arXiv:1112.4078v2 [math.LO] 16 May 2012
CLOSED FIELDS
ANNALISA CONVERSANO, PAOLA D’AQUINO, AND SALMA KUHLMANN
Abstract. In [KKMZ02] the authors gave a valuation theoretic characterization for a real closed field to beκ-saturated, for a cardi- nalκ≥ ℵ0. In this paper, we generalize the result, giving necessary and sufficient conditions for certain o-minimal expansion of a real closed field to beκ-saturated.
1. Introduction
A totally ordered structure M = hM, <, . . .i (in a countable first order language containing <) is o-minimal if every subset of it which is definable with parameters in M is a finite union of intervals in M.
These structures have many interesting features. We focus here on the following: For α >0, Mis ℵα-saturated if and only if the underlying orderhM, <i isℵα-saturated as a linearly ordered set ([AK94]). IfMis an o-minimal expansion of a divisible ordered abelian group (DOAG), then hM, <i is a dense linear order without endpoints (DLOWEP).
Now,ℵα-saturated DLOWEP are well understood, they are Hausdorff’s ηα - sets, see [R]. The above equivalence provides therefore a charac- terization ofℵα-saturation of such o-minimal expansions forα6= 0. We are reduced to characterising ℵ0-saturation. This problem was solved in [Ku90] and in [KKMZ02] for DOAG and for real closed fields, re- spectively.
In this paper we generalize this result to power bounded o-minimal expansions of real closed fields, see Theorem 5.2. Miller in [M1] proved a dichotomy theorem for o-minimal expansions of the real ordered field by showing that for any o-minimal expansion Rof Rnot polynomially bounded the exponential function is definable in R. Later, Miller ex- tended this result to any o-minimal expansion of a real closed field (see [M2]) by replacing polynomially bounded by power bounded.
In [DKS10] it was shown that a countable real closed field is recur- sively saturated if and only if it has an integer part which is a model
Date: May 17, 2012.
2000 Mathematics Subject Classification. Primary: 06A05, 12J10, 12J15, 12L12, 13A18; Secondary: 03C60, 12F05, 12F10, 12F20.
Key words and phrases. natural valuation, value group, residue field, pseudo- Cauchy sequences, o-minimal expansion of a real closed field, definable closure, dimension,ℵ0-saturation.
1
Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-212602
of Peano Arithmetic (see [DKS10] for these notions). In a forthcoming paper, we give a valuation theoretic characterization of recursively sat- urated real closed fields (of arbitrary cardinality), and their o-minimal expansions.
2. Background on o-minimal structures
We recall some properties of o-minimal structures. LetLbe a count- able language containing <, and letM=hM, <, . . .i be an o-minimal L-structure. If A ⊂ M then the algebraic closure acl(A) of A is the union of the finite A-definable sets, and the definable closure dcl(A) is the union of the A-definable singletons. In general, dcl(A) ⊆ acl(A), but in an o-minimal structure M they coincide. For example, if M is a divisible abelian group and A ∈ M then the definable closure of A coincides with the Q vector space generated by A, dcl(A) = hAiQ. If M is a real closed field then the definable closure of A ⊂M is the relative real closure of the field Q(A)in M, i.e. dcl(A) =Q(A)rc.
Notice that over a countable languageL the cardinality of the defin- able closure of a set A is:
(1) |dcl(A)|=
ℵ0 if |A| ≤ ℵ0
|A| if |A|>ℵ0
In [PS] it is proved that in any o-minimal structure Mthe operator dcl is a pregeometry, i.e. it satisfies the following properties:
(1) for any A⊆M, A⊆dcl(A);
(2) for any A⊆M, dcl(A)⊆dcl(dcl(A));
(3) for any A⊆M, dcl(A) = S
{dcl(F) :F ⊆A, F finite}
(4) (Exchange Principle) for any A ⊆ M, a, b ∈ M if a ∈ dcl(A∪ {b})−dcl(A)then b ∈dcl(A∪ {a}).
The Exchange Principle guarantees that in any o-minimal structure M there is a good notion of independence:
A subset A ⊂M isindependent if for alla∈A,a 6∈dcl(A− {a}). If B ⊂M we say thatAisindependent overBifa6∈dcl(B∪(A−{a})). A subset A⊆ M is said to generate M if M = dcl(A). An independent set A that generates M is called a basis. The Exchange Principle guarantees that any independent subset of M can be extended to a basis, and all basis for M have the same cardinality. So a basis for M is any maximal independent subset. The dimension of M is the cardinality of any basis. It is easy to extend the notion of a basis of M′ over Mwhen M M′. Note that
(2) dim(M′)≤ |A|
We recall the notion ofprimemodel of a theoryT. LetA⊆ M |=T. The model M is said to be prime over A if for any M′ |= T with A ⊆ M′ there is an elementary mapping f : M → M′ which is the
identity on A. For exampe, if T is the theory of real closed fields the real closure of an ordered field F is prime over F. It is well known, see [PS], that if M is an o-minimal structure, and A ⊆ M then T h(M) has a prime model over A, and this is unique up toA-isomorphism. For any subset A⊆ M it coincides with dcl(A). If A=∅ then dcl(∅) =P is the prime model of T.
Let us notice that if Mis a real closed field, then the dimension of M over the prime field coincides with the transcendence degree of M over Q.
3. ℵα-saturated divisible ordered abelian groups We summarize the required background (see [Ku01] and [Ku90]).
Let (G,+,0, <) be a divisible ordered abelian group. For any x ∈ G let |x| = max{x,−x}. For non-zero x, y ∈ G we define x ∼ y if there exists n ∈ N such that n|x| ≥ |y| and n|y| ≥ |x|. We write x << y if n|x| < |y| for all n ∈ N. Clearly, ∼ is an equivalence relation. Let Γ :=G− {0}/∼={[x] :x∈G− {0}}. We can define an order on <Γ
in terms of << as follows, [y] <Γ [x] if x << y (notice the reversed order).
Fact 3.1. (a) Γ is a totally ordered set under <Γ, and we will refer to it as the value set of G.
(b) The map
v:G −→ Γ∪ {∞}
0 7→ ∞
x 7→ [x] (if x6= 0)
is a valuation on Gas a Z-module, i.e. for every x, y ∈G:
v(x) = ∞ if and only ifx = 0,v(nx) =v(x) for all n∈ Z, n6= 0, and v(x+y)≥min{v(x), v(y)}.
(c) For every γ ∈ Γ the Archimedean component associated to γ is the maximal Archimedean subgroup of G containing some x ∈γ. We denote it by Aγ. For each γ, Aγ ⊆(R,+,0, <).
Definition 3.2. Let λ be an infinite ordinal. A sequence (aρ)ρ<λ con- tained in G is said to be pseudo Cauchy (or pseudo convergent) if for every ρ < σ < τ we have
v(aσ −aρ) < v(aτ−aσ).
Fact 3.3. If (aρ)ρ<λ is pseudo Cauchy sequence then for all ρ < σ we have
v(aσ−aρ) =v(aρ+1−aρ).
Definition 3.4. Let (aρ)ρ<λ be a pseudo Cauchy sequence in G. We say that x∈Gis a pseudo limit of S if
v(x−aρ) = v(aσ−aρ) =v(aρ+1−aρ) for all ρ < σ.
We now recall the characterization of ℵα-saturation for divisible or- dered abelian groups, see [Ku90].
Theorem 3.5. [Ku90] Let G be a divisible ordered abelian group, and let ℵα ≥ ℵ0. Then G isℵα-saturated in the language of ordered groups if and only
(1) its value set is an ηα-set
(2) all its Archimedean components are isomorphic to R
(3) every pseudo Cauchy sequence in a divisible subgroup of value set <ℵα has a limit in G.
Notice that in the case of ℵ0-saturation the necessary and sufficient conditions reduce only to (1) and (2), see [Ku90].
4. ℵα-saturated real closed fields
If (R,+,·,0,1, <)is an ordered field then it has a natural valuation v, that is the natural valuation associated to the ordered abelian group (R,+,0, < ). We will denote by G the value group of R with respect to v, i.e. G = v(R). If (R,+,·,0,1, <) is a real closed field then G is divisible, and we will refer to the rational rank of G, rk(G), for the linear dimension of G as a Q-vector space.
For the natural valuation on R we use the notations OR={r ∈R : v(r) ≥ 0} and µR = {r ∈ R : v(r) > 0}, for the valuation ring and the valuation ideal, respectively. The residue field k is the quotient OR/µR, and we recall that it is a subfield of R. Notice that in the case of ordered fields there is a unique archimedean component up to isomorphism, and if the field is real closed the archimedean component is the residue field.
A notion of pseudo Cauchy sequence is easily extended to any ordered field as in the case of ordered abelian groups.
The following characterization of ℵα-saturated real closed fields was obtained in [KKMZ02].
Theorem 4.1. [KKMZ02, 6.2] LetRbe a real closed field, v its natural valuation, G its value group and k its residue field. Let ℵα ≥ ℵ0. Then R is ℵα-saturated in the language of ordered fields if and only if
(1) G is ℵα-saturated, (2) k ∼=R,
(3) every pseudo Cauchy sequence in a subfield of absolute tran- scendence degree less than ℵα has a pseudo limit in R.
In the proof of Theorem 4.1 the dimension inequality (see [P]) is crucially used in the case of ℵ0-saturation. This says that the rational rank of the value group of a finite transcendental extension of a real closed field is bounded by the transcendence degree of the extension.
5. ℵα-saturated expansions of a real closed field We show now a generalization of Theorem 4.1 to o-minimal expan- sions of a real closed field M= (M,+,·,0,1, <, . . .).
The proof follows the lines of the previous characterizations. Also in this case some care is needed for ℵ0-saturation. We need to bound the rational rank of the value group of a finite dimensional extension.
(Recall from (1) that the cardinality of the definable closure of a finite set is infinite.) Analogues of the dimension inequality have been proved by Wilkie and van den Dries in more general cases.
Let T be the theory of an o-minimal expansion of R and assume T is smooth, see [W]. In [W] Wilkie showed that if R is a model T, and dim(R) is finite then rk(R) ≤ dim(R). This result has been further generalized by van den Dries in [vdD] to power bounded o-minimal expansions of a real closed field. We recall that M is power bounded if for each definable function f : M → M there is λ ∈ M such that
|f(x)| ≤xλ for all sufficiently large x >0 inM.
Theorem 5.1. [vdD] Suppose the dimension of Mis finite. Then the rational rank of the value group G of M is bounded by dim(M).
Theorem 5.2. LetM=hM, <,+,·, . . .ibe a power bounded o-minimal expansion of a real closed field, v its natural valuation, G its value group, k its residue field, P ⊆ M its prime model.
Then M is ℵα-saturated if and only if (1) (G,+,0, <) isℵα-saturated, (2) k ∼=R,
(3) for every substructureM′ with dim(M′/P)<ℵα, every pseudo Cauchy sequence in M′ has a pseudo limit in M.
Proof. We assume conditions (1), (2) and (3) and we show that M is ℵα-saturated.
Letq be a complete1-type overMwith parameters inA⊂M, with
|A|< ℵα. Let M0 be an elementary extension of Min which q(x) is realized, and x0 ∈M0 such thatM0 |=q(x0).
To realize qinMit is necessary and sufficient to realize the cut that x0 makes in M′ = dcl(A)⊆ M
q′(x) :={b≤x; b ∈M, q⊢b≤x} ∪ {x≤c; c∈M, q⊢x≤c}.
As we will see in realizing the cut q′ instead of type q some care is needed in the case of ℵ0-saturation. If q′(x) contains an equality, the result is obvious. So suppose that in q′(x) there are only strict inequalities.
Set
B :={b∈M′; q ⊢b < x}and C:={c∈M′; q ⊢x < c}
and consider the following subset of v(M0):
∆ ={v(d−x0)|d∈M′}.
There are three cases to consider:
(a) Immediate transcendental case: ∆ has no largest element.
(b) Value transcendental case: ∆has a largest element γ 6∈v(M′).
(c) Residue transcendental case: ∆has a largest elementγ ∈v(M′).
(a) ∆ has no largest element. Then
∀d∈M′ ∃d′ ∈M′ :v(d′−x0)> v(d−x0).
Let{v(dλ−x0)}λ<µbe cofinal in∆, then{dλ}λ<µ is a pseudo Cauchy sequence in M′ anddim(M′/P)≤ |A|<ℵα. Condition (3) implies the existence of a pseudolimit a∈M of {dλ}λ<µ. We claim thata realizes q′(x) inM. The ultrametric inequality gives
v(a−x0) =v(a−dλ+dλ −x0)≥min{v(a−dλ), v(dλ−x0)}.
Moreover, from properties of pseudo Cauchy sequences we have v(a−dλ) =v(dλ+1−dλ) =v(x0−dλ),
which implies that for allλ,v(a−x0)≥v(dλ−x0). Thus for alld∈ M′, v(a−x0)> v(d−x0). We want to show that a fills the cut determined by B and C, and so a realizes q′. Let b∈B, if a≤b then a≤b < x0, and this implies v(a−x0)≥v(b−x0), which is a contradiction. Hence b < a. In a similar way we can how that if c∈C then a < c.
(b) ∆ has a largest element γ 6∈ v(M′). Fix d0 ∈ M′ such that v(d0 − x0) = γ is the maximum of ∆. Assume d0 ∈ B (the case d0 ∈ C is treated similarly). Let ∆1 = {v(c −d0) : c ∈ C} and
∆2 ={v(b−d0) :b∈B, b > d0}.
Claim. ∆1 < γ <∆2.
From d0 ∈B it follows v(c−x0)< γ for all c∈C. Thus
v(c−d0) = v(c−x0+x0−d0) = min{v(c−d0), v(x0−d0)}= v(c−x0)< γ
Let b ∈ B and b ≥ d0 then v(x0 −b) ≥ v(x0 −d0) = γ, and by the maximality of γ the equality must hold. Thus,
v(b−d0) =v(b−x0+x0 −d0)≥min{v(b−d0), v(x0−d0)}=γ.
Since γ 6∈v(M′) we have v(b−d0)> γ, which completes the proof of the Claim.
Consider the set of formulas
t(y) ={v(c−d0)< y;c∈C} ∪ {y < v(b−d0);b∈B, b > d0}.
This is a type over Gwith parameters in v(M′). Let G′ =v(M′). If ℵα >ℵ0 then card(G′)<ℵα and by hypothesis (1) we can realize t(y) in G.
If ℵα=ℵ0 thenM′ has finite dimension over the prime field P, and Theorem 5.1 implies that the rational rank of G′ is bounded by the dimension of M′ over P. So, we can transform the typet(y)in a type t′(y) where the parameters vary over the finite Q-basis of G′. Since G is ℵ0-saturated we can realize t′(y) in G. Let a∈ M, a > 0 such that v(a) =g. We claim that a+d0 ∈M realizes q′. From the definition of the type t(y), it follows that for all c∈C and for all b ∈ B such that b > d0,
v(c−d0)< v(a)< v(b−d0),
and by order property of the valuation v we have that for all c ∈ C and for all b∈B such that b > d0
b−d0 < a < c−d0
which implies for all c∈C and for all b∈B b < a+d0 < c, hence a realizes the type q′ inM.
(c) ∆ has a largest element γ ∈ v(M′). Let d0 ∈ M′ and a ∈ M′ such that v(d0 −x0) = γ = v(a) (without loss of generality we may assume a >0).
Claim. There exist b0 ∈ B and c0 ∈ C such that for all b ∈ B with b ≥b0 and for all c∈C with c≤c0 we have
v(b−d0) =γ =v(a) =v(c−d0).
From v(d0 −x0) = v(a) it follows that there exists n ∈ N such that na > |x0−d0|> na. We distinguish the two cases according to d0 ∈B and d0 ∈ C. Assume d0 ∈ B, and let b0 = d0+ na and c0 = d0 +na.
Clearly, b0 < x0, so b0 ∈ B, and x0 < c0, so c0 ∈ C. Moreover, v(b0−d0) = v(an) = v(a) = v(na) = v(c0−d0). If b ∈ B, b > b0 and c∈C,c < c0, then the following inequalities holdd0 < b0 < b < c < c0. Thus, v(b−d0) ≤ v(b0 −d0) = γ = v(c0 −d0) ≤ v(b −d0). Hence, γ =v(b−d0). Similarly, one shows that γ =v(c0−d0)≤v(c−d0)≤ v(b0−d0) =γ, and so γ =v(c−d0).
Assume d0 ∈ C, and let b0 = d0 −na and c0 = d0 − an. Similar calculations show that v(c−d0) =γ =v(b−d0) forc∈C, c < c0, and b ∈B, b > b0.
Our aim is to show that there is an element r ∈ M which realizes the cut q′(x). It is enough to show that there is r′′ ∈M realizing (3)
b−d0
a < x;b∈B, b≥b0
∪
x < c−d0
a ;c∈C, c ≤c0
.
Indeed, r′ =r′′a∈M realizes
(4) {b−d0 < x;b ∈B, b≥b0} ∪ {x < c−d0;c∈C, c≤c0}
and so r = r′ +d0 ∈ M realizes q′(x). Assume d0 ∈ B. The claim implies that for all b ∈B, b≥b0, and for all c∈C, c≤c0 we have
v
b−d0
a
=v
x0−d0
a
=v
c−d0
a
= 0,
and taking residues the following inequalities hold in R, the residue field
b−d0
a < x0−d0
a < c−d0
a .
(Notice that the inequalities are strict because of the maximality of v(a)in ∆.) The cut in R
b−d0
a ;b ∈B, b≥b0
∪
c−d0
a ;c∈C, c≤c0
is realized in R by x0−ad0. If r′′ ∈ M is such that r′′ = x0−ad0 then r′′
realizes (3) in M. The proof in the case d0 ∈C is similar and we omit it.
We now assume thatMisℵα-saturated and we show that conditions (1),(2) and (3) hold.
(1) Let q(x) be a type with set of parameters A ⊂ G such that card(A)<ℵα, e.g. suppose A={gµ:µ < λ}, whereλ <ℵα. We have to show that q(x) is realized in G. Without loss of generality we can assume that q(x) is a complete type. Let H be the divisible hull of A in G. Notice that card(H)<ℵα forℵα >ℵ0.
It is enough to realize in G the set
{g ≤x; g ∈H, q(x)⊢g ≤x} ∪ {x≤g ; g ∈H, q(x)⊢x≤g}.
If the set contains an equality, we are done. So suppose that we only have strict inequalities.
For every µ ∈ λ fix an element aµ ∈ M, aµ >0, such that v(aµ) = gµ. If g ∈ H and g = q1gi1 +· · ·+qmgim with q1, . . . , qm ∈ Q, then g = v(aqi11 · · · · ·aqimm) where for simplicity we choose aqijj > 0 for all j ∈ {1, . . . , m}. Let
H1={g ∈H;q(x)⊢g < x} and H2 ={g ∈H;q(x)⊢x < g}
and consider
q′(x) ={kaqi11 · · · · ·aqikk < x;k∈N, v(aqi11 · · · · ·aqikk)∈H2}∪
{kx < aqi11 · · · · ·aqikk;k ∈N, v(aqi11 · · · · ·aqikk)∈H1}.
SinceMis a dense linear ordering without endpoints,q′(x)is finitely realizable in M. Thus q′(x) is a type in the parameters {aµ}µ<λ.
Since M is ℵα-saturated it follows that q′(x) is realized in M, say by a. Then v(a)realizes q(x).
(2) Since (M,+,0, <) is ℵα-saturated Theorem 3.5 implies that all Archimedean components are isomorphic to R, but there is only one Archimedean component and this is the residue field, so k ∼=R.
(3) Let (aν)ν<µ be a pseudo Cauchy sequence in M′, whereM′ is a substructure of M and dim(M′/P) = λ < ℵα. Let {bα;α < λ} be a basis of M′ over the prime field P. Then all elements aν are definable in terms of finitely many elements of the basis with coefficients in the prime fieldP. Recall that the prime fieldP coincides withdcl(∅)hence every element ofP is definable by a formula without paramenters. This is crucial in the case of ℵ0-saturation. Let
q1(x) ={n|x−aν+1|<|aν −aν+1|;ν < µ, n∈N}.
Then q1(x) is a set of formulas in λ parameters (in the case of ℵ0- saturation the parameters are only finitely many). Moreover, q1(x) is finitely satisfied in Msince (aµ)µ<λ is pseudo Cauchy. Hence q1(x) is a type, and a realization of q1(x) in M (which is ℵα-saturated) is a
pseudo limit of the sequence.
6. ℵα-saturated o-minimal expansions
If we take any o-minimal expansion of a real closed field (not nec- essarily power bounded) we obtain the following analogue of Theorem 4.1.
Theorem 6.1. Let M = hM, <,+,·, . . .i be an o-minimal expansion of a real closed field, v its natural valuation, G its value group, k its residue field, P ⊂ M its prime model.
ThenMisℵα-saturated ⇐⇒ for every substructureM′ ⊂ M such that dim(M′/P)<ℵα, then
(1) (G, <,+, v(M′)) is ℵα-saturated, (2) k ∼=R,
(3) every pseudo Cauchy sequence in M′ has a pseudo limit inM.
The proof is analogous to that of Theorem 5.2, and we omit it.
We just point out that in the value transcendental case the expansion (G, <,+, v(M′))of the value group is needed forℵ0-saturation. In the
power bounded case the valuation inequality allows us to get rid of the parameters inv(M′). By Miller’s dichotomy (see [M2]) the exponential function is definable if we are not in the power bounded case. In a forthcoming paper we further analyze Theorem 6.1 in that particular case. Finally, note that if in Theorem 5.2 we assume M is just a real closed field, then we obtain exactly Theorem 4.1: the prime model P is the field of real algebraic numbers, and M′ is a submodel of finite dimension over P if and only if it is of finite absolute transcendence degree.
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FB Mathematik & Statistik, Universität Konstanz, Germany E-mail address: annalisa.conversano@uni-konstanz.de
Dipartimento di Matematica, Seconda Università di Napoli, Italy E-mail address: paola.daquino@unina2.it
FB Mathematik & Statistik, Universität Konstanz, Germany E-mail address: salma.kuhlmann@uni-konstanz.de
