . It is left to show that T ∗ t is complete. Let M and N bemodel of T ∗ t of size κ > ℵ . By Corollary 3.8, 1 M and 1 N satisfy the same L t -type.Thus there is an L t -isomorphism h mapping the L t -substructure of M generatedby 1 M to the L t -substructure of N generated by 1 N . By the proof of Proposition3.7 this isomorphism h extends to an L t -isomorphism between M and N . (cid:3) Exchange property.
In this subsection we establish that every model of T t has the exchange property. We will do so by showing that the definable closure insuch a model is equal to the Q ( t )-linear span. Lemma 3.10.
Let
M | = T t and let A ⊆ M be an L t -substructure. Then A isdefinably closed.Proof. Without loss of generality, we can assume that M is | A | + -saturated. Let b ∈ M \ A . It is enough to show that there exists c ∈ M such that b = c andtp L t ( b | A ) = tp L t ( c | A ). By Corollary 3.8 it is sufficient to find c ∈ M such that b = c and tp L ( (cid:0) λ p ( t ) ( b ) (cid:1) p ( t ) ∈ I | A ) = tp L ( (cid:0) λ p ( t ) ( c ) (cid:1) p ( t ) ∈ I | A ) . Let ϕ ( x, y ) be an L -formula, p ( t ) , . . . , p m ( t ) ∈ I and a ∈ A n such that M | = ϕ ( λ p ( t ) ( b ) , . . . , λ p m ( t ) ( b ) , a ) . By saturation of M , we only need to find c ∈ M such that c = b and M | = ϕ ( λ p ( t ) ( b ) , . . . , λ p m ( t ) ( b ) , a ). By Lemma 3.6, (cid:0) λ p ( t ) ( b ) (cid:1) p ( t ) ∈ I is Q -linear indepen-dent over A . Thus the set X := { d ∈ M m : M | = ϕ ( d, a ) } has interior. By axiom (T2) the intersection { ( λ p ( t ) ( c ) , . . . , λ p m ( t ) ( c )) : c ∈ M } ∩ X is dense in X . (cid:3) Corollary 3.11.
Let
M | = T t and let Z ⊆ M . Then the L t -definable closure of Z is the Q ( t ) -subspace of M generated by Z .Proof. By Lemma 3.10 the definable closure of Z is the L t -substructure generatedby Z . However, the latter is just the Q ( t )-subspace of M generated by Z . (cid:3) The exchange property for T t follows immediately from Corollary 3.11 and theclassical Steinitz exchange lemma for vector spaces. Proposition 3.12.
The theory T t has the exchange property. ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 9
Open core.
Let
M | = T t . Then by Axiom (T2) it defines functions from M to M whose graph is dense. We already know that M has EP by Proposition 3.12.To give a negative answer to Question 3, it is left to show that every open subsetof M n definable in M is already definable in the reduct ( M, <, + , , Theorem 3.13.
The theory T is an open core of T t .Proof. Let
M | = T t . We prove that every open set is L -definable. Without loss ofgenerality, we can assume that M is ℵ -saturated. Let C be a finite subset. UsingBoxall and Hieronymi [1, Corollary 3.1], we will show that for every n ∈ N andevery subset of M n that is L t -definable over C , is also L -definable over C . Let n ∈ N and p ( t ) , . . . , p n − ( t ) ∈ I be distinct and non-constant. We define D := { (cid:0) a, λ p ( t ) ( a ) , . . . , λ p n − ( t ) ( a ) (cid:1) : a / ∈ dcl L t ( C ) } . From (T2) and saturation of M , it follows easily that D is dense in M n . ThusCondition (1) of [1, Corollary 3.1] is satisfied. Condition (3) of [1, Corollary 3.1]holds by Corollary 3.8. It is only left to establish Condition (2).Let b ∈ D and a / ∈ dcl L t ( C ) be such that b = (cid:0) a, λ p ( t ) ( a ) , . . . , λ p n − ( t ) ( a ) (cid:1) . Let U ⊆ M n be open and suppose that tp L ( b | C ) is realized in U . We need to show thattp L ( b | C ) is realized in U ∩ D . By Lemma 3.6 we know that the coordinates of b are Q -linearly independent over dcl L t ( C ). Thus the set of realizations of tp L ( b | C )is open, and so is its intersection with the open set U . Denote this intersectionby V . By (T2) and ℵ -saturation of M , we find a ′ / ∈ dcl L t ( C ) such that b ′ = (cid:0) a ′ , λ p ( t ) ( a ′ ) , . . . , λ p n − ( t ) ( a ′ ) (cid:1) . Now b ′ is the desired realization of tp L ( b | C ) in U ∩ D . (cid:3) By Theorem 3.13 every model of T t has o-minimal open core and thus is definablycomplete.3.5. Neostability results.
We will now show that T t is NIP, but not strong. Weuse an equivalent definition of the independence property in the theorem below,namely that in a monster model M of T t there is no formula ϕ ( x, y ) and no element a ∈ M such that for some indiscernible sequence ( b i ) i<ω of tuples in M | y | we have M | = ϕ ( a, b i ) if and only if i < ω is even . For a proof that this is equivalent to the classic definition of NIP, see [8].
Theorem 3.14.
Every completion of the theory T t has NIP .Proof.
We let
M | = T t be a monster model of T t . We suppose for a contradictionthat there is an L t -formula ϕ ( x, y ) along with an element a ∈ M and indiscerniblesequence ( b i ) i<ω of elements in M | y | that witnesses IP, i.e. M | = ϕ ( a, b i ) preciselyif i is even. Let | y | = n and for each i < ω we denote the j -coordinate of b i by b i,j . By quantifier elimination in the language L t , we can assume that the formula ϕ ( a, y ) is equal to a boolean combination of formulas of the form(1) a − P nj =1 λ q j ( t ) ( y j ) = 0(2) a − P nj =1 λ q j ( t ) ( y j ) > q , . . . , q n ( t ) ∈ Q ( t ). Since NIP is preserved under boolean combinations,we can assume that ϕ is of the form (1) or (2). For ease of notation, let f ( b i ) = P nj =1 λ q j ( t ) ( b i,j ) for each i < ω . Suppose that ϕ is of form (1). We suppose without loss of generality that a − f ( b i ) =0 holds if and only if i < ω is odd. Then we have that f ( b ) = f ( b ), but f ( b ) = f ( b ). Thus we conclude tp L t ( b b ) = tp L t ( b b ), contradicting indis-cernability.Now assume that ϕ is of the form (2). Without loss of generality assume that a − f ( b i ) > i < ω is odd. Then for all i < ω , we have a < f ( b i )and a > f ( b i +1 ) . However, this means that f ( b ) < f ( b ) and f ( b ) > f ( b ). Sowe again obtain tp L t ( b b ) = tp L t ( b b ), contradicting indiscernability. (cid:3) Thus R t | = DC + EP + NIP, but R t is not o-minimal. This gives a negative answerto Question 2. Proposition 3.15.
No completion of the theory T t is strong.Proof. Fix a family ( q j ( t )) j ∈ N of distinct elements of I . Consider the family of L t -formulas given by ( λ q j ( t ) ( x ) ∈ ( a k , b k )) j,k ∈ N such that • ( a k , b k ) ∩ ( a ℓ , b ℓ ) = ∅ , for all ℓ = k ∈ N , and • the tuples ( a k , b k ) k ∈ N form an indiscernible sequence.In the array that corresponds to varying j ∈ N along the rows and the k ∈ N alongthe columns, it is easy to see that formulas in the same row are pairwise inconsistent.However, for every path ( λ q γ ( k ) ( t ) ( x ) ∈ ( a γ ( k ) , b γ ( k ) )) k ∈ N , every finite subset of theseformulas are consistent by our axiom scheme (T2). So by compactness, every paththrough the entire array is consistent. (cid:3) Question 4
Let T be a theory extending the theory of dense linear orders without endpointsin an language L . We write T | = UF if every model of T satisfies UF. The maingoal of this section is to establish the following theorem. Theorem 4.1.
Suppose that T | = DC + UF . Let T ′ be an open core of T . Thereis a theory T ∞ Sk extending T such that T ∞ Sk has definable Skolem functions and T ′ isan open core of T ∞ Sk . This immediately gives a negative answer to Question 4, as there are many docu-mented examples of a theory T with T | = DC + UF and o-mininal open core thatis not o-minimal itself. To prove Theorem 4.1 we follow a strategy of Kruckmanand Ramsey [4] and rely on a construction due to Winkler [9] allowing us to succes-sively add definable Skolem functions to the language L of a given theory T whilepreserving uniform finiteness. As explained below, this construction preserves theopen definable sets by a result from [1]. We begin by recalling notations and resultsfrom [9].4.1. Skolem expansions.
Let L be a language and Θ = { θ t ( x, y ) : t < |L|} be anenumeration of all L -formulas ϕ ( x, y ) where the variable y has length 1. Define L Sk to be L ∪ { f t : t < |L|} , where the arity of f t is the length of the tuple x appearingin θ t ( x, y ).The Skolem expansion T + of T is the L Sk -theory T + = T ∪ {∀ x ∃ y ( θ t ( x, y ) → θ t ( x, f t ( x ))) : t < |L|} . ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 11
We refer to the f t ’s as Skolem functions. From here on we assume that T has quantifier elimination in the language L and assume that for each L -definable function f there is an L -term t such that T | = ∀ x f ( x ) = t ( x ). Let M + | = T + , and denote its reduct to L by M . For A ⊆ M we denote by h A i Sk the L Sk -substructure generated by A .Following [9], we say an L Sk -formula χ ( x , . . . , x n ) is a uniform configuration ifit is a conjunction of equalities of the form f t ( x i , . . . , x i m ) = x i involving Skolemfunctions. We need the following result about uniform configurations from [9]. Fact 4.2 ([9, p. 448]) . Let χ ( x ) be a uniform configuration. Then there exists an L -formula χ ′ ( x ) such that for all A | = T + and a ∈ A | x | the following are equivalent: • A | = χ ′ ( a ) . • The result of altering the Skolem structure of A precisely so that A | = χ ( a ) is again a model of T + . In the case of Fact 4.2, we say that χ ′ ( x ) codes the eligibility of the configu-ration χ ( x ). Lemma 4.3.
Let t ( x ) , . . . , t n ( x ) be L Sk -terms such that for every i ≤ n there isan L Sk -function symbol f i with t i ( x ) = f i ( x, t ( x ) , . . . , t i − ( x )) . Then there is an L -formula ϕ ( x, y ) and a uniform configuration χ ( x, y ) such that T + | = ∀ x ∀ y (cid:16)(cid:0) ϕ ( x, y ) ∧ χ ( x, y ) (cid:1) ↔ (cid:0) n ^ i =1 y i = f i ( x, y , . . . , y i − ) (cid:1)(cid:17) . Proof.
Let J ⊆ { , . . . , n } be the set of all i such that f i ∈ L Sk \ L . Let χ ( x, y ),where y = ( y , . . . , y n ), be the uniform configuration given by ^ i ∈ J f i ( x, y , . . . , y i − ) = y i and let ϕ ( x, y ) be the L -formula given by ^ i ∈{ ,...,n }\ J f i ( x, y , . . . , y i − ) = y i . It is easy check this pair of formulas has the desired property. (cid:3)
One of the main results in [9] is that if T | = UF, then the Skolem expansion has amodel companion. Indeed, more is true: Fact 4.4 ([9, Theorem 2, Corollary 3]) . Let T | = UF . Then the Skolem expansion T + has a model companion T Sk that satisfies UF . From here on we assume that T | = UF. We have the following axiomatization ofthe model companion of the Skolem expansion. Fact 4.5 ([9, p. 447]) . The theory T Sk is axiomatized as the expansion of T + bythe set Φ of all sentences of the form ∀ x . . . ∀ x k ψ ( x ) , where x = ( x , . . . , x n ) and(i) ψ ( x ) = ∃ ∞ x k +1 . . . x n ϕ ( x ) ∧ χ ′ ( x ) → ∃ x k +1 , . . . x n ϕ ( x ) ∧ χ ( x ) ,(ii) ϕ ( x ) is a quantifier free L -formula, (iii) χ ( x ) is a uniform configuration,(iv) χ ′ ( x ) codes the eligibility of the configuration χ ( x ) . Let M Sk be an |L| + -saturated model of T Sk with underlying set M , and denote itsreduct to L by M . We need following easy corollary of the axiomatization of T Sk . Fact 4.6.
Let I be the set of partial L -elementary maps ι : X → Y between M Sk and itself such that • ι is a partial L Sk -isomorphisms and • X = h X i Sk and Y = h Y i Sk .Then I is a back-and-forth system.Proof. Let ι : X → Y in I . Let a ∈ M \ X . By symmetry, it is enough tofind a ′ ∈ M such that there exists ι ′ ∈ I extending ι such that ι ( a ) = a ′ . Bysaturation of M Sk , we just need to find a ′ ∈ M such that for all L Sk ( X )-terms t ( x ) = ( t ( x ) , . . . , t n ( x )) tp L ( a ′ , t ( a ′ ) | Y ) = ι tp L ( a, t ( a ) | X ) . Without loss of generality, we can assume that there is c ∈ X m such that for every i ∈ { , . . . , n } there is a function symbol f i ∈ L Sk with t i ( x ) = f i ( x, t ( x ) , . . . , t i − ( x ) , c ) . Let ϕ ( x, y , . . . , y n , z ) be the L -formula and χ ( x, y , . . . , y n , z ) be the uniform con-figuration given by Lemma 4.3. Let the L -formula χ ′ ( x, y , . . . , y n , z ) code theeligibility of χ ( x, y , . . . , y n , z ). For ease of notation, set y := ( y , . . . , y n ).Consider an L -formula ψ ( x, y, z ) and c ′ ∈ X | c ′ | such that ψ ( x, y, c ′ ) ∈ tp L ( a, t ( a ) | X ).Extending c , we can assume that c = c ′ . By saturation of M Sk it suffices to find a ′ ∈ M such that M Sk | = ψ ( a ′ , t ( a ′ ) , ι ( c )). Since M | = ψ ( a, t ( a ) , c ) ∧ ϕ ( a, t ( a ) , c ) ∧ χ ′ ( a, t ( a ) , c ) and a X , we have that M | = ∃ ∞ xy ψ ( x, y, c ) ∧ ϕ ( x, y, c ) ∧ χ ′ ( x, y, c ) . Since ι is L -elementary, M | = ∃ ∞ xy ψ ( x, y, ι ( c )) ∧ ϕ ( x, y, ι ( c )) ∧ χ ′ ( x, y, ι ( c ))) . Thus from the axiomatization of T Sk we know that there is ( a ′ , a ′ , . . . , a ′ n ) ∈ M n such that M | = ψ ( a ′ , a ′ , . . . , a ′ n , ι ( c )) ∧ ϕ ( a ′ , a ′ , . . . , a ′ n , ι ( c )) ∧ χ ( a ′ , a ′ , . . . , a ′ n , ι ( c )) . By our choice of ϕ and χ , we have that a ′ i = t i ( a ′ ) for each i . Thus M | = ψ ( a ′ , t ( a ′ ) , c ) . (cid:3) We now collect the following easy corollary of Fact 4.6.
Fact 4.7.
Let a, a ′ ∈ M n and let σ : M → M be an L Sk -automorphism fixing C such that σ ( a ) = a ′ and for all L Sk -terms t ( x ) = ( t ( x ) , . . . , t n ( x ))tp L ( t ( a ) | C ) = tp L ( t ( a ′ ) | C ) . Then tp L Sk ( a | C ) = tp L Sk ( a ′ | C ) . ATHOLOGICAL EXAMPLES OF STRUCTURES WITH O-MINIMAL OPEN CORE 13
No new definable open sets in T Sk . Let M Sk be an |L Sk | + -saturated modelof T Sk with underlying set M , and denote its reduct to L by M . Fix a subset C ⊆ M of cardinality at most |L Sk | . Theorem 4.8.
Let C = h C i Sk . Then every open set definable over C in M Sk isdefinable in M .Proof. By [1, Therorem 2.2] it is enough to show that for every a ∈ M n for whichthe set of realisations of tp L ( a | C ) is dense in an open set, the set of realisationsof tp L Sk ( a | C ) is dense in the set of realizations of tp L ( a | C ). Let U ⊆ M n be anopen definable set such that the set of realizations of tp L ( a | C ) intersected with U is dense in U . It is left to show that there is a ′ ∈ U such that a ′ | = tp L Sk ( a | C ). ByFact 4.7 and saturation of M Sk , it is enough to find for • every tuple t = ( t , . . . , t m ) : M n → M m of L Sk ( C )-terms and • every L ( C )-definable set X ⊆ M n + m with ( a, t ( a )) ∈ X an a ′ ∈ U such that ( a ′ , t ( a ′ )) ∈ X . Fix t and X . After increasing m , we canassume that there is c ∈ C ℓ such that X is L ( c )-definable and for every i ≤ mt i ( x ) = f i ( x, t ( x ) , . . . , t i − ( x ) , c )where f i is a function symbol in L Sk . Let ϕ ( x, y , . . . , y n , c ) be the L -formulaand χ ( x, y , . . . , y n , c ) be the uniform configuration given by Lemma 4.3. Set y = ( y , . . . , y n ). Let the L -formula χ ′ ( x, y, c ) code the eligibility of χ ( x, y, c ).We now prove the existence of a ′ . Let d be a realization of tp L ( a | C ) in U . Let d , . . . , d m ∈ M m be such that ( d , d , . . . , d m ) ∈ X and M | = ( ϕ ∧ χ ′ )( d , d , . . . , d m , c ) . Since there are infinitely many realizations of tp L ( a | C ) in U , there are infinitelymany e ∈ M n + m such that e ∈ X ∩ U and M | = ( ϕ ∧ χ ′ )( e, c ). Thus by Fact 4.5,there is e = ( e , e , . . . , e m ) ∈ M n + m such that( e , e . . . , e m ) ∈ X ∩ U and M Sk | = χ ( e , e . . . , e m , c ) . Thus ( e , . . . , e m ) = t ( e ) and we can set a ′ = e . (cid:3) Corollary 4.9.
Let T ′ be an open core of T . Then T ′ is an open core of T Sk .Proof. Let L ′ be the language of T ′ . Without loss of generality, we can assumethat L ′ ∩ L Sk = ∅ . Let L ∗ be the union of L ′ and L Sk . Let M Sk | = T Sk . Since T ′ isan open core of T , we can expand M Sk to a model M ∗ of the L ∗ -theory T ′ ∪ T Sk .Let X ⊆ M n be an open set given by X := { a ∈ M n : M Sk | = ϕ ( a, c ) } , where ϕ is an L Sk -formula with parameters c ∈ M m . Let N be an elementaryextension of M ∗ that is |L| + -saturated. Set Y := { a ∈ N n : N | = ϕ ( a, c ) } . Since X is open, so is Y . By Theorem 4.8 there is an L ′ -formula ψ ( x, y ) such that thereis d ∈ M ℓ with Y = { a ∈ N n : N | = ψ ( a, d ) } . Since M ∗ (cid:22) N , there is d ′ ∈ M ℓ such that X = { a ∈ M n : M ∗ | = ψ ( a, d ′ ) } . Thus X is L ′ -definable. (cid:3) Proof of Theorem 4.1.
We are now able to complete the proof of Theorem 4.1 usingthe same argument as in [4, Corollary 4.9]. Suppose T | = DC + UF and let T ′ bean open core of T. Set T be the Morleyization of T in a language L . For every n >
0, we will now construct a language L n and an L n -theory T n such that (1) T n has quantifier-elimination,(2) T n | = UF, and(3) T ′ is an open core of T n .Let n ≥
0, and suppose we already constructed a language L n and an L n -theory T n with the properties (1)-(3). Let Φ be the set of L n -formulas ϕ ( x, y ) such that | y | = 1 and T n | = ∀ x ∃ ! y ϕ ( x, y ) , For each ϕ ( x, y ) ∈ Φ we introduce a new function symbol f ϕ of arity | x | . Let e L bethe union of the L n and { f ϕ : ϕ ∈ Φ } . Let e T be the union of T n with the set ofall e L -sentence of the form ∀ x ∀ y ( f ϕ ( x ) = y ) ↔ ϕ ( x, y ) , where ϕ ∈ Φ. Since e T is an expansion of T n by definitions, it is easy to check that e T satisfies (1)-(3). Now consider the model companion ( e T ) Sk of the Skolem expansion( e T ) + . Let T n +1 be the Morleyization of ( e T ) Sk in an expanded language L n +1 . Weknow T n +1 | = UF by Fact 4.4. By Corollary 4.9, the theory T ′ is an open core of T n +1 .Now set T ∞ Sk := S i ∈ N T n . From the construction, it follows immediately that T ′ isan open core of T ∞ Sk and that T ∞ Sk has definable Skolem functions. (cid:3) References [1] Gareth Boxall and Philipp Hieronymi, Expansions which introduce no new open sets.
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Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801
Email address : [email protected] Department of Mathematics & Statistics, McMaster University, Hamilton Hall, 1280Main Street West, Hamilton, Ontario, Canada, L8S 4K1
Email address : [email protected] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801
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