Definable choice for a class of weakly o-minimal theories
aa r X i v : . [ m a t h . L O ] A p r DEFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES
MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW
Abstract.
Given an o-minimal structure M with a group operation, we show that for a properly convex subset U , thetheory of the expanded structure M ′ = ( M , U ) has definable Skolem functions precisely when M ′ is valuational. As acorollary, we get an elementary proof that the theory of any such M ′ does not satisfy definable choice. §
1. Introduction. An o-minimal structure is an ordered structure in which all subsets definablewith parameters consist of a finite collection of points and open intervals. There is a large bodyof work in support of the conclusion that o-minimal structures form the tamest possible class ofordered models. Much of this background is developed in the series of papers by J. Knight, A. Pillay,and C. Steinhorn (see [10], [13], [14]); a comprehensive treatment of the subject is also found inthe survey text [6]. Throughout this paper, we will be considering expansions of o-minimal groups.It is well known that such a group must be both abelian and divisible (see, e.g., [13]), hence theunderlying order is necessarily dense without endpoints.More generally, weakly o-minimal refers to the class of ordered structures in which each definablesubset of the model is a finite union of points and convex sets, the important distinction being thata 1-dimensional convex set S may be contained in a linear order M in which the supremum orinfimum may not be elements of M ∪ {±∞} . The most straightforward way of obtaining a weaklyo-minimal structure is to begin with an o-minimal group ( M , + , <, . . . ) and let C ⊆ M be anysubset which is downward closed (for any c ∈ C , if d < c and d ∈ M , then d ∈ C ). In [1] Y.Baisalov and B. Poizat prove that the expansion M ′ = ( M , U ) formed by adding a predicate U whose interpretation is C is weakly o-minimal, and in fact has a weakly o-minimal theory (i.e., every N ′ elementarily equivalent to M ′ is also weakly o-minimal). The motivation for such expansionsarose from the work of G. Cherlin and M. Dickmann [2], who obtained a quantifier elimination forthe theory RCVF of real closed valued fields.In addition to the work of [1], D. Macpherson, D. Marker, and C. Steinhorn [11] studied theclass of weakly o-minimal structures that arose by adding a convex predicate to an o-minimal field.There, they noted that the analysis of the resulting theory varied greatly depending on whetherthe cut was valuational or nonvaluational . Here, we show that the divide still appears in themore general setting where we expand a group instead of a field. We exhibit this distinction byconsidering the question of whether the resulting theory has definable Skolem functions or definablechoice. Definition . A theory T with language L has definable Skolem functions if, for every model M of T and for any L -formula ϕ (¯ x, y ) with lg(¯ x ) = n , there is a definable (possibly with parameters)function F ϕ : M n −→ M , such that for every ¯ b from M , if M | = ∃ yϕ (¯ b, y ), then M | = ϕ (¯ b, F ϕ (¯ b )). Definition . A theory T has definable choice if, for any M | = T and for any formula ϕ (¯ x, y ),there is a definable unary function F such that: Key words and phrases. model theory, Skolem functions, definable choice, o-minimal, weakly o-minimal.A portion of this material was submitted in partial fulfillment of the second author’s PhD thesis [15] at the University ofMaryland under the advisement of the first author. During this time both authors received support from NSF grant DMS-0600217. Laskowski has also been partially supported by NSF grant DMS-1308546.1
MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW
1. If ¯ a ∈ M and M | = ∃ yϕ (¯ a, y ), then M | = ϕ (¯ a, F (¯ a )) ( F is a Skolem function for ϕ );2. If { b ∈ M : M | = ϕ (¯ a, b ) } = { b ∈ M : M | = ϕ (¯ a ′ , b ) } , then F (¯ a ) = F (¯ a ′ ).Obviously, definable choice implies definable Skolem functions. It is well known, see e.g., [6], thatany o-minimal expansion of a group has both definable Skolem functions and definable choice. Themain results of this paper are summarized in the following theorem. Theorem . Suppose that M is an o-minimal group and M ′ = ( M , U ) is an expansion bya unary predicate that is interpreted as a downward closed, proper subset of C ⊆ M for which sup( C ) M . Then: If C describes a nonvaluational cut, then T h ( M ′ ) does not have definable Skolem functions; If C describes a valuational cut, then T h ( M ′ ) has definable Skolem functions.In either case, T h ( M ′ ) does not have definable choice. It is remarkable that the positive and negative results about Skolem functions arise from verydifferent analyses of the definable subsets in the two cases. The outline of the paper is as follows.In Section 2, we define valuational and nonvaluational cuts and prove a number of equivalentformulations in the presence of a group. In Section 3, we prove Theorem 1.3(1), with the major toolbeing the analysis of dense pairs of o-minimal structures. In Section 4, we prove Theorem 1.3(2). Allof the arguments in this section borrow heavily from work of L. van den Dries and A. Lewenberg [7]where they formed the theory T convex to study valuational expansions of o-minimal fields. Here,however, we show that many of their arguments go through with only a group operation present.Finally, in Section 5, we complete the proof of Theorem 1.3 by showing why we never have definablechoice in any expansion of this ilk. §
2. Valuational and nonvaluational cuts.
In this section, we define valuational and non-valuational cuts in ordered structures that are endowed with an ordered group operation. Ourgoal is Proposition 2.10, which enumerates a number of equivalents of this notion in the realm ofexpansions of o-minimal groups.
Definition . A proper cut h C, D i of a linear order ( M, < ) is a partition of M into non-empty,disjoint convex sets C and D such that C < D . We say that a proper cut h C, D i is irrational ifsup( C ) does not exist as an element of M . Definition . Given a linearly ordered group M , we say that a cut h C, D i of M is valuational if there exists ε ∈ M such that ε > C + ε = C . The model M is valuational if there is adefinable C such that h C, M \ C i is a valuational cut, and nonvaluational otherwise.Note that a weakly o-minimal structure M is valuational if and only if it it has a definable propernontrivial subgroup: if M has a valuational definable cut h C, D i , then the the set { ε : ε + C = C } isa definable proper nontrivial subgroup; conversely, if M has a definable proper nontrivial subgroup G , then G must be convex (see, e.g. , [15]), in which case the elements bounded above by G formthe downward portion of a valuational cut. Example . Fix a nonstandard expansion R ∗ of the reals with infinite elements; let M =( R ∗ , + , <, U ), where U M is the convex hull of R in R ∗ , and M = ( Q , + , <, P ), where P M = { x ∈ Q : − π < x < π } . Then M is valuational: define C , the left portion of a cut, by the formula U ( x ) ∨ ∃ y ( x < y ∧ U ( y )), and use the fact that any two standard reals sum to a standard real. Incontrast, since M remains archimedean, it is nonvaluational. If sup( C ) ∈ M , then M ′ is o-minimal, so T h ( M ′ ) has both definable Skolem functions and definable choice. EFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES It is shown in [11] that nonvaluational weakly o-minimal fields satisfy monotonicity and cellulardecomposition properties which correspond strongly to the o-minimal versions. R. Wencel extendedtheir results to weakly o-minimal nonvaluational groups (see [16]) but we will not use any of thoseresults here.Now suppose that M is an o-minimal structure and U ⊆ M is the downward closed half of anirrational cut of M . Then there is an associated 1-type generated by the formulas x > a for all a ∈ U and x < b for all b ∈ M \ U . It is a well known fact that these formulas generate a completetype p ∈ S ( M ), which we dub tp M (sup U ), the type generated by U . There are two behaviors ofsuch a type, which we will see precisely correspond to whether or not the cut determined by U isvaluational. Definition . Let T be o-minimal, M | = T , and p ∈ S ( M ) be the complete type generatedby an irrational cut of M . Then p is uniquely realizable if there is an elementary extension M ′ of M which realizes p with precisely one element. Equivalently, given b ∈ C which realizes p , p isrealized uniquely by b in every prime model over M ∪ { b } .The following result is proved by D. Marker in [12]. Theorem . Let M be o-minimal, p ∈ S ( M ) the complete type generated by an irrational cut.If p is uniquely realizable, b ∈ C realizes p , and M ( b ) is prime over M ∪ { b } , then every element of M ( b ) \ M realizes a uniquely realizable type over M . Lemma . Let M be an o-minimal group, U a downward closed subset of M describing anirrational cut of M . Then p = tp M (sup U ) is nonuniquely realizable if and only if the expandedstructure M ′ = ( M , U ) is valuational. Proof.
For the forward direction, suppose a, b ∈ C , with a < b and both a and b realizing p insome prime model over M ∪ { b } . Because M is a divisible ordered abelian group, we may define ε = b − a , where ε ∈ M , and 0 < ε < b − a .Let c ∈ U M . Then c + ε < a + ε < b . Since c, ε ∈ M , then c + ε ∈ M ; and tp M ( b ) is generatedby p , which implies that c + ε ∈ U M ′ . Thus M ′ is valuational.For the converse, suppose M ′ is valuational. Then let ε ∈ M with 0 < ε be such that for any a ∈ U M ′ , a + ε ∈ U M ′ . Note that this means for any d ∈ M , if d > U M ′ , then d − ε > U M ′ (otherwise ( d − ε ) ∈ U M ′ and ( d − ε ) + ε U M ′ ). Choose a realization b of p .Observe that b + ε | = p : if not, then there is d ∈ M such that d > U M ′ and b + ε > d , so clearly b > d − ε . But by the above comment, d − ε > U M ′ , a contradiction. So p is realized in everyprime model over M ∪ { b } by b and b + ε , and in particular b is not the unique realization. ⊣ Corollary . With M , U , and p as above, and ( M , U ) nonvaluational, let b ∈ C be a real-ization of p , and N be a prime model over M ∪ { b } . Then M is dense in N . Proof.
By Lemma 2.6, p is uniquely realized in N by b . Thus any irrational cut whose completetype over M is realized in N is uniquely realized in N (else there are d < d ′ such that tp C ( d/M ) = tp C ( d ′ /M ), whence d + ( b − d ) and d ′ + ( b − d ) both realize p ). Thus for any b < b ′ ∈ N , b and b ′ must realize different cuts over M ; as such, there is a ∈ M such that b < a < b ′ . ⊣ We note that there is nothing particularly magic about the operation of addition in defining avaluational cut. We indicate here a class of surrogate functions.
Definition . Let ( M , < ) be an ordered group, and F : M → M a definable function. F is pluslike on M if the following hold: • F is continuous on M ; • For every b ∈ M , the unary function F b ( x ) := F ( x, b ) is strictly increasing; and MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW • For every a ∈ M , the unary function a F ( y ) := F ( a, y ) is strictly increasing.Note that + M itself is pluslike. Pluslike functions can be seen as capturing the topological prop-erties of the additive structure of M without concern for the group structure. Just as valuationalconvex sets are defined in terms of + M , we define an F − valuational set in terms of a binary pluslikefunction F . Definition . Let M be a weakly o-minimal expansion of an ordered group, h C, D i a definablecut of M , and F : M → M a definable pluslike function. Then h C, D i is F -valuational if thereis ε > M such that for all a ∈ C , F ( a, ε ) ∈ C . A definable downward-closed convex set U ⊆ M is F -valuational if the cut h U M , { x ∈ M : x > U }i is F -valuational. Proposition . Let M be an o-minimal expansion of an ordered group, U a predicate for aproperly convex downward-closed subset, and M ′ = ( M , U ) . The following are equivalent: (i) U is F -valuational for some definable pluslike function F . (ii) U is F -valuational for all definable pluslike functions F . (iii) U is valuational. (iv) p := tp M (sup U ) is nonuniquely realizable. Proof. (ii) ⇒ (i) is trivial, and we previously showed (iii) ⇔ (iv); thus it suffices to show (i) ⇒ (iv) ⇒ (ii).(i) ⇒ (iv) is essentially the same as the forward direction in Lemma 2.6: suppose F and ε witnessthe fact that U is F -valuational. First note that since F is pluslike, it is strictly increasing inboth variables; thus for a fixed a ∈ M , a F is a bijection onto some subset of M , and has adefinable continuous partial inverse a F − . Now note that for any d ∈ M , d > U M implies that d F − ( ε ) > U M (or else d > U M and d F − ( ε ) ∈ U M , so F ( d F − ( ε ) , ε ) = d ∈ U M , a contradiction).Suppose p is realized by b in some prime model M over M ∪ { b } . We claim that F ( b, ε ) | = p as well. If not, then there is d ∈ M such that d > U M , and F ( b, ε ) > d . Thus, b > F − d ( ε ). But F − d ( ε ) > U M , contradicting the statement in the above paragraph. So, b | = p and F ( b, ε ) | = p .And because F is definable, we have F ( b, ε ) ∈ M . Since F is strictly increasing, F ( b, ε ) = b ,contradicting unique realizability of p .To show (iv) ⇒ (ii), let F : M −→ M be the pluslike function defined by F ( x , x ) = y if and only if ϕ ( x , x , y )Suppose b | = p , and let a < b such that a | = p and a is an element of a prime model M over M ∪ { b } . That ϕ defines a pluslike function is a first-order property, so let F ′ be the pluslikefunction defined by ϕ M . Since M (cid:22) M , we have M | = ∀ y ∀ x ∃ ! x ( ϕ ( x , x , y )). Let ε be theunique value such that ϕ ( a, ε, b ).If ε ∈ M , then let c ∈ U M such that F ′ ( c, ε ) < F ′ ( a, ε ) = b . Then c ∈ M and ε ∈ M implies F ( c, ε ) ∈ M , and thus F ( c, ε ) ∈ U M , so U is F -valuational. And if ε M , then since p is the typeof an irrational cut, then by Theorem 2.5, there is ε ∈ M such that 0 < ε < ε . We may thenredo the above argument with ε . ⊣ §
3. Failure of Skolem functions for nonvaluational expansions.
As a first attempt todetermine whether or not a weakly o-minimal model has Skolem functions, we describe a usefulcharacterization that follows easily by compactness. Alternatively, it follows as a consequence ofvan den Dries’ ‘Rigidity condition’ i.e., Theorem 2.1 of [4].
Lemma . Suppose T is a theory in a language with at least one constant symbol that admitsquantifier elimination. Then T has definable Skolem functions if and only if T has a definitionalexpansion T df that is universally axiomatizable. EFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES This test cannot be used conveniently in the case of a nonvaluational structure. Because “ M is nonvaluational” is expressed by an ∀∃ axiom, substructures of nonvaluational weakly o-minimalstructures are not well-behaved, as illustrated by the following example. Example . Let M = ( Q , + , <, U ), where < M is lexicographic, + M is componentwise ad-dition in Q , and P M = { ¯ x ∈ Q : ¯ x < (1 , , π ) } = { ( n, p, q ) : n ≤ , p ≤ , q < π } . That M is weakly o-minimal is clear, since ( Q , + , < ) ≡ ( Q , + , < ) and P M is convex. M is non-valuational, since every ‘small’ element is of the form (0 , , q ) for some q ∈ Q , and Q is itselfarchimedean. We note that M has substructures M ′ and M ′′ such that M ′ is valuational, and M ′′ is o-minimal: Let M ′ be the substructure generated by + M with universe { ( n, p, q ) : p = 0 } .Then U M ′ = { ( n, , p ) : n ≤ } , which is valuational in M ′ , witnessed by ε = (0 , , M ′′ be the substructure with universe { ( n, p, q ) : q = 0 } . Then U M ′′ = { ( n, p,
0) : ( n, p, ≤ (1 , , } ,which is an interval with endpoints in M ′′ ; thus M ′′ is o-minimal.To show that the obstruction alluded to above is in fact fatal, we state what can be discernedabout definable sets and functions in a nonvaluational expansion. Corollary 2.7 allows us to use thevan den Dries analysis of definable functions in dense pairs which appears in [5]. Formally, a densepair is a pair ( N , M ) such that M and N are o-minimal ordered abelian groups with M (cid:22) N ,and such that M is dense in N . We shall make use of the following, stated as Theorem 3, part (3)in [5]: Theorem . Let ( N , M ) be a dense pair, and let f : M n → M be definable in ( N , M ) . Thenthere are f , . . . , f k : M n → M definable in M such that for each m ∈ M n , we have f ( m ) = f i ( m ) for some i ∈ { , . . . , k } . Finally, we note that working in a weakly o-minimal structure obtained by adding a nonvalu-ational convex predicate to an o-minimal structure is essentially the same as working in a densepair. The following lemma makes this notion explicit:
Lemma . Let M be o-minimal, U be a downward-closed nonvaluational properly convex subset,and M ′ = ( M , U ) . Let N be a prime model over M ∪ { b } , where b realizes tp M (sup U ) . Then forany X ⊆ M definable in M ′ , there is a formula ϕ X (¯ x, y ) such that X = ϕ X ( N n , b ) ∩ M n . Proof.
Given the formula ψ ( x ) which defines X in M ′ , replace all instances of U ( x ) in ψ with x < y . ⊣ Theorem . Let M be an o-minimal structure with a group operation, U be a downward-closedconvex subset of M , and M ′ = ( M , U ) . If M ′ has a definable, nonvaluational cut which is notdefinable in M , then M ′ does not have definable Skolem functions. Proof.
The proof will analyze p = tp M (sup U ). We proceed by showing there is no M ′ -definablefunction f : M → M such that for every a ∈ U M ′ , we have a < f ( a ) and f ( a ) ∈ U .Let f : M → M be such a function, and assume f is M ′ -definable. Consider Γ( f ) ⊆ M , thegraph of f on M . Let b ∈ C be a realization of p , and N be a prime model over M ∪ { b } . Thenby Lemma 3.4, there is an L -formula ϕ ( x, y, ¯ z ), and a tuple ¯ c from N such that { ( x, y ) : N | = ϕ ( x, y, ¯ c ) } ∩ M = Γ( f ). Note that the solution set of ϕ ( x, y, ¯ c ) in N need not be the graph ofa function, but the formula defines Γ( f ) in the pair ( N , M ). Hence, by Theorem 3.3, there are M -definable functions f , . . . , f k such that for each a ∈ U , f ( a ) = f i ( a ) for some i ≤ k . Each ofthese functions f i is also definable in M ′ , so by weak o-minimality, there is a convex set I , whichis unbounded in U , and i ≤ k such that f ↾ I = f i . Restricting the domain of f , we may assume f = f i .The main point of the dense pair argument was to get this function f to be definable in M .Thus by monotonicity for o-minimal structures, we may assume I is an open interval . Now I is MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW unbounded in U M ′ and tp M (sup U ) is omitted in M ; thus, I must be ( c, d ) for some c, d ∈ M , c ∈ U M ′ and d > U M ′ .By the hypotheses on f , we may shrink the domain further to assume f is strictly increasing on I , and for all a ∈ U , a < f ( a ) and f ( a ) ∈ U . Let g ( x ) := f ( x ) − x . Then we may also assume byshrinking that g is strictly decreasing and positive on I .Again by the hypotheses on f , we can say in a certain sense that lim x → sup U g ( x ) = 0: since M ′ is nonvaluational, for any ε >
0, there is a ∈ U M ′ and b > U M ′ such that b − a ≤ ε . Then a < f ( a ) < b implies that f ( a ) − a < b − a , and thus g ( a ) = f ( a ) − a < ε .But since I is an interval with right endpoint d ∈ M , d > U M ′ , there is d ′ ∈ I ∩ U M ′ . Let ε = f ( d ′ ) − d ′ = g ( d ′ ). Then by the above, there is a ′ ∈ U M ′ such that g ( a ′ ) < ε . g was strictlydecreasing on I , so we must have g ( d ′ ) < ε , an impossibility. ⊣ The authors note that recent, independent work by Eleftheriou, Hasson, and Keren (see [9]and [8]), has generalized the above result, showing that if M ∗ is any proper expansion of M bynon-valuational cuts, then M ∗ does not have definable Skolem functions. § T -resistance and Skolem functions for valuational expansions. To explore Skolemfunctions in the valuational case, we first consider the specialization to T -resistant structures, ananalogue of the T -convex structures discussed in [7] and [3]. In those works, the authors weremotivated by a desire to generalize the theory of real-closed valued fields (RCVF), which is a keyexample of a T -convex theory. For that reason, the authors restrict their study to that of o-minimalfields; however, here we only assume that T expands a group. Our proof of Theorem 4.2, below, isis based on the proof of Theorem 3.3 and Corollary 3.13 of [7]. Definition . Let M be an o-minimal expansion of a group with complete theory T , and U be a proper, convex subgroup of M . We say that the model ( M , U ) is T -resistant , if the followingconditions hold:1. The language of M contains at least one constant symbol c interpreted by a non-zero elementof U .2. U is closed under every 0-definable continuous total function with domain M .The following results, as well as the proof structure and the supporting lemmas, are closelypatterned after similar results for T -convex theories in [7] and [3]. Suppose the L -structure G is ano-minimal expansion of a group, let T = T h ( G ), and suppose that U ⊆ G is a proper, T -resistantsubgroup. Let L ∗ = L ∪ { U, c } and let T ∗ = T h ( G ) ∪ ‘ U is T -resistant’ ∪ ‘ c > U ’ Theorem . Suppose G is an o-minimal expansion of a group, such that T = T h ( G ) has quan-tifier elimination, and is universally axiomatizable. Then the L ∗ -theory T ∗ defined above admitselimination of quantifiers and is universally axiomatizable. Proof.
As axioms asserting that ‘ U is T -resistant’ and ‘ c > U ’ are visibly universal, the factthat T ∗ is universally axiomatizable is automatic. For the quantifier elimination, our proof follows,with only minor changes, the proof of Theorem 3.10 from [7]. As in that article, we use a varianton the Robinson-Shoenfield substructure-completeness test for quantifier elimination of T ∗ . Weprovide a series of lemmas constituting a sketch of the argument. In many cases, the proofs areidentical to corresponding items from [7]; in those cases, we refer the reader to the article for furtherdetails. Lemma . To show that a model ( G , U ) is T -resistant, it suffices to show that U is closed underall functions f which: EFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES • are -definable in G , strictly increasing, continuous, even, • and which satisfy lim x → + ∞ f ( x ) = + ∞ . This is proved as Lemma 3.4 from [7], and functions satisfying the conditions of the lemma arereferred to as T -functions . Lemma . Let f be a T -function on G and U a T -resistant subgroup of G . Adopt the notation | A | = {| a | : a ∈ A } . Then f ( | U | ) = | U | , and f ( | G \ U | ) = | G \ U | . The proof of Lemma 4.4 is identical to that of Lemma 3.5 from [7]. The main lemma whichfollows is adapted from Lemma 3.6 of [7].
Lemma . Let U be a T -resistant subgroup of G (cid:22) H , and a ∈ H such that | U | < a < | G \ U | .There is exactly one T -resistant subgroup U a of Gh a i (the prime model of T over G ∪ { a } , whichin an o-minimal field corresponds to the definable closure of a over G ) containing a such that ( G , U ) ⊆ ( Gh a i , U a ) , namely: U a = { x ∈ Gh a i : | x | < |G \ U |} Proof of main lemma:
First, we show that U a satisfies the conclusion of the lemma. Define U a in terms of a as U a := { x ∈ Gh a i : | x | ≤ f (¯ u, a ) for some 0-definablecontinuous function f : H m +1 → H , ¯ u ∈ U m } That U a is convex is clear from the definition. To see that U a is in fact T -resistant, choose b ∈ U > a with | b | ≤ f (¯ u, a ), and any T -function g on Gh a i . Because g is strictly increasing in absolute value,we have | b | < | g ( b ) | ; since g is 0-definable, then so is g ◦ f , giving | b | < g ◦ f (¯ u, a ). By (4.3 - 4.4)above, U a is T -resistant.Note also that U a is the minimal T -resistant subset of Gh a i containing a , since any T -resistantsubgroup U of Gh a i must be closed under all 0-definable functions f : U n → U .We claim that ( G , U ) is a submodel of ( Gh a i , U a ). This can be reduced to showing that U a ∩ G = U .First, we know that there is an expansion ( G ∗ , U ∗ ) (cid:23) ( G , U ) such that U ∗ ∩ G = U . To wit, define p ( x ) to be the type { x > b : b ∈ U } ∪ { x < b : b > U } , and let G ∗ be an elementary extension p with an element a ∗ ∈ C . If G ∗ = Gh a ∗ i and U ∗ = U G ∗ , wehave U ∗ ∩ G = U , and U < a ∗ < | G \ U | . Because a and a ∗ have the same type over G , we mayassume a ∗ = a , and G ∗ = G h a i . Then U a ⊆ U ∗ by minimality of U a , so U a ∩ G ⊆ U ∗ ∩ G = U ;thus, U a ∩ G = U .For uniqueness, we rely upon a technical result, which appears as Lemma 2.13 in [7]: Lemma . Let U be a T -resistant subgroup of G containing G ′ (cid:22) G , and define IN F G ′ := { x ∈ G : | x | < ε for each ε in ( G ′ ) > } , the set of elements of G which are infinitesimal relative to G ′ .Then, if a ∈ U with a
6∈ G ′ + IN F G ′ , we have G ′ h a i ⊆ U . Given the technical lemma, suppose that there is U ∗ ) U a with a ∈ U ∗ and ( G , U ) ⊆ ( G h a i , U ∗ ), i.e. , U ∗ ∩ G = U . Then, because U ∗ properly extends U , there is a ∈ U ∗ \ U a with | U a | < a < | G \ U | ; in particular, | U | < a < | G \ U | . Since tp G ( a ) =tp G ( a ), there is an automorphism h of G h a i fixing G pointwise such that h ( a ) = a . Write a = t (¯ g, a ) for some term t of L df (whichconsists of L together with all 0-definable functions of G ), and a tuple ¯ g from G . Applying h successively, define a i +1 = h i +1 ( a ) = t (¯ g, a i ). For each n , define: U n := { x ∈ Gh a i : | x | ≤ f (¯ u, a n ) for some 0-definablecontinuous function f : H m +1 → H , ¯ u ∈ U m } MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW
Let Ph ¯ g, a i be the prime model of T over the tuple ¯ ga . Then for all n , a n is an element of Ph ¯ g, a i .Our construction yields that a < a < a < · · · and U a ( U ( U · · · ; and all U i are T -resistantsubgroups of Gh a i .By Lemma 4.6, each ( U n ∩ Ph ¯ g, a i ) must also be a T -resistant subgroup G n of Ph ¯ g, a i , and G n ≺6 = G n +1 , with G n ⊆ U n ∩ Ph ¯ g, a i and a n +1 ∈ G n +1 \ G n . This chain is impossible, since Ph ¯ g, a i was assumed to be prime and thus have finite rank (where rank here refers to the number ofgenerators of Ph ¯ g, a i over the prime model Ph∅i ). This concludes the proof of Lemma 4.5.
Corollary . With T , G , U , and a as in the statement of the main lemma, the set U ∗ := { x ∈ Gh a i : | x | ≤ u for some u ∈ U } is the unique T -resistant subgroup of Gh a i such that ( G , U ) ⊆ ( Gh a i , U ∗ ) and U ∗ does not contain a . With this in mind, given a T -resistant subgroup U of G (cid:22) H and an element a ∈ H with | U | < a < | G \ H | , there are two T -resistant subgroups W of Gh a i , such that ( G , U ) (cid:22) ( Gh a i , W ):one for which a ∈ W (by Lemma 4.5), and one for which a W (by Corollary 4.7).For the proof of Theorem 4.2, it suffices to show, using the Robinson-Shoenfield test for quantifierelimination, that:1. For any substructure ( G , U ) of a model of T ∗ , there is a T ∗ -closure ( e G , e U ), such that: • ( G , U ) ⊆ ( e G , e U ) | = T ∗ , and • ( e G, e U ) can be embedded over ( G , U ) into every model of T ∗ extending ( G , U ).2. If ( G , U ) ⊆ ( G , U ) are models of T ∗ with G 6 = G , there is a ∈ G \ G such that ( Gh a i , U ∩ G h a i )can be embedded over ( G , U ) into some elementary extension of ( G , U ).To prove (1), suppose that ( G , U ) is a submodel of a model ( G ∗ , U ∗ ) of T ∗ . Then G ⊆ G ∗ | = T .Since T is universally axiomatizable, this implies that G | = T . Additionally, U ⊆ U ∗ , which is a T -resistant subgroup of G . If U ∗ is closed under all 0-definable continuous functions f , then so is U . Thus, there are two cases to consider: Case 1: U = G . In this case, ( G , U ) is T -resistant, and is its own T ∗ -closure. Case 2: U = G . In this case, take an elementary extension H (cid:23) G with a new infinite element a ∈ H and G < a . Define W to be the convex hull of G in Gh a i . Then ( Gh a i , W ) is a T ∗ -closure of( G , U ).For the proof of (2), let ( G , U ) ⊆ ( G , U ) be T -resistant with G 6 = G . We proceed by finding avalue a ∈ G \ G such that ( Gh a i , U ∩ G h a i ) is embeddable over ( G , U ) into an elementary extensionof ( G , U ). Case 1: G contains an element a with | U | < a < | G \ U | .If a ∈ U , then choose an elementary extension ( G ∗ , U ∗ ) of ( G , U ), and an element b ∈ U ∗ suchthat | U | < b < | G \ U | . Then a and b have the same type over G , so there is a G -isomorphism h : Gh a i → Gh b i ⊆ G ∗ taking a to b . By the Lemma 4.5, h maps U ∩ G h a i to U ∗ ∩ G h b i , thus a and h satisfy condition (2) of the Robinson-Shoenfield test.If a U , let b be in an elementary extension ( G ∗ , U ∗ ) of ( G , U ) such that | U | < b < | G \ U | , andsuch that b U ∗ . Then b , together with h , defined as above, satisfy condition (2). Case 2: G does not contain a such that | U | < a < | G \ U | . If this is the case, then U is theconvex hull of U in G . Now, by the model-completeness of T , G is an elementary extension of G and there is an elementary extension ( G ∗ , U ∗ ) of ( G , U ) such that G is embedded in G ∗ over G by some map i : G → G ∗ . In this case, we have that U ∗ is the convex hull of U in i ( G ); thus, i ( U ) = U ∗ ∩ i ( G ). Thus i embeds ( G , U ) into ( G ∗ , U ∗ ) over ( G , U ). ⊣ Corollary . Let
M | = T be an o-minimal L -structure, with U ⊆ M such that ( M , U ) is T -resistant. Given c ∈ M \ U M , the L ( U, c ) -theory of ( M , U, c ) has definable Skolem functions. EFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES Proof. As M is an o-minimal expansion of a group, T h ( M ) has definable Skolem functions.Thus, by Lemma 3.1, T h ( M ) has a definitional expansion T df that is universally axiomatizable.Now apply Theorem 4.2 with respect to T df to obtain the universally axiomatizable, quantifiereliminable T ∗ . Thus, by Lemma 3.1 again, T ∗ has definable Skolem functions. As having definableSkolem functions is invariant under definitional expansions, the L ( U, c )-theory of (
M, U, c ) also hasdefinable Skolem functions. ⊣ It is clear that, in the context of an o-minimal structure M expanding a group with a new convexsubset U , if M ′ = ( M , U ) is T -resistant, then M ′ is valuational; as such Corollary 4.8 is a partialconverse to Theorem 3.5. The remainder of this subsection is devoted to proving Theorem 4.10,which extends this converse to show that a structure which fails to be T -resistant in a particularway must in fact be nonvaluational. Definition . For an o-minimal structure M and a convex U ⊂ M , we say that U forms an infinite cut of if dcl M ( ∅ ) ⊆ U .Note that if M does not define a positive element, as may be the case if the language of M doesnot contain multiplication, it is possible to have an infinite cut whose supremum is a finite realnumber. Theorem . Let M be an o-minimal expansion of a group with a definable positive element,and let U ⊂ M be a properly convex symmetric subset such that the expanded structure ( M , U ) forms an infinite cut. If ( M , U ) is not T -resistant, then ( M , U ) is nonvaluational. Proof.
We begin by showing that if there is some definable function which contradicts T -resistance, then there is a counterexample of a special form. Claim . Suppose that U forms an infinite cut of M , and ( M , U ) is not T -resistant. Thenthere is H : M → M such that • H is 0-definable in M , continuous, total, and strictly increasing, and • There are α, β ∈ U such that H ( α ) ∈ U and H ( β ) > U . Proof of Claim.
Suppose ( M , U ) is not T -resistant, witnessed by the 0-definable total con-tinuous function G : M → M , such that U is not closed under G . Note the condition that U bean infinite cut is necessary: if there a 0-definable element c ∈ M \ U , then the constant function G ( x ) = c is a counterexample to T -resistance. However, because we require that U is an infinitecut, there is no definable element of M \ U ; thus the range of G has a nonempty intersection with U (witnessed, at a minimum, by G (0) ∈ U ).By monotonicity in M , there are disjoint open intervals I , I , . . . , I n , such that I i < I i +1 foreach i , M \ S ni =1 I i is finite, and G ↾ I i is strictly monotone for each i . Because M is o-minimal,inf I i and sup I i must lie inside U , as must G (inf I i ) and G (sup I i ). Thus we may assume thatthe image under G of at least one of I or I n has nonempty intersection with M \ U . We assumewithout loss of generality that G ( I n ) ∩ ( M \ U ) is nonempty (if not, consider − G ( − x )); further wemay assume that G ( I n ) ∩ ( M \ U ) > U (else we consider − G ( x )). G ↾ I n cannot be constant, so itmust be strictly increasing. Choose i largest so that G ↾ I i is strictly decreasing or constant, and G ↾ I i +1 is strictly increasing, and define G ∗ as follows: G ∗ ( x ) = G ( x ) x < I i − G ( x ) + 2 G (inf I i ) x ∈ I i G ( x ) + 2 G (inf I i ) − G (inf I i +1 ) x > I i Proceed reductively on the next largest i at which the function changes behavior. Since inf I i =sup I i +1 , this process preserves continuity, and at each stage, the modified function is strictlyincreasing from I i to positive infinity. After repeating the adjustment finitely many times, the MICHAEL C. LASKOWSKI & CHRISTOPHER S. SHAW resultant function G ∗ is strictly increasing on all of M except for possibly I , on which G may takea constant value from U . If this is the case, define G ∗∗ as G ∗∗ ( x ) = (cid:26) − G ∗ ( x ) + 2 G (sup I ) x ∈ I G ∗ ( x ) otherwiseThen G ∗∗ ( x ) satisfies the requirements for H stated in the claim. ⊣ Now suppose H is 0-definable in M , continuous, total, and strictly increasing, and let α, β ∈ U with H ( α ) ∈ U and H ( β ) > U . By Proposition 2.10, it suffices to find a definable pluslike function F : M −→ M such that the cut defined by the downward closure of U is not F -valuational. Define F ( x, y ) := H ( x + y )That F is pluslike is immediate. To see that the cut is not F -valuational, choose any ε >
0. Then β ∈ U , but we have F ( β, ε ) > H ( β ) > U . ⊣ We are finally ready to prove the difficult direction of Theorem 1.3.
Theorem . Suppose that M is an o-minimal group and M ′ = ( M , U ) is an expansion by aunary predicate that is interpreted as a downward closed, valuational C ⊆ M . Then T h ( M ′ ) hasdefinable Skolem functions. Proof.
First, by compactness there is always an elementary extension M ∗ = ( M ∗ , U ∗ ) of M ′ where M ∗ has a positive element e ∗ > dcl M ∗ ( ∅ ) = dcl M ′ ( ∅ ). As M ∗ and M ′ have the same theory,we may assume that M ∗ = M ′ and hence e ∗ ∈ M . Next, as ‘having definable Skolem functions’ isinvariant under definable expansions, we can freely adjust U . In that vein, we may freely assumethat 0 ∈ U . If this were not the case, then consider U ′ = { a ∈ M : − a U } Then as (
M, U ) and (
M, U ′ ) are bi-definable, we could choose ( M, U ′ ) in place of ( M, U ). So,assume that 0 ∈ U . Next, by replacing U by U + e ∗ if needed, then we may assume that U describes an infinite cut. Finally, we can make U symmetric by replacing U by V = { a ∈ M : both a, − a ∈ U } Thus, we have reduced to the case where we can apply Theorem 4.10 to (
M, U ) to conclude that(
M, U ) is T -resistant. So, by Theorem 4.2, T h ( M, U, c ) has definable Skolem functions, where theinterpretation of c is any element a ∈ M with a > U . But, as Definition 1.1 allows formulas F ϕ with parameters from the model, it follows that T h ( M, U ) also has definable Skolem functions. ⊣ §
5. Failure of definable choice.
In this brief section we complete the proof of Theorem 1.3by proving that no expansion ( M , U ) of an o-minimal group by a downward closed subset of M can satisfy definable choice, unless of course, the interpretation U M ′ were already M -definable. Theorem . Suppose that M is an o-minimal group and C ⊆ M is downward closed and not M -definable. Then the weakly o-minimal expansion M ′ = ( M , U ) where U M ′ is interpreted as C does not satisfy definable choice. Proof. As C is not M -definable, h C, ( M \ C ) i is an irrational cut of M . We split into casesdepending on whether or not the cut is valuational or nonvaluational. If the cut is nonvaluational,then by Theorem 3.5 T h ( M ′ ) does not have definable Skolem functions, hence it cannot satisfydefinable choice. Thus, we may assume that the cut is valuational. But then, directly from thedefinition of a valuational cut, the infinitesimal set I = { ε ∈ M : ε + C = C } EFINABLE CHOICE FOR A CLASS OF WEAKLY O-MINIMAL THEORIES is a proper, 0-definable convex subgroup of M . This subgroup I induces a 0-definable equivalencerelation ∼ I on M , defined by a ∼ I b if and only if a − b ∈ I . Let ϕ ( x, y ) denote the 0-definableformula x − y ∈ I , and assume by way of contradiction that there were a definable F for ϕ ( x, y ) asin Definition 1.2. But then, as the ∼ I -classes are convex, the set R =range( F ) would be an infinite,discrete subset of M , directly contradicting the fact that M ′ is weakly o-minimal. ⊣ §
6. Future directions.
There are several opportunities for incremental advancement in ourunderstanding of Skolem functions for models ( M , U ), where M is o-minimal. First, in the val-uational case, is to find an explicit algorithm for calculating Skolem functions corresponding toa formula. In [15], Shaw gave an explicit algorithm for calculating Skolem functions in the caseof a model ( M , U ) that satisfies a condition called T -immunity. However, T -immunity is strictlystronger than T -resistance, and in particular is not satisfied if multiplication is definable in M . Asecond question is whether the counterexample in § M , U ), could we add a unary function f tothe language which satisfies x < f ( x ) ∧ U ( x ) whenever U ( x ) holds, such that the resulting structure( M , U, f ) has definable Skolem functions? REFERENCES [1]
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