A dichotomy for T$T$ ‐convex fields with a monomial group

We prove a dichotomy for o‐minimal fields R$\mathcal {R}$ , expanded by a T$T$ ‐convex valuation ring (where T$T$ is the theory of R$\mathcal {R}$ ) and a compatible monomial group. We show that if T$T$ is power bounded, then this expansion of R$\mathcal {R}$ is model complete (assuming that T$T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if R$\mathcal {R}$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model‐theoretic tameness.

1. Let ℝ an be the expansion of the real field ℝ by functions which are real analytic on a neighborhood of the box [−1, 1]  , restricted to this box.Let  an ∶= Th(ℝ an ), in the language extending the language of ordered rings by these function symbols.This theory is o-minimal and model complete [9,15].The field of Puiseux series ℝ(( 1∕∞ )) ∶= ⋃  ℝ(( 1∕ )) admits an expansion to a model of  an , where each restricted analytic function on ℝ is extended to the corresponding box in ℝ(( 1∕∞ )) via Taylor series expansion.The convex hull of ℝ in ℝ(( 1∕∞ )), consisting of all series in which only non-negative exponents of  appear, is  an -convex, and the subgroup  ℚ = {  ∶  ∈ ℚ} ⊆ ℝ(( 1∕∞ )) > is a monomial group.2. Let ℝ an,exp further expand ℝ an by the unrestricted exponential function and let  an,exp ∶= Th(ℝ an,exp ).Again, this theory is o-minimal and model complete [12].The field  of logarithmic-exponential transseries admits an expansion to a model of  an,exp ; cf.[13,Corollary 2.8].This field is essentially obtained from the field of Puiseux series over ℝ by "closing off" under exponentials and logarithms.Once again, the convex hull of ℝ is  an,exp -convex, and the subgroup of transmonomials (transseries obtained by exponentiating the purely infinite elements of ) is a monomial group.

MATHEMATICAL LOGIC QUARTERLY
In this paper, we show that ℝ(( 1∕∞ )), as a model of  an expanded by predicates for the convex hull of ℝ and the monomial group  ℚ , is still model complete.Additionally, this structure admits quantifier elimination in a slightly extended language and it is distal.While this structure is no longer weakly o-minimal (the subgroup  ℚ is discrete), all definable unary subsets in ℝ(( 1∕∞ )) are the union of an open set and finitely many discrete sets.In contrast, the field , as a model of  an,exp expanded by a predicate for the group of transmonomials, was shown to be highly untame by Camacho [4,Theorem 4.11].Explicitly, this structure defines the natural numbers ℕ and is therefore at least as complex as Peano arithmetic.
As it turns out, the precise dividing line in our setting is whether the theory  defines an exponential function.If  does not define an exponential, then any definable function in any model of  is eventually bounded by a power function by Miller's dichotomy [19].We can then use results of van den Dries [10] and Tyne [24] to prove a quantifier elimination result, thereby showing that the tameness properties enjoyed by the Puiseux series hold for any model of  with a monomial group (assuming that the monomial group is compatible with the power functions).If  does define an exponential, then ℕ is externally definable in any model of  with a monomial group that is compatible with the exponential.
We define exactly what we mean by a "compatible" monomial group in § 2, where we also provide the necessary background on power boundedness and -convex subrings.Our quantifier elimination result for power bounded  is established in § 3, and we use this result to show that the value group and residue field are still stably embedded, even after adding a monomial group (stable embeddedness without the monomial group was shown by van den Dries [10]).In § 4, we use our quantifier elimination result to show that the unary definable sets in these expansions are unions of an open set and finitely many discrete sets, and in § 5 we show that the theory of these expansions is distal.We turn our attention to exponential  in § 6, where we show that the natural numbers are externally definable in any model of  expanded by a monomial group.

Notation and conventions
We always use , , and  to denote elements of ℕ = {0, 1, 2, …}.If  is a totally ordered set, then by a cut in , we mean a downward closed subset of .If  is a cut in  and  is an element in an ordered set extending , then we say that  realizes the cut  if  <  <  ⧵ .For an arbitrary subset  ⊆ , we let  ↓ denote the downward closure of , so  ↓ = { ∈  ∶  ≤  for some  ∈ } is a cut in .Given an ordered abelian group Γ, we let Γ > denote the set { ∈ Γ ∶  > 0}.
Given a ring , we let  × denote the multiplicative group of units in .

O-minimality
Throughout,  is a language extending the language {0, 1, <, +, −, ⋅} of ordered rings, and  is a complete o-minimal theory extending the theory of real closed ordered fields.It is well-known that  has definable Skolem functions, and consequently, we may arrange that  has quantifier elimination and a universal axiomatization just by extending  by function symbols for all (∅)-definable functions.Let  ⊧ .Then for  ⊆ , we have that dcl  () (the -definable closure of ), is an elementary substructure of  (again, as a consequence of definable Skolem functions).It follows that  has a prime model , which admits a unique embedding into any other model of  with image dcl  (∅).Given an elementary extension  of  and a subset  ⊆ , we denote by ⟨⟩ the intermediate extension dcl  ( ∪ ) ⊆ .When  is just a singleton {}, we write ⟨⟩ for this extension.The definable closure dcl  is a pregeometry, and we define rk  (|) to be the cardinality of a dcl  -basis for  over  (that is, a subset  ⊆  which is dcl  -independent over  such that  = ⟨⟩).If rk  (|) = 1, then  is said to be a simple -extension of , and  = ⟨⟩ for some  ∈  ⧵ .
A power function is an ()-definable endomorphism of the ordered multiplicative group  > .Each power function  can be thought of as the function  ↦   , where  ∶=  ′ (1) ∈ .The collection Λ of all such  is a subfield of , called the field of exponents of .By Miller's dichotomy [19], either  is power bounded (every definable function is eventually bounded by a power function) or  defines an exponential function (an ordered group isomorphism exp ∶  →  > which is equal to its own derivative).If  is power bounded, then every power function is (∅)-definable, and every other model of  is also power bounded with the same field of exponents as  (we just say  is power bounded, and we call Λ the field of exponents of ); cf.[19,Propositions 4.2 & 4.3].If  defines an exponential function exp, then exp is (∅)-definable; see the beginning of § 2 in [19].

𝑻-convex subrings
As stated in the introduction, a -convex subring of  is a convex subset of  which is closed under all (∅)-definable continuous functions  ∶  → .Let   ∶=  ∪ {} and let   be the   -theory which extends  by axioms stating that  is a proper -convex subring.Let (, ) ⊧   .Then  is a convex subring of  (hence a valuation ring), and (, ) is a convexly valued ordered field.As each element of the prime model  is (∅)-definable, we always have  ⊆ .We let Γ ∶=  × ∕ × denote the value group of (, ), written additively, and we let  ∶  × → Γ denote the surjective valuation map.If  is power bounded with field of exponents Λ, then Γ has the structure of an ordered Λ-vector space, where  ⋅ () ∶= (  ) for  ∈ Λ and  ∈  × .In fact, Γ is stably embedded as an ordered Λ-vector space [10,Theorem 4.4].We write O for the unique maximal ideal of , and we let  ∶= ∕O denote the residue field of (, ).We let  ∶  →  be the corresponding residue map; then  is order-preserving as  is convex.The residue field , considered with its induced structure, is a model of ; cf.[11,Remark 2.16] and [25,Remark 2.3].Moreover,  is stably embedded as a model of  [10,Corollary 1.13].Accordingly, we always construe  as an -structure.We sometimes include the subscript  on , , and Γ when confusion may otherwise arise.Let (,   ) be a   -extension of (,   ), so (,   ) is a model of   which is also an   -extension of (,   ).If  is a simple -extension of , then we say that (,   ) is a simple   -extension of (,   ).
Fact 1.1 [11,Remark 3.8].Let (,   ) ⊧   and let  be a simple -extension of .There are at most two -convex valuation rings  1 and  2 of  which make  a   -extension of : Proof.This is clear if no element in  realizes the cut  ↓  .Suppose that  ∈  realizes this cut.If The corollary follows by Fact 1.1.□ The theory   is tame, regardless of whether  is power bounded.However, when  is power bounded, van den Dries showed that we have an analog of the Abhyankar-Zariski inequality, called the Wilkie inequality: Fact 1. 3 The Wilkie inequality [10, § 5].Suppose that  is power bounded with field of exponents Λ.Let (,   ) ⊧   , let (,   ) be a   -extension of (,   ), and suppose that rk  (|) is finite.Then We will often use the following consequence of this inequality: Of course, it may be the case that for an extension (,   ) ⪯ (,   ) ⊧   , we have that both   =   and Γ  = Γ  .In this case, (,   ) is said to be an immediate extension of (,   ).
Let (  ) be a well-indexed sequence of elements of , so  ranges ordinals less than  for some limit ordinal .We say that (  ) is pseudocauchy if there is an index  0 such that for all  0 <  <  <  < .An element  in some   extension of  is called a pseudolimit of (  ) if there is an index  0 such that for all  0 <  <  < .If (,   ) is an immediate extension of (,   ), then for any  ∈  ⧵ , there is a pseudocauchy sequence (  ) in  with pseudolimit  and with no pseudolimits in ; cf.[17].

2.4
Monomial groups is a monomial group, and if  is a monomial group, then (|  ) −1 is a section of .Note that any monomial group is necessarily a subgroup of  > , as  is real closed and Γ is divisible.
In this paper, we will restrict our attention to monomial groups that are compatible with the o-minimal structure on .These monomial groups should respect the power functions in the case that  is power bounded and the exponential function when  is not power bounded.More precisely, we say that a monomial group  ⊆  > is -compatible if either 1.  is power bounded and  is closed under all power functions, or 2.  defines an exponential function exp and  ≻ ∶= { ∈  ∶  > 1} is closed under exp.
We say that a section  of  is -compatible if the corresponding monomial group (Γ) is -compatible.If  ∶ Γ →  > is -compatible and  is power bounded with field of exponents Λ, then  is an ordered Λ-vector space embedding.Lemma 1.5.Any model (, ) ⊧   admits a -compatible monomial group .
Proof.If  defines an exponential function, then the existence of a -compatible monomial group follows from Ressayre's dyadic representation of real closed exponential fields [21,Theorem 4].Though Ressayre builds a compatible monomial group with respect to the archimedean valuation and the exponential function 2  , his methods adapt to the construction of a compatible monomial group for any exponential and any -convex valuation ring.Suppose  is power bounded with field of exponents Λ.Then  > is an ordered Λ-vector space and ( × ) > ∶= { ∈  > ∶ () = 0} is a Λ-subspace of  > , so we may take a Λ-subspace  ⊆  > which is a Λ-vector space complement to ( × ) > .Then every  ∈  × can be uniquely represented as a product of some  ∈  and some  ∈  × , so  is a -compatible monomial group for (, ).□ Let   ∶=   ∪ {}, and let   be the   -theory which extends   by axioms stating that  is a -compatible monomial group.

QUANTIFIER ELIMINATION FOR POWER BOUNDED 𝑻
In this section, we assume that  is power bounded with field of exponents Λ.We will show that   is model complete.This model completeness is a by-product of a quantifier elimination proof in an extended language: Let  Γ,, be the three- MATHEMATICAL LOGIC QUARTERLY sorted language with sorts for  and the residue field   , both in the language , and a sort for the value group Γ  in the language of ordered Λ-vector spaces.We include a function symbol  ∶  →   for the residue map (defined to be zero off of the valuation ring), a function symbol  ∶  × → Γ  for the valuation map, and a function symbol  ∶ Γ  →  > for the -compatible section corresponding to .We do not include relation symbols for  and  in the sort for , but these predicates are  Γ,, (∅)-definable: the monomial group  is defined by { ∈  > ∶ (()) = } and the valuation ring  is defined by { ∈  ∶ () ≥ 0}.Any model  = (, , ) ⊧   admits a unique expansion to an  Γ,, -structure (, Γ  ,   ), and if  ⊆  are models of   , then the expansion (, Γ  ,   ) is an  Γ,, -substructure of the expansion (, Γ  ,   ).It is therefore harmless to refer to the three-sorted  Γ,, -structure (, Γ  ,   ) also as a model of   .
Corollary 2.4.The value group Γ is purely stably embedded as an ordered Λ-vector space and orthogonal to the residue field, which is purely stably embedded as a model of .

DEFINABLE SETS
In this section and the next, we will establish some consequences of our quantifier elimination result.For this, it will be more convenient to work in a one-sorted language in which we still have quantifier elimination.Let Proof.Let  and  be models of   , and assume that  is || + -saturated.Let  be a common   -substructure of  and .As in the proof of Corollary 2.2, we augment by additional sorts for the value group and residue field to get that (, ( × ), ()) is a common  Γ,, -substructure of (, Γ  ,   ) and (, Γ  ,   ), where the sections from the value group sort to the field sort are defined using the map .Theorem 2.1 gives an  Γ,, -embedding (, Γ  ,   ) → (, Γ  ,   ) over (, ( × ), ()), which restricts to an   -embedding  →  over . □ For the rest of this section, let  ⊧   and let  ⊆  be a set of parameters.Proof.We proceed by induction on the complexity of terms.If  is a variable or a constant symbol, then we take  = 0, () = (), and  =  (here  0 is the one-point space).

It remains to show that
for   -terms  0 , … ,   .For each  ≤ , take   ∈ ℕ, an ()-definable function   ∶    +1 → , and an   ()-definable set   ⊆    +1 as in Lemma 3.2.Let  ∶=  0 + ⋯ +   and for  = ( 0 , … ,   ) ∈ is a finite union of   ()-definable discrete sets, and for each , we have As each   is ()-definable and   is open, we see that  ∩   is finite (otherwise,  ∩   has interior).Thus,  ∩ ⋃ ∈    is   ()-definable and discrete.As  ⧵ ⋃ ∈    is a finite union of   ()-definable discrete sets, we conclude that  is a finite union of   ()-definable discrete sets.□ Structures in which all unary definable sets are a union of an open set and finitely many discrete sets are sometimes called d-minimal, though d-minimal structures are often additionally assumed to be definably complete.Of course, our structure is not definably complete, as the valuation ring is bounded but has no supremum in .In [14, § 9], Fornasiero gives a more relaxed definition of a d-minimal structure which does not include definable completeness.We do not know whether  is d-minimal in this sense.
The first example of a d-minimal structure is the expansion of the real field ℝ by a predicate for the multiplicative subgroup 2 ℤ [8].D-minimality was further developed by Miller and others; cf.[20].The theory   is quite similar to the theory of (ℝ, 2 ℤ ): our proof of Theorem 3.3 has the same structure as the proof in [20] that (ℝ, 2 ℤ ) is d-minimal.Moreover, the proof we give in the next section that   is distal follows the proof in [16] that (ℝ, 2 ℤ ) has a distal theory.It is also worth mentioning here previous work of Scowcroft [22], who proved a version of Theorem 3.3 for the field of -adic numbers with the canonical section of the -adic valuation (which is interdefinable with a predicate for the monomial group  ℤ ).

DISTALITY
Distality is a model-theoretic dividing line introduced by Pierre Simon in [23], that aims to capture order-like behavior within dependent (or NIP) theories.A theory is distal if for every indiscernible (  ) ∈ and any parameter set  such that (1)  =  1 + () +  2 where  1 ,  2 are infinite without endpoints, and Proof.Let ( ,   ,   ) ⊧   be a monster model.As in the previous section, we assume that  has quantifier elimination and a universal axiomatization, and we work in the language   , so ( × ) =   .We will use the Hieronymi-Nell criterion for distality [16,Theorem 2.1], applied to our theory  with additional function symbol .We need to verify the following: 1.The theory   has quantifier elimination in the language   .2. For every   -substructure  ⊆  and every  ∈   , there is a tuple  ∈ (⟨⟩)  for some  such that (⟨⟩) ⊆ ⟨(), ⟩.
Proof.All distal theories are dependent.To see that   is not strongly dependent, let ( ,   ,   ) ⊧   be sufficiently saturated, and note that for each  ∈  >0 , the set   ∩ (0, ) is definable, discrete, and infinite.By [6, Theorem 2.11],   is not strong.□

EXPONENTIAL 𝑻
In this section, we assume that  defines an exponential function exp.Let  = (, , ) ⊧   .Recall our assumption that  ≻ is closed under exp.We denote the compositional inverse of exp by log.
Proof.We claim that exp() = ( × ) > .For one inclusion, let  ∈ , and note that both exp  and (exp ) −1 = exp(−) belong to  > by -convexity and (∅)-definability of exp, so  ∈ ( × ) > .For the other inclusion, let  ∈ ( × ) > .If  ≥ 1, then log  ∈ , since  is convex and 0 ≤ log  < .If  < 1, then  −1 > 1 and so log( −1 ) ∈  by the previous case.Therefore log  = − log( −1 ) ∈  as well.The decomposition  =  ⊕ log() follows from our claim and the fact that  > is an internal (multiplicative) direct sum of ( × ) > and .□ We now follow Camacho's strategy for showing that Hahn fields with a predicate for the subring of purely infinite elements are undecidable [4, § 4.2].Let  ∈  and  ∈ .By Lemma 5.1, there is a unique  ∈  log() with  −  ∈ .We define |  to be this element , so (, ) ↦ |  is an   (∅)-definable function.We also define supp() ∶= { ∈  ∶ ( − |  ) = }, so supp() is an   ()-definable subset of .The element |  functions as a sort of the "truncation of  at ," and supp() serves as an analog of the support.Indeed, viewing the field of transseries  as a model of  an,exp , the element |  is exactly the truncation of an element  ∈  at a transmonomial , and the set supp() is exactly the support of .If  is an infinitesimal transmonomial in , then the support of (1 − ) −1 is the set {  ∶  ∈ ℕ}.Thus, ℕ = {log ∕ log  ∶  ∈ supp(1 − ) −1 } is definable in .We will show that something similar holds in our model .
Proof.Since  ∈  is less than 1, it belongs to the maximal ideal O, so (1 − ) and its inverse (1 − ) −1 both belong to  × .We have (1 − ) −1 | 1 = 0, so 1 ∈ supp((1 − ) −1 ) and if  ∈ supp((1 − ) −1 ), then  ≤ 1.Let us now fix  ∈ ℕ and  ∈  with  +1 ≤  <   .We will show that  ∈ supp((1 − ) −1 ) if and only if  =  +1 .We have Since  −1 , … ,  −1   are all in  ≻ ⊆ log(), their sum is in log() as well, so 1 +  + ⋯ +   belongs to  log().We have The definability of such a set  as above precludes the possibility of any "tame" model-theoretic behavior such as distality, dependence, or even NTP 2 .By taking parameters from the initial segment ℕ ⊆ , we can transfer model-theoretic combinatorial properties from ℕ to .As an illustration, we will show that the theory of  has the antichain tree property (ATP), as described in [2].A theory has ATP if there is a formula (, ) and a tuple of parameters (  ) ∈2 < such that {(,   ) ∶  ∈ } is consistent if and only  ⊆ 2 < is an antichain.Among theories with the strict order property (SOP), the antichain tree property implies all the other "non-tame" combinatorial properties studied in model theory so far, such as TP 2 [1] (of course, the theory of  has SOP as well).
Proof.We argue as in [2,Example 4.31].Assume that  is sufficiently saturated, let  be the definable set from Corollary 5.3, and let (, ) be the formula which states that  ∈  ⧵ {1} and that  ⋅  =  for some  ∈ .We need to find a M L Q MATHEMATICAL LOGIC QUARTERLY tuple of parameters (  ) ∈2 < such that {(,   ) ∶  ∈ } is consistent if and only  ⊆ 2 < is an antichain.By saturation, it is enough to find for each , a tuple of parameters (  ) ∈2 < such that {(,   ) ∶  ∈ } is consistent if and only  ⊆ 2 < is an antichain.Fix , let  1 , … ,   enumerate the antichains in 2 < , and let  1 , … ,   enumerate the first  prime numbers.For each  ∈ 2 < , let   be the product of the primes   for which  ∈   .Now let  ⊆ 2 < be an arbitrary subset.If  is an antichain, then  =   for some  ∈ {1, … , } and so {(,   ) ∶  ∈ } is consistent, as witnessed by   .Conversely, suppose that {(,   ) ∶  ∈ } is consistent, as witnessed by some  ∈  ⧵ {1}.Then  is less than each   , so  belongs to ℕ since each   is in ℕ.Take a prime factor  of .Then  is also a prime factor of each   , so  =   for some  ∈ {1, … , }.But then  ⊆   , so  is an antichain.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The author declare no conflicts of interest.
Research for this paper was conducted in part at the Fields Institute for Research in Mathematical Sciences.The first author was supported by the National Science under Award No. 2103240.We thank the referee for their helpful feedback.
Claim 1.By quantifier elimination for  and saturation, we can extend the inclusion   ⊆   to an (  )-embedding   →   .Claim 2. We can extend the inclusion Γ  ⊆ Γ  to an embedding Γ  → Γ  .Suppose  ∈ Γ  ⧵ Γ  .Using saturation, take  ∈ Γ  , realizing the same cut as  over Γ  .Put  ∶= () and  ∶= (), and note that  realizes the same cut over  as , so we get an -embedding  ∶ ⟨⟩ →  which sends  to .By Corollary 1.4, we have (⟨⟩) = (⟨⟩) = ().It follows from Corollary 1.2 that for  ∈ ⟨⟩, we have [3, theory   is complete.If  is model complete, then   is also model complete in the language   .Proof.Let  be the prime model of .Then (, {0}, ) admits an  Γ,, -embedding into any model of   since  ⊧ , so   is complete; cf.[3, Corollary B.11.7].For model completeness in the language   , let  and  be models of   and assume that  is || + -saturated.Let  ⊧   be a common   -substructure of  and .By a variant of Robinson's model completeness test[3, Corollary B.10.4], it is enough to show that the inclusion  ⊆  extends to an embedding  → .
is a finite union of   ()-definable discrete sets, so it suffices to show that the   ()-definable set ⋃   is open, it is enough to show that   ⧵ ⋃   , is finite for each , and this holds since   ⧵ ⋃   , is contained in union of { 1 (), … ,   ()} and the boundary of the set { ∈  ∶ (, ) = 0}.□ Using this lemma, we can describe the subsets of .Every   ()-definable subset of  is the union of an   ()-definable open set and finitely many   ()definable discrete sets.
Then  ∈  if and only if  ∈ ℕ or  > ℕ, by Proposition 5.2.It follows that ℕ is externally definable in  since it is the intersection of  with a convex subset of .Suppose now that  has an archimedean model.Then there is a model of  ⊧   where   = { ∈  ∶ || <  for some  ∈ ℕ}.Defining  in this model as above, we have ℕ =  ∩   .□ we conclude that  ∈ supp((1 − ) −1 ) if and only if  =  +1 .□ Corollary 5.3.There is a definable set  ⊆  with ℕ ⊆  such that if  ∈  ⧵ ℕ, then  > ℕ.Consequently, ℕ is externally definable in any model of   , and if  has an archimedean model, then ℕ is definable in some model of   .Proof.Fix  ∈  with  < 1 and let  be the definable set  ∶= { ∈  ∶ exp( log ) ∈ supp((1 − ) −1 )}.