Fields Institute Communications Volume 00, 0000

Model theory of real closed rings

Alexander Prestel

Fachbereich Mathematik und Statistik Universität Konstanz 78457 Konstanz Germany

Niels Schwartz

Fakultät für Mathematik und Informatik Universität Passau 94032 Passau Germany

Dedicated to Manfred Knebusch on the Occasion of his 60th Birthday.

Abstract. We show that the class of real closed rings is an elementary class, i.e., it can be axiomatized in the first order language of rings. Adding a binary ‘radical relation’ which encodes information about the spectrum, we prove that the theory of real closed rings as well as the theory of real closed domains admit a model companion. In the first case the model companion consists of certain von Neumann regular rings, in the second case it consists of certain real closed valuation rings.

1 Introduction

Real closed rings occur in algebra (as real closed fields and as convex subrings of real closed fields, cf. [Kn-Sch] or [PC], as well as [Ch-D] for model-theoretic properties), in topology (as rings of continuous functions into the real numbers, [Sch8]), in semi-algebraic geometry (as rings of continuous semi-algebraic functions on a semialgebraic space, [Sch1], [Sch3], [Sch5]) and in model theory (as real closures of commutative regular f-rings, [Li]). A completely different notion of real closed ring was introduced by M. Knebusch in [Kn1], [Kn2]. We start with the following

Definition (cf. [Sch2], [Sch6]): A ring A is real closed if and only if it satisfies the following conditions:

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14Pxx, 03C60, 13J30; Secondary 13A99.

The work on this paper has been done partly during the authors’ joint visit to Santander, hosted by Tomas Recio and supported by the HCM-project on Real Algebraic and Analytic Geometry (contract no. CHRX-CT94-0506).

(a) A is a reduced commutative ring with 1.

(b) The squares of A form the positive cone of a partial order ≤; together with this partial order, A is an f-ring.

(c) If 0 ≤ a ≤ b, then a^{2} ∈ Ab.

(d) For every prime ideal p ⊆ A, the quotient field qf(A/p) is real closed and the ring A/p is integrally closed.

This is an axiomatic description of the class of real closed rings, but not in first order model theory since condition (d) contains a quantification over the set of prime ideals. So, the first and most obvious question is whether the class of real closed rings is elementary in the sense of model theory. Conceivably, the answer for this and other model theoretic questions depends on the language that is being used. The smallest appropriate language is the language of ring theory, L = {+, −, ·, 0, 1}. An extension that is natural in our context is the language of partially ordered rings, L^{≤} = {+, −, ·, ≤, 0, 1}. Other extensions involve the lattice operations (note that real closed rings are f-rings) and radical relations (which will be discussed below). In Section 2 we prove the

Theorem The class of real closed rings is elementary using either of the languages L or L^{≤}.

The category of real closed rings is known to be cocomplete ([Sch-M],[Sch5]), hence it is an inductive class in the sense of model theory. Elementary inductive classes always have an ∀∃-axiomatization. We propose such an axiom scheme for the class of real closed rings.

The theorem opens the door to further model theoretic considerations. Some natural questions are the following:

• Does there exist a model companion, or even a model completion?

• If so, does every real closed ring have a prime extension in the model companion ?

• Supposing that a model companion exists: Is it complete, does it have quantifier elimination?

All of these questions have something to do with substructures. This is a point where it becomes important which language is being used; the choice of the language determines what is a substructure of a given structure.

In Section 3 we show that with respect to each of the languages L, L^{≤}, L^{∧∨}, L^{∧∨≤} there is a model companion for the theory of real closed rings. In fact, the model classes of all these model companions coincide: it is the class of von Neumann regular real closed f-rings without minimal idempotents. However, the model companions are not model completions and there do not exist prime extensions in the model companions. To improve the results in this direction we use radical relations (which were first introduced by Prestel and Schmid, cf. [P-S1], [P-S2]). These are binary relations on rings that encode information about the prime spectrum.

In Section 4 we discuss radical relations in general. Every radical relation on a ring can be defined by a subset of its prime spectrum ([P-S1]). We study this relationship between subsets of the prime spectrum and radical relations. In fact, for real closed rings there is a bijective correspondence between radical relations and pro-constructible subsets of the spectrum, i.e., the subsets that are closed with respect to the constructible topology.

In Section 5 we return to model theory. We extend the languages used before by a binary relation symbol ⪯ and then interpret this symbol by radical relations. Now the theory of real closed rings with radical relation has a model completion with respect to each of the languages considered. The model completion has a prime model and is therefore complete. With respect to the language containing the lattice operations it has quantifier elimination.

Sections 6 to 8 are devoted to the model theory of real closed (integral) domains with radical relations.

The quotient field K of a real closed domain A is a real closed field. In case K is non-archimedean, it carries a canonical (henselian) valuation with divisible value group Γ. Every radical relation on A is reflected by what we call a convexity relation ⪯ on Γ. Any convexity relation ⪯ associates a linearly ordered set L(Γ, ⪯) with Γ. With these notations, we show in Section 8 that the class of real closed domains with radical relation admits a model companion: the class of real closed valuation rings A such that L(Γ, ⪯) is densely ordered without extreme points.

The main tools for proving this result are the celebrated Ax-Kochen-Ershov Theorem and our elementary classification of divisible ordered abelian groups Γ with convexity relation ⪯, presented in Section 7. It should be mentioned that there is an intimate connection between such groups and the ‘contraction groups’ studied by F.-V. Kuhlmann in [Ku].

2 Axiomatization of real closed rings

Real closed rings in full generality were first introduced as a tool for semi-algebraic geometry ([Sch1], [Sch3], [Sch5]); for a related construction which, however, does not have the necessary functorial properties, see [Br]. They can be used to extend large parts of the theory of semi-algebraic spaces (as developed by Delfs and Knebusch, [D-K], [Kn3]) to the more general setting of real spectra. In particular, they are vital for studying the topology of real spectra ([Sch9]). These substantial applications spawned a sequence of papers on algebraic and functorial properties of real closed rings ([Sch2], [Sch6], [Sch4], [Sch7], [Sch-M]). The class of real closed rings is closed under a large number of ring theoretic constructions. Explicitly we mention only that direct limits of real closed rings are always real closed ([Sch5], Theorem I 4.8). This fact has an immediate impact on the axiomatization of real closed rings.

We shall use the following notation: If A is a ring and α ∈ Sper(A) then α ∩ −α is a prime ideal, called the support of α, supp(α). The subset α/supp(α) of A/supp(α) is the positive cone of a total order; this totally ordered domain is denoted by A/α. Its totally ordered quotient field is κ(α) = κ_{A}(α), and the real closed field of A at α is the real closure ρ(α) = ρ_{A}(α) of κ(α). The canonical homomorphism A → A/α ⊆ κ(α) ⊆ ρ(α) is denoted by ρ_{α}.

In addition to the axiomatic description of real closed rings given in the introduction we shall need an alternative characterization which may be found in [Sch5]: For any ring A and any pro-constructible subset X ⊆ Sper(A) there is a canonical homomorphism τ_{A,X} : A ⟶ ∏_{α∈X} ρ(α). Its components are the canonical homomorphisms ρ_{α} : A → ρ(α) into the real closed residue fields of A. An element c = (c(α))_{α∈X} ∈ ∏_{α∈X} ρ(α) is said to be a constructible (or definable) section if there is a formula φ(x) in the language L_{A}^{≤} of partially ordered rings augmented by constant symbols \underline{a}, a ∈ A, such that

ρ(α) ⊧ φ(c(α)),
ρ(α) ⊧ ∀y,z(φ(y)&φ(z) → y = z)

hold for every real closed residue field ρ(α) of A, α ∈ X. (The constant symbol \underline{a} is interpreted as ρ_{α}(a) in ρ(α).) The constructible sections form a von Neumann regular real closed subring σ(A, X) ⊆ ∏_{α∈X} ρ(α) which contains τ_{A,X}(A). Restricting the codomain of τ_{A,X} we obtain a homomorphism σ_{A,X} : A ⟶ σ(A, X). The support map supp : Sper(σ(A, X)) ⟶ Spec(σ(A, X)) is a homeomorphism. The image of the functorial map

Sper(σ_{A,X}) : Sper(σ(A, X)) ⟶ Sper(A)

is X. Restricting the codomain of this map one obtains a homeomorphism onto X with its constructible topology.

For any two prime cones α, β ∈ X with α ⊂ β the prime ideal supp(β)/supp(α) of the totally ordered ring A/α is convex. In ρ(α) there is a largest convex valuation ring C_{βα} with center supp(β)/supp(α) in A/α, i.e., such that M_{βα} ∩ A/α = supp(β)/supp(α) where M_{βα} ⊂ C_{βα} is the maximal ideal.

The restriction λ_{βα}|_{A/α} of the canonical place λ_{βα} : C_{βα} ⟶ C_{βα}/M_{βα} factors uniquely as shown by the diagram:

[figure: A/α; C_{βα}; ⊆; ρ(α); A/β; ⊆; ρ(β); C_{βα}/M_{βα}]

(Note that C_{βα}/M_{βα} is a real closed field containing canonically an isomorphic copy of ρ(β).) An element c = (c(α))_{α∈X} ∈ ∏_{α∈X} ρ(α) is a compatible section if, for all α, β ∈ X with α ⊂ β, the following conditions are satisfied:

c(α) ∈ C_{βα},
λ_{βα}(c(α)) = c(β).

The constructible and compatible sections form a real closed subring ρ(A, X) ⊆ σ(A, X) which contains σ_{A,X}(A). The homomorphism ρ_{A,X} : A ⟶ ρ(A, X) obtained from σ_{A,X} by restriction of the codomain is called the real closure of A on X.

Concerning the pro-constructible subset X ⊆ Sper(A) there are two particularly important special cases. The first one is that X = Sper(A). In this case we drop X from the notation and call ρ_{A} : A ⟶ ρ(A) the real closure of A. The ring A is real closed if and only if ρ_{A} is an isomorphism ([Sch1], [Sch2], [Sch3], [Sch5], [Sch6], [L]). The rings ρ(A, X) are always real closed.

The other important special case concerning X arises in the context of partially ordered rings. If (A, P_{A}) is a partially ordered ring (with positive cone P_{A}), then

Sper(A, P_{A}) = {α ∈ Sper(A)|P_{A} ⊆ α}

is a closed subset of Sper(A) and is called the real spectrum of (A, P_{A}). The real closure of A over Sper(A, P_{A}) is called the real closure of (A, P_{A}) and is denoted by ρ_{(A,P_{A})} : (A, P_{A}) ⟶ ρ(A, P_{A}). A very detailed discussion of real closures of partially ordered rings is contained in [Sch-M], section 12.

Our first main result is:

Theorem The L^{≤}-structure A is a real closed ring if and only if the following conditions are satisfied:

(i) axioms for reduced commutative partially ordered rings with 1; (ii) ∀X ∃Y (0 ≤ X → X = Y^{2}); (iii) ∀X ∃Y (0 ≤ Y & 0 ≤ Y − X & Y (Y − X) = 0); (iv) (convexity condition) ∀X, Y ∃Z (0 ≤ X & X ≤ Y ⇒ X^{2} = ZY);

(v) for every n ∈ ℕ the statement ∀A_{2n}, …, A_{0} ∃Z : Z^{2n+1} + D^{2}A_{2n}Z^{2n} + ⋯ + D^{4n+2}A_{0} = 0, where D denotes the discriminant of the polynomial X^{2n+1} + A_{2n}X^{2n} + ⋯ + A_{0};

(vi) for every n ∈ ℕ the statement ∀A_{n−1}, …, A_{0} ∀X, Y ∃Z_{1}, …, Z_{n} (X^{n} + A_{n−1}X^{n−1}Y + ⋯ + A_{0}Y^{n} = 0 ⇒ Y (X − Z_{1}Y) · . . . · (X − Z_{n}Y) = 0).

Proof: First we suppose that A is real closed. From the definition given in the introduction it is clear that (i), (ii), (iii) (with Y = X ∨ 0) and (iv) are satisfied. For the proof of (v), pick n ∈ ℕ and a_{0}, …, a_{2n} ∈ A, define P = X^{2n+1} +a_{2n}X^{2n} + ⋯+a_{0}, Q = X^{2n+1} +d^{2}a_{2n}X^{2n} +⋯+d^{4n+2}a_{0}, where d is the discriminant of P. If F = c_{0} + ⋯ + c_{k}X^{k} ∈ A[X] and if α ∈ Sper(A) then F (α) denotes the polynomial c_{0}(α) + ⋯ + c_{k}(α)X^{k} ∈ A/α[X]. If i = \sqrt{−1} then ρ(α)[i] is algebraically closed, hence contains 2n + 1 roots of each of P (α) and Q(α). If t(α) is a root of P (α) then d(α)^{2}t(α) is a root of Q(α). Since ρ(α) is real closed both P (α) and Q(α) have roots belonging to ρ(α). In fact, every root belonging to ρ(α) is actually a member of A/α (by condition (d)). For every α ∈ Sper(A) let t(α) be the largest root of P (α) in A/α, z(α) the largest root of Q(α) in A/α. It is claimed that z = (z(α))_{α} ∈ ∏_{α∈Sper(A)} ρ(α) belongs to A. For this it suffices to prove that z is a constructible and compatible section. Constructibility is clear, only compatibility needs to be discussed.

Suppose that α ⊂ β in Sper(A). Because of z(α) ∈ A/α ⊆ C_{βα} and because the canonical homomorphism f : A/α → A/β is the restriction of the place λ_{βα} : C_{βα} → C_{βα}/M_{βα} it suffices to prove that f(z(α)) = z(β). If d(α) = 0 then also d(β) = 0, hence z(α) = 0 and z(β) = 0, and the claim is trivial in this case. Now suppose that d(α) ≠ 0. The roots of Q(α) in ρ(α)[i] are obtained from those of P (α) through multiplication by d(α)^{2}. In particular, z(α) = d(α)^{2}t(α). If also d(β) ≠ 0 then the same holds with β instead of α. But if d(β) = 0 then f(d(α)) = d(β) = 0, hence

f(z(α)) = f(d(α)^{2}t(α)) = d(β)^{2}f(t(α)) = 0 = z(β),

and the claim also holds in this case. It remains to consider the case that also d(β) ≠ 0. The roots of P (α) and Q(α) in ρ(α)[i] are all distinct, and so are the roots of P (β) and Q(β) in ρ(β)[i]. Let \overline{A/α} and \overline{A/β} be the integral closures of A/α and A/β in ρ(α)[i] and ρ(β)[i]. Because of (d) (note that the support map supp : Sper(A) → Spec(A) is a homeomorphism) we have \overline{A/α} = A/α[i], \overline{A/β} = A/β[i], and f extends uniquely to a homomorphism \overline{f} : \overline{A/α} → \overline{A/β} with \overline{f}(i) = i. Then \overline{f} maps the 2n + 1 roots of Q(α) bijectively onto the 2n + 1 roots of Q(β). Since the roots are all distinct, the real roots correspond bijectively to each other. Thus, f maps all real roots of Q(α) bijectively onto the real roots of Q(β). Since f is order preserving, the largest roots are mapped to each other. This finishes the proof of compatibility. Altogether, z ∈ A is the desired root of Q.

Next we prove that (vi) holds in every real closed ring. For any a_{0}, …, a_{n−1} ∈ A and any α ∈ Sper(A) we are concerned with the roots of the polynomial X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α) ∈ ρ(α)[X]. In ρ(α)[i] there are n roots u_{1} = u_{11} + iu_{12}, …, u_{n} = u_{n1} + iu_{n2}. They are characterized by the property that

σ_{k}(u_{1}, …, u_{n}) = a_{k}(α), k = 0, …, n − 1,

where σ_{k} is the k-th elementary symmetric function. We view the elementary symmetric functions as defined on ρ(α)^{n} × ρ(α)^{n} and set

σ_{k1} = Re σ_{k}(v_{11}, …, v_{n1}; v_{12}, …, v_{n2}),
σ_{k2} = Im σ_{k}(v_{11}, …, v_{n1}; v_{12}, …, v_{n2}).

Then a tuple (v_{11}, …, v_{n1}; v_{12}, …, v_{n2}) ∈ ρ(α)^{n} × ρ(α)^{n} determines the complete set of roots of X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α) if and only if

σ_{k1}(v_{11}, …, v_{n1}; v_{12}, …, v_{n2}) = a_{k}(α), σ_{k2}(v_{11}, …, v_{n1}; v_{12}, …, v_{n2}) = 0

for all k = 0, …, n − 1. By symmetry we can restrict our attention to tuples (v_{11}, …, v_{n1}; v_{12}, …, v_{n2}) with v_{11} ≤ … ≤ v_{n1}. There is exactly one sequence (v_{11}, …, v_{n1}) ∈ ρ(α)^{n} that satisfies this condition and is the sequence of real parts of a complete set of complex roots of X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α).

We consider the polynomial ring A[U_{11}, …, U_{n1}] and the functorial map

π : SperA[U_{11}, …, U_{n1}] → SperA

between the real spectra. In SperA[U_{11}, …, U_{n1}] let

C = {γ | U_{11}(γ) ≤ ⋯ ≤ U_{n1}(γ) & ∃ U_{12}, …, U_{n2}

⋀_{k=0}^{n−1} (σ_{k1}(U_{11}(γ), …, U_{n1}(γ); U_{12}, …, U_{n2}) = a_{k}(π(γ))

& σ_{k2}(U_{11}(γ), …, U_{n1}(γ); U_{12}, …, U_{n2}) = 0) }

This set is clearly constructible. The above discussion shows that the only element of C ∩ π^{−1}(α), α ∈ Sper(A), is the ordered sequence of real parts of a complete set of complex roots of X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α). For r = 1, …, n, let j_{r} : A[X] → A[U_{11}, …, U_{n1}] be defined by j_{r}(X) = U_{r1}. The corresponding functorial map between the real spectra is denoted by p_{r}; geometrically it is the projection onto the r-th component. The ‘Real Theorem of Chevalley’ [C-R], Proposition 2.3, shows that p_{r}(C) ⊆ SperA([X]) is constructible for every r = 1, …, n. If p : SperA[X] → SperA is the functorial map induced by the inclusion A → A[X] then p_{r}(C) ∩ p^{−1}(α) = p_{r}(C ∩ π^{−1}(α)) has only one element. Identifying ρ(α) with the set of ρ(α)-rational points in Sperρ(α)[X] ≅ p^{−1}(α), [C-R], Proposition 4.3, the point of p_{r}(C) ∩ p^{−1}(α) belongs to ρ(α). Thus, we have shown that p_{r}(C) is the graph of a semi-algebraic function z_{r} ∈ ∏_{α∈Sper(A)} ρ(α). For every α ∈ Sper(A) the tuple (z_{1}(α), …, z_{n}(α)) is the ordered sequence of real parts of a complete set of complex roots of the poynomial X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α).

Our next step is to show that each z_{r} is also a compatible section. We have to show that, for all α, β ∈ Sper(A) with α ⊂ β, z_{r}(α) ∈ C_{βα} and λ_{βα}(z_{r}(α)) = z_{r}(β). Being a convex valuation ring in a real closed field, C_{βα} is Henselian, hence it extends uniquely to a valuation ring of ρ(α)[i]; explicitly, the extension is C_{βα}[i].

The canonical place of C_{βα}[i] is denoted by λ_{βα}[i]. Suppose that z_{r}(α)+it_{r}(α), r = 1, …, n, is a complete set of complex roots of X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α). All the coefficients belong to C_{βα}, hence the roots belong to the (integrally closed) valuation ring C_{βα}[i]. Therefore λ_{βα}[i](z_{r}(α) + it_{r}(α)), r = 1, …, n, is a complete set of roots of X^{n} + a_{n−1}(β)X^{n−1} + … + a_{0}(β) = X^{n} + λ_{βα}[i](a_{n−1}(α))X^{n−1} + … + λ_{βα}[i](a_{0}(α)) in C_{βα}[i]/M_{βα}[i] = (C_{βα}/M_{βα})[i]. Since X^{n} + a_{n−1}(β)X^{n−1} + … + a_{0}(β) ∈ ρ(β)[X] and ρ(β)[i] is the algebraic closure of ρ(β) we conclude that λ_{βα}[i](z_{r}(α)+it_{r}(α)) = λ_{βα}(z_{r}(α))+iλ_{βα}(t_{r}(α)), r = 1, …, n, is a complete set of roots of X^{n}+a_{n−1}(β)X^{n−1}+…+a_{0}(β) in ρ(β)[i]. Because λ_{βα} is order preserving, the sequence (λ_{βα}(z_{1}(α)), …, λ_{βα}(z_{n}(α))) is the ordered sequence of real parts of this complete set of complex roots. But there is only one such sequence, namely (z_{1}(β), …, z_{n}(β)). We conclude that z_{r}(β) = λ_{βα}(z_{r}(α)) for each r. Thus, the z_{r} are constructible and compatible sections; since the ring A is assumed to be real closed we conclude that z_{1}, …, z_{n} ∈ A.

To finish the proof of condition (vi), suppose that x, y ∈ A satisfy the identity X^{n} + a_{n−1}X^{n−1}Y + … + a_{0}Y^{n} = 0. We claim that y(x − z_{1}y) · . . . · (x − z_{n}y) = 0. So, pick α ∈ Sper(A) and consider the product y(α)(x(α)−z_{1}(α)y(α))·. . .·(x(α)− z_{n}(α)y(α)). If y(α) = 0 then the product is 0. If y(α) ≠ 0 then \frac{x(α)}{y(α)} ∈ ρ(α) is a root of the polynomial X^{n} + a_{n−1}(α)X^{n−1} + … + a_{0}(α). Every real root of this polynomial occurs in the sequence (z_{1}(α), …, z_{n}(α)), i.e., \frac{x(α)}{y(α)} = z_{r}(α) for some r. But then x(α) − z_{r}(α)y(α) = 0 and the product is 0 as well. Altogether we have shown that every component of y(x − z_{1}y) · . . . · (x − z_{n}y) in ∏_{α∈Sper(A)} ρ(α) is 0, hence y(x − z_{1}y) · . . . · (x − z_{n}y) = 0, and the proof of (vi) is finished.

Now we assume that the L^{≤}-structure A satisfies the conditions (i)–(vi) and we wish to show that A is a real closed ring. This will be done by checking the conditions (a)–(d) of the definition of real closed rings (cf. the introduction). Conditions (i) and (ii) show that A is a partially ordered reduced commutative ring with 1 whose positive elements are exactly the squares. We use condition (iii) to show that A is an f-ring: First of all, the additive group of A is a partially ordered group, and for every element x ∈ A there is an element y ∈ A with 0 ≤ y, y−x and y(y−x) = 0. Assume by way of contradiction that there is some z ∈ A, 0 < z ≤ y, y − x. Then 0 < z^{2} ≤ y(y − x) = 0 (recall that A is reduced) gives a contradiction. Thus, the infimum of y and y − x exists and is 0. Now [B-K-W], Théorème 1.2.9, implies that the additive group of A is lattice-ordered. Then A is even a lattice-ordered ring ([B-K-W], Definition 8.1.3). By [B-K-W], Théorème 9.3.1, A is an f-ring if a ∧ b = 0 implies ab = 0. Set x = a − b and choose y according to condition (iii). It has been shown already that y ∧ (y − x) = 0. Thus, x = a − b with a ∧ b = 0 and x = y − (y − x) with y ∧ (y − x) = 0. But there is only one such decomposition ([B-K-W], Proposition 1.3.4), hence a = y, b = y−x and, consequently, ab = 0. The convexity condition (iv) is exactly the condition (c) of the definition. It remains to prove condition (d). It will be shown that A/p is a real closed domain for every minimal prime ideal. Then condition (d) holds by the following argument: Let q ⊆ A be any prime ideal. Pick a minimal prime ideal p ⊆ A that is contained in q. Then A/p is a real closed domain and A/q is a reduced factor ring, hence is a real closed ring as well ([Sch5], Theorem I 4.5). Since A/q is a domain it follows that the quotient field is real closed and A/q is integrally closed (by the definition in the introduction).

We proceed in several steps.

1) Every prime ideal q ⊆ A is an l-ideal, the factor domain A/q is totally ordered, the positive elements of A/q are the squares.

Proof: Suppose that 0 ≤ a ≤ b, b ∈ q. Condition (ii) supplies elements c, d ∈ A with c^{2} = a, d^{2} = b. Because of c^{2} = |c|^{2} and d^{2} = |d|^{2} these elements may be chosen to be positive. Then we have 0 ≤ c ≤ d, d ∈ q. By the convexity condition, a = c^{2} ∈ (d) ⊆ q, i.e., a ∈ q, and q is a convex ideal. To prove that it is an l-ideal we show that a ∈ q implies a ∨ 0 ∈ q (cf. [B-K-W], Definition 8.2.1, Théorème 2.2.1): Since (a ∨ 0) ∧ (−a ∨ 0) = 0 and A is an f-ring it follows that (a ∨ 0)(−a ∨ 0) = 0 ([B-K-W], 9.1.10(iv)), thus, one of the factors belongs to q. Now (a ∨ 0) − (−a ∨ 0) = a ∈ q shows that even both factors are in q.

Being an l-ideal, the prime ideal q is l-prime ([B-K-W], Definition 8.5.1), hence is irreducible ([B-K-W], 8.5.2) and the factor ring is totally ordered ([B-K-W], Théorème 9.1.5). The canonical homomorphism π_{q} : A → A/q preserves the lattice operations. Thus, if 0 ≤ \overline{a} in A/q and if a ∈ A is any representative of \overline{a} then π_{q}(a ∨ 0) = \overline{a}, hence there exists a positive representative a. By condition (ii) one finds an element b ∈ A such that a = b^{2}. But then \overline{a} = π_{q}(b)^{2}, and this finishes the proof of the first step.

2) The support map supp : Sper(A) → Spec(A) is a homeomorphism.

Proof: By 1), the map is bijective. It is always continuous. Openness follows from

supp{α ∈ Sper(A) | a(α) > 0}
= supp{α ∈ Sper(A) | (a ∨ 0)(α) ≠ 0}
= {q ∈ Spec(A) | a ∨ 0 ∉ q}.

(Note that the sets {α ∈ Sper(A) | a(α) > 0}, a ∈ A, are a basis of the topology of Sper(A) because A is an f-ring.)

3) For every prime ideal q ⊆ A the totally ordered domain A/q satisfies the convexity condition.

Proof: Suppose that 0 ≤ \overline{a} ≤ \overline{b} in A/q and pick representatives a, b ∈ A of \overline{a} and \overline{b}. Then π_{q}((a ∨ 0) ∧ (b ∨ 0)) = \overline{a}, π_{q}(b ∨ 0) = \overline{b} and 0 ≤ (a ∨ 0) ∧ (b ∨ 0) ≤ b ∨ 0, hence we may assume that 0 ≤ a ≤ b. The convexity condition for A yields an element c ∈ A with a^{2} = cb, and this implies \overline{a}^{2} = π_{q}(c)\overline{b}.

4) Given any prime ideal q ⊆ A, in the totally ordered field qf(A/q) the positive elements are the squares.

Proof: This follows immediately from the last statement in 1).

5) Given a prime ideal q ⊆ A, every odd degree polynomial over qf(A/q) has a root in qf(A/q).

Proof: Suppose that \overline{F} = X^{2n+1} + \overline{a_{2n}}X^{2n} + … + \overline{a_{0}} ∈ qf(A/q)[X] is irreducible. Writing \overline{a_{i}} = \frac{\overline{b_{i}}}{\overline{c}} with \overline{b_{0}}, …, \overline{b_{2n}}, \overline{c} ∈ A/q, \overline{c} ≠ 0, the polynomial \overline{G} = X^{2n+1} + \overline{c}\overline{b_{2n}}X^{2n} + … + \overline{c}^{2n+1}\overline{b_{0}} ∈ A/q[X] has a root in qf(A/q) if and only if so does \overline{F}. Thus, it suffices to deal with the case that \overline{F} ∈ A/q[X]. We choose a set of representatives a_{0}, …, a_{2n} ∈ A for the coefficients of \overline{F} and define F = X^{2n+1} + a_{2n}X^{2n} + … + a_{0} ∈ A[X]. If d_{F} denotes the discriminant of F then π_{q}(d_{F}) ≠ 0 since \overline{F} (being irreducible) does not have any multiple roots in its algebraic closure. Setting

G = X^{2n+1} + d_{F}^{2} a_{2n}X^{2n} + … + d_{F}^{4n+2}a_{0} = X^{2n+1} + b_{2n}X^{2n} + … + b_{0} ∈ A[X]

and

\overline{G} = X^{2n+1} + π_{q}(b_{2n})X^{2n} + … + π_{q}(b_{0}) ∈ A/q[X],

the polynomial \overline{G} has a root in qf(A/q) if and only if so does \overline{F}.

According to condition (v) the polynomial G has a root z ∈ A. But then π_{q}(z) ∈ A/q is a root of \overline{G}, hence \overline{F} has a root in qf(A/q).

6) If p ⊆ A is a minimal prime ideal then A/p is integrally closed.

Proof: Let \overline{F} = X^{n} + \overline{a_{n−1}}X^{n−1} + … + \overline{a_{0}} ∈ A/p[X] be a polynomial having a root \overline{a} ∈ qf(A/p). We write \overline{a} = \frac{\overline{b}}{\overline{c}}, \overline{b}, \overline{c} ∈ A/p, \overline{c} ≠ 0 and choose representatives a_{0}, …, a_{n−1}, b, c ∈ A. Then b^{n} + a_{n−1}b^{n−1}c + … + a_{0}c^{n} ∈ p. Since p is a generic point of Spec(A) and since p belongs to the constructible set

C = {q ∈ Spec(A) | b^{n} + a_{n−1}b^{n−1}c + … + a_{0}c^{n} ∈ q}

there exists some t ∈ A with

p ∈ D(t) = {q ∈ Spec(A) | t ∉ q} ⊆ C.

Now (tb)^{n} + a_{n−1}(tb)^{n−1}(tc) + … + a_{0}(tc)^{n} belongs to every prime ideal, hence (tb)^{n}+a_{n−1}(tb)^{n−1}(tc)+…+a_{0}(tc)^{n} = 0 (note that the ring is reduced). Condition (vi) can be applied to this situation: There exist z_{1}, …, z_{n} ∈ A such that (tc)(tb − z_{1}(tc)) · . . . · (tb − z_{n}(tc)) = 0 ∈ p. One of the factors, but not tc, belongs to p, say tb − z_{i}(tc) ∈ p. Reducing modulo p one obtains

\overline{a} = \frac{\overline{b}}{\overline{c}} = \frac{π_{p}(tb)}{π_{p}(tc)} = π_{p}(z_{i}) ∈ A/p,

and the proof of 6) is finished.

7) If p ⊆ A is a minimal prime ideal then A/p is a real closed domain.

Proof: We use the criterion of [Sch6], Satz 1. The quotient field of A/p is real closed (by 4) and 5)), the ring A/p is integrally closed (by 6)) and the convexity condition holds by 3).

Note that the theorem exhibits a ∀∃-axiomatization for the class of real closed rings in the language L^{≤}.

The theorem can be used to prove axiomatizability of real closed rings also for other languages:

Corollary The real closed rings are an elementary class of L-structures. There is an ∀∃-axiomatization.

Proof: Let Th^{≤} be the theory of real closed rings in the language L^{≤}, let Th be the intersection of Th^{≤} with the set of all L-sentences. Clearly, every real closed ring is a model of Th. The sentence

(∗) ∀X, Y (X ≤ Y ↔ ∃Z : Z^{2} = X − Y)

belongs to the theory Th^{≤}, hence Th together with the sentence (∗) is an axiom system for the partially ordered real closed rings. Now let A be a model of Th in the language L. If the relation ≤ is defined by

a ≤ b if and only if ∃c : c^{2} = b − a

then the ring A with the relation ≤ is a model of Th together with (∗), hence A is real closed. Thus, it has been shown that the real closed rings are the model class of Th. To see that there is also an ∀∃-axiomatization in the language L, recall that the class of real closed rings is inductive.

Concerning the language used in the theorem and the corollary, there are other options available. Since real closed rings are f-rings, in particular lattice-ordered rings (l-rings), the partial order defines the lattice operations ∧ and ∨. Explicitly, x ∧ y is the unique element z satisfying the following condition:

z ≤ x & z ≤ y & ∀t (t ≤ x & t ≤ y → t ≤ z);

x ∨ y can be defined similarly or can be derived from ∧ by using the identity x ∨ y = −((−x) ∧ (−y)). Conversely, either of the lattice operations defines the partial order, e.g., x ≤ y if and only if x = x ∧ y. Thus, the real closed rings are an elementary class also with respect to the languages L^{∧} and L^{∧∨} obtained from L by adjoining only the inf-operation or both lattice operations. Also, there exist ∀∃-axiomatizations for both languages since the real closed rings are an inductive class. Knowing that they are an inductive class with respect to L^{≤} this follows from the remark that order preserving homomorphisms between reduced f-rings always preserve the lattice operations ([Dz-M], Lemma 2.2).

As far as the axiomatization is concerned, the choice of the language does not make much of a difference. But it should be noted that the notion of a substructure changes with the language. The substructures of real closed rings with respect to L are the real rings ([Kn-Sch], p. 103, Definition 1), the substructures with respect to L^{≤} are the representable partially ordered rings (because of the universal axiomatization of this class), the substructures with respect to L^{∧} or L^{∧∨} are the reduced f-rings (note that the f-rings are an equationally definable class, cf. [Bi], p. 404, Theorem 8, hence they have a universal axiomatization).

3 Von Neumann regular real closed rings

A small and very special subclass of all real closed rings is the class of von Neumann regular real closed rings. Their model-theoretic properties are studied in [Li], [M], and [W]. Using the language L^{∧} or the language L^{∧∨} , the von Neumann regular f-rings have a model-completion, namely the class of von Neumann regular real closed f-rings without minimal idempotents [M], Section 6; [W], Theorem 4.17). According to [Li], Theorems 1,2, every von Neumann regular f-ring has a unique prime extension (a real closure) in the class of von Neumann regular real closed f-rings and also in the class of von Neumann regular real closed f-rings without minimal idempotents.

It is an easy consequence of these results that the theory of real closed rings has a model companion with respect to the language L^{∧}: The description of the real closed ring A as the subring of constructible and compatible sections in ∏_{α∈Sper(A)} ρ(α) presents A as a substructure of a von Neumann regular f-ring. Each one of these, in turn, is a substructure of a von Neumann regular real closed f-ring without minimal idempotents. The theory of this class is model-complete, hence it is the model companion of the theory of real closed rings. We extend this result to the other languages that are natural in connection with real closed rings:

Theorem The theory of von Neumann regular real closed f-rings without minimal idempotents is the model companion of the theory of real closed rings with respect to each of the languages L, L^{≤}, L^{∧}, and L^{∧≤}.

Proof: Let Σ, Σ^{≤}, Σ^{∧}, Σ^{∧≤} be the theories of von Neumann regular real closed f-rings without minimal idempotents with respect to the different languages. Every real closed ring can be presented as a substructure of a model of Σ, Σ^{≤}, Σ^{∧}, or Σ^{∧≤}. Therefore it suffices to show that Σ, Σ^{≤}, Σ^{∧} and Σ^{∧≤} are model-complete. For Σ^{∧} this has already been mentioned. Starting from this fact we prove the claim for each of the other theories.

Σ^{∧≤}: The sentence

∀X ∀Y (X ≤ Y ↔ X = Y ∧ X)

belongs to Σ^{∧≤}. Therefore every formula belonging to the language L^{∧≤} is equivalent to a formula belonging to L^{∧}. Using this fact and the model-completeness of Σ^{∧} one concludes that every extension A ⊆ B of models of Σ^{∧≤} is elementary. Using [P1], Lemma 3.11, we conclude that Σ^{∧≤} is model-complete.

Σ^{≤}: Let A ⊆ B be an extension of models of Σ^{≤}. Since the inclusion is an order preserving homomorphism of reduced f-rings it follows that the lattice operations are also preserved ([Dz-M], Lemma 2.2), hence it is also an extension of models of Σ^{∧≤}. Since every L^{≤}-formula φ(X_{1}, …, X_{n}) is also an L^{∧≤}-formula, we conclude exactly as above that Σ^{≤} is model complete.

Σ: Once again, let A ⊆ B be an extension of models of Σ. We prove that A ⊆ B is a substructure also with respect to L^{≤}: Let i : A → B be the inclusion homomorphism. Because the positive elements are exactly the squares in both rings it is clear that i is order preserving. It remains to check that a ∈ A is positive whenever i(a) ∈ B is positive. If a is not positive then a has a unique representation of the form a = b − c with 0 ≤ b, c and bc = 0 (since we know that A is an f-ring). Similarly, there is a unique representation i(a) = x − y with 0 ≤ x, y and xy = 0. From i(a) = i(b) − i(c), 0 ≤ i(b), i(c), and i(b)i(c) = 0 it follows that x = i(b), y = i(c). Because of a ≱ 0 we know that c ≠ 0, hence i(c) ≠ 0 (note that i is injective), hence i(a) ≱ 0.

Having identified A as a substructure of B also with respect to L^{≤} we can use exactly the same arguments as in the previous proof (together with the modelcompleteness of Σ^{≤}); the details are omitted.

Corollary Each of the theories Σ, Σ^{≤}, Σ^{∧}, Σ^{∧≤} is complete.

Proof: It suffices to prove that there is a prime model for each theory ([P1], Korollar 3.12). The model is the same for all theories; it is the ring of locally constant functions from Cantor space to the field R_{0} of real algebraic numbers.

It is an obvious question whether the model companions are even model completions. To check this one needs to look at the amalgamation property for real closed rings.

Example: The class of real closed rings does not have amalgamation.

Let A be a real closed ring, let X = Spermin(A). Suppose that X is not dense in Sper(A) with respect to the constructible topology. (This is the case, for example, if A is a real closed domain, but not a field.) Then there are pro-constructible subsets Y, Z ⊆ Sper(A) such that X ⊆ Y ∩ Z, Y ≠ Z. The ring of constructible sections over A in ∏_{α∈Sper(A)} ρ(α) is denoted by σ(A), the images of σ(A) under the projections

p_{Y} : ∏_{α∈Sper(A)} ρ(α) ⟶ ∏_{α∈Y} ρ(α), p_{Z} : ∏_{α∈Sper(A)} ρ(α) ⟶ ∏_{α∈Z} ρ(α),

are denoted by σ(A, Y) and σ(A, Z); let i_{Y} : A ⟶ σ(A, Y) and i_{Z} : A ⟶ σ(A, Z) be the canonical maps. Both i_{Y} and i_{Z} are embeddings of real closed rings (because X ⊆ Y ∩ Z) with respect to any of the languages we are considering. If amalgamation holds then there exists a commutative diagram

A ⟶^{i_{Y}} σ(A, Y)
↓^{i_{Z}}  ↓^{f}
σ(A, Z) ⟶^{g} B

with embeddings f and g. Since f and g are injective and since σ(A, Y) and σ(A, Z) are von Neumann regular rings it follows that the maps Sper(f) and Sper(g) must be surjective. But then

Y = Sper(fi_{Y}) = Sper(gi_{Z}) = Z,

a contradiction. □

Using the fact that von Neumann regular f-rings have amalgamation one can show that the existence of prime extensions ([Li], p. 174) of real closed rings in the class of von Neumann regular real closed f-rings without minimal idempotents would imply amalgamation for real closed rings. Thus, the example shows that prime extensions cannot exist.

The example gives a hint as to why amalgamation and the existence of prime extensions fail. Using different pro-constructible subsets of the real spectrum there are too many different ways to embed a given real closed ring in a von Neumann regular real closed ring. To remedy this situation one can try to include information about the spectrum in the language. We do this via radical relations, which were first introduced in [P-S1] (see [P-S2] for applications). Preparatory material about radical relations is contained in the next section, after that we study radical relations in the context of real closed rings.

4 Radical relations

In [P-S1] a binary relation ⪯ on a ring is called a radical relation if it satisfies the following conditions:

(1) a ⪯ a ;

(2) a ⪯ b, b ⪯ c ⇒ a ⪯ c ;

(3) a ⪯ c, b ⪯ c ⇒ a + b ⪯ c ;

(4) a ⪯ b ⇒ ac ⪯ bc ;

(5) a ⪯ 1, 1 ⪯̸ 0 ;

(6) b ⪯ b^{2} .

The following discussion shows how radical relations can be constructed. Given a ring A and a subset X ⊆ Spec(A), the sets

D_{X}(s_{1}, …, s_{n}) = {p ∈ X | (s_{1}, …, s_{n}) ⊈ p}, n ∈ ℕ, s_{1}, …, s_{n} ∈ A,

form a sublattice with largest and smallest element in the distributive lattice of all subsets of X. We denote the lattice by D_{X}(A) or D(A) if X = Spec(A). If Y ⊆ X then the map D_{X}(A) → D_{Y} (A) : M ↦ M ∩ Y is a homomorphism of distributive lattices. The set of principal sets D_{X}(s), s ∈ A, is denoted by D′_{X}(A).

There are two important cases in which D_{X}(A) = D′_{X}(A):

(i) If A is a Bézout ring, i.e., finitely generated ideals are principal, then for every sequence s_{1}, …, s_{n} in A one finds an element s ∈ A with (s) = (s_{1}, …, s_{n}). It is clear that D_{X}(s_{1}, …, s_{n}) = D_{X}(s).

(ii) Suppose that every ideal in X is real. Then

D_{X}(s_{1}, …, s_{n}) = {p ∈ X | (s_{1}, …, s_{n}) ⊈ p}
= {p ∈ X | (s_{1}^{2}, …, s_{n}^{2}) ⊈ p}
= {p ∈ X | s_{1}^{2} + … + s_{n}^{2} ∉ p}
= D_{X}(s_{1}^{2} + … + s_{n}^{2}).

It is an easy matter to show that the relation ⪯_{X} on A defined by

a ⪯_{X} b if and only if D_{X}(a) ⊆ D_{X}(b)

is a radical relation. It is less obvious that every radical relation arises in this way ([P-S1], Theorem 2.5). If X_{i}, i ∈ I, is a family of subsets of Spec(A) defining the same radical relation ⪯ then one checks easily that ⪯ is also defined by ∪_{i∈I}X_{i}. Thus, there exists a largest subset defining a given radical relation.

Proposition If ⪯ is a radical relation of the ring A then the largest subset X ⊆ Spec(A) defining ⪯ is a pro-constructible subset.

Proof: Given any subset X ⊆ Spec(A), the closure with respect to the constructible topology is denoted by X̃. It suffices to show that X and X̃ always define the same radical relation. Because DX̃(a) ⊇ DX̃(b) implies D_{X}(a) ⊇ D_{X}(b) it is clear that b ⪯X̃ a implies b ⪯_{X} a. For the converse suppose that b ⪯_{X} a. If b ⪯̸X̃ a then there must exist some p ∈ X̃ \ X with b ∉ p, a ∈ p. The subset D(b) ∩ V (a) ⊆ Spec(A) is constructible and D(b) ∩ V (a) ∩ X = ∅, where V (a) = {p ∈ Spec(A) | a ∈ p}. But then D(b) ∩ V (a) ∩ X̃ = ∅ as well. On the other hand, p belongs to this set, a contradiction that finishes the proof.

This result immediately raises the question whether it is possible to classify the radical relations of the ring A by the pro-constructible subsets of Spec(A). The next theorem clarifies completely when this can be done:

Theorem Let X be a pro-constructible subset of Spec(A). Then the following conditions are equivalent:

(a) For all pro-constructible subsets Y, Z ⊆ X the radical relations ⪯_{Y} and ⪯_{Z} coincide if and only if Y = Z.

(b) The sets X ∩ D(a) ∩ V (b) form a basis for the constructible topology of X.

Proof: (b)⇒(a) One direction of the equivalence in (a) is obvious. For the other one assume that Y ≠ Z, say p ∈ Z \ Y. Since Y is pro-constructible there exist a, b_{1}, …, b_{n} ∈ A such that

p ∈ D(a) ∩ V (b_{1}, …, b_{n}) ⊆ Spec(A) \ Y.

Condition (b) yields elements b, c ∈ A with

p ∈ X ∩ D(c) ∩ V (b) ⊆ D(a) ∩ V (b_{1}, …, b_{n}).

Because of p ∈ D_{Z}(c), p ∉ D_{Z}(b) we conclude that c ⪯̸_{Z} b. But

Y ∩ D(c) ∩ V (b) ⊆ Y ∩ D(a) ∩ V (b_{1}, …, b_{n}) = ∅

implies that D_{Y} (c) ⊆ D_{Y} (b), i.e., c ⪯_{Y} b.

(a)⇒(b) Suppose that (b) is false. Then there exists a constructible subset C ⊆ X that cannot be represented as a union of sets of the form X ∩ D(a) ∩ V (b), a, b ∈ A, hence there is some p ∈ C with X ∩ D(a) ∩ V (b) ⊈ C for all a, b ∈ A with p ∈ D(a) ∩ V (b). Since C ⊆ X is constructible, so is Y = X \ C; the set Z = Y ∪ {p} is pro-constructible. We claim that ⪯_{Y} =⪯_{Z}. Because of Y ⊆ Z the implication b ⪯_{Z} a → b ⪯_{Y} a holds trivially. To prove the converse, assume by way of contradiction that b ⪯̸_{Z} a and b ⪯_{Y} a. Then D_{Y} (b) ⊆ D_{Y} (a) and D_{Z}(b) ⊈ D_{Z}(a). This means that Y ∩ D(b) ∩ V (a) = ∅ and Z ∩ D(b) ∩ V (a) ≠ ∅, i.e., p ∈ D(b) ∩ V (a), X ∩ D(b) ∩ V (a) ⊆ X \ Y = C. However this contradicts the choice of p, and the proof is finished.

The equivalent conditions are satisfied if D_{X}(A) = D′_{X}(A), e.g., if A is a Bézout ring and X = Spec(A) (a case which was considered in [P-S1]) or if A is any ring and X ⊆ Spec(A) is the pro-constructible subset of real prime ideals. Note that every prime ideal of a real closed ring is real. More generally, if A is an f-ring with the convexity property then every prime ideal is real: Let p ⊆ A be a prime ideal and suppose that a_{1}^{2} + … + a_{n}^{2} ∈ p. Then 0 ≤ a_{i}^{2} ≤ a_{1}^{2} + … + a_{n}^{2} implies that a_{i}^{4} ∈ (a_{1}^{2} + … + a_{n}^{2}) ⊆ p (by the convexity condition), hence a_{i} ∈ p. In [P-S1], p. 180, the largest set of prime ideals defining a radical relation ⪯ on the ring A is called the spectrum of (A, ⪯), denoted by Spec(A, ⪯). Our results show that Spec(A, ⪯) is a pro-constructible subset of Spec(A), hence Spec(A, ⪯) is a spectral space ([H]) as well. We briefly discuss two alternative presentations of this space.

Given any subset X ⊆ Spec(A), we define the relation

R_{X} = {(a, b) ∈ A × A | D(a) ∩ V (b) ∩ X = ∅}.

For every pair (a, b) ∈ R_{X} the set C(a, b) = Spec(A)\(D(a)∩V (b)) is constructible and contains X. Therefore the set Y = ∩_{R_{X}} C(a, b) is pro-constructible and contains X. As in the proof of the theorem one shows that for every p ∈ Y, X ∪ {p} defines the same radical relation as X. But then Y = ∪_{p∈Y} (X ∪ {p}) also defines the same radical relation. On the other hand, if Y ⊂ Z then the arguments used in the proof of the theorem show that ⪯_{Y} ≠⪯_{Z}. We conclude that Y = Spec(A, ⪯_{X}).

The ring A with a radical relation ⪯ is quasi-ordered ([F], p. 13). If the equivalence relation ∼ is defined by: a∼ b if and only if a ⪯ b and a ⪰ b, then the set of equivalence classes, A/∼, is partially ordered by: [a] ≤ [b] if and only if a ⪯ b. The partially ordered set A/∼ is isomorphic to the partially ordered set D′_{X}(A), where X is any subset of Spec(A) defining ⪯. As in [H-K], Definition 3.16, we define L to be the category of partially ordered sets with smallest element (denoted by 0) and largest element (denoted by 1); the morphisms preserve 0, 1, the partial order and all existing finite infima and suprema. A particularly important object of this category is the totally ordered set 2 = {0, 1}. We also consider 2 as a topological space with the spectral topology {∅, {1}, 2}. This is a spectral space, [H], its patch (or: constructible) topology is the discrete topology. If we consider this topology then we write 2_{d}. The category of spectral spaces is closed under the formation of products in the category of topological spaces, [H]. Thus, given any set M, 2^{M} is a spectral space as well; with respect to the constructible topology this is the same as 2_{d}^{M}. If X ⊆ 2_{d}^{M} is a closed subset then X with the topology inherited from 2^{M} is a spectral space. Now suppose that L ∈ ob(L) and consider Hom_{L}(L, 2) ⊆ 2^{L}. As a subspace of 2_{d}^{L} it is closed, hence Hom_{L}(L, 2) is a spectral space, which will be denoted by Spec(L). Clearly

Spec : L → Sp : L ↦ Spec(L)

defines a contravariant functor into the category Sp of spectral spaces.

These general considerations can be applied to the partially ordered set A/ ∼ belonging to a radical relation. Then Spec(A/ ∼) is yet another spectral space associated with (A, ⪯). In fact, if D_{X}(A) = D′_{X}(A), then the maps

Spec(A/∼) → Spec(A, ⪯) : f ↦ (fπ_{(A,⪯)})^{−1}(0),
Spec(A, ⪯) → Spec(A/∼) : p ↦ f_{p},

where π_{(A,⪯)} : A → A/ ∼ maps a ∈ A to its equivalence class and f_{p} : A/ ∼→ 2 is defined by f_{p}([a]) = 0 or 1 depending on whether a ∈ p or a ∉ p, are well-defined mutually inverse and continuous.

To extend our model-theoretic considerations to rings, partially ordered rings, etc., that are equipped with a radical relation we augment the appropriate language by the new relation symbol ⪯. To better understand the impact of this new symbol it is useful to note that some radical relations are definable. Our next result exhibits two such cases.

Proposition (a) Suppose that A is a reduced ring and let X be the set of minimal prime ideals, let ⪯ be the radical relation defined by X. Then b ⪯ a is equivalent to: ∀x (xa = 0 → xb = 0), i.e., Ann(a) ⊆ Ann(b).

(b) Let A be a ring, X ⊆ Spec(A) the set of maximal ideals, ⪯ the corresponding radical relation, and assume that A/∼ is a lattice. Then b ⪯ a is equivalent to:

∀x ((∃y, z : yb + zx = 1) ⇒ (∃u, v : ua + vx = 1)).

Proof: (a) First suppose that b ⪯ a, i.e., D_{X}(b) ⊆ D_{X}(a). Pick x ∈ A such that xa = 0. If p ⊆ A is a minimal prime ideal then either a ∈ p or x ∈ p, hence b ∈ p or x ∈ p. We see that xb ∈ p for every p ∈ X, hence xb = 0 because A is reduced. Conversely, suppose that D_{X}(b) ⊈ D_{X}(a), say p ∈ D_{X}(b) \ D_{X}(a). Because p is a minimal prime ideal that is not contained in the constructible subset D(a) ⊆ Spec(A) there exists some s ∈ A such that p ∈ D(s) and D(s) ∩ D(a) = ∅. Then bs ∉ p, hence bs ≠ 0, but as belongs to every prime ideal, hence as = 0.

(b) Pick a, b, x ∈ A such that b ⪯ a and (b, x) = A. Whenever m ⊆ A is a maximal ideal then b ∉ m or x ∉ m. Since D_{X}(b) ⊆ D_{X}(a) we conclude that a ∉ m or x ∉ m. Thus, (a, x) is not contained in any maximal ideal, i.e., (a, x) = A. Now assume that b ⪯̸ a, say m ∈ D_{X}(b) \ D_{X}(a). There are finitely many elements x_{1}, …, x_{n} ∈ A such that m ∈ V (x_{1}, …, x_{n}) and V (x_{1}, …, x_{n})∩V (b) = ∅. Because of D_{X}(A) = D′_{X}(A) there is an element x ∈ A such that D_{X}(x) = ∪_{i=1}^{n}D_{X}(x_{i}). Then x ∈ m and a ∈ m, hence (a, x) ≠ A. On the other hand, if m′ ⊆ A is any maximal ideal and if x ∈ m′ then b ∉ m′. This implies that (b, x) = A.

If the ring A is von Neumann regular then every prime ideal is minimal and maximal. Because it is a Bézout ring, both (a) and (b) of the proposition can be applied. Thus, the radical relation corresponding to the entire prime spectrum is definable in the language of rings. This is the only radical relation that we shall consider on von Neumann regular rings.

Next we consider the behavior of radical relations under homomorphisms. Let A and B be rings, f : A → B a homomorphism, f̂: Spec(B) → Spec(A) the induced functorial map between the prime spectra. If X ⊆ Spec(A) and Y ⊆ Spec(B) are pro-constructible subsets then they define radical relations ⪯_{X} on A and ⪯_{Y} on B. We want to compare these relations via f. The subset f̂(Y) ⊆ Spec(A) is proconstructible and yields another radical relation ⪯_{f̂(Y)} on A. Obviously, a ⪯_{f̂(Y)} a′ if and only if f(a) ⪯_{Y} f(a′). We call ⪯_{f̂(Y)} the inverse image of ⪯_{Y} under f.

Proposition Suppose that X = Spec(A, ⪯_{X}) and Y = Spec(B, ⪯_{Y}).

(a) f preserves the radical relation (i.e., a ⪯_{X} a′ implies f(a) ⪯_{Y} f(a′)) if and only if f̂(Y) ⊆ X.

(b) Assume that for A the correspondence between radical relations and pro-con-structible subsets of Spec(A) is bijective. Then f preserves and reflects the radical relation (i.e., a ⪯_{X} a′ is equivalent to f(a) ⪯_{Y} f(a′)), if and only if f̂(Y) = X.

Proof: (a) f preserves the radical relation if and only if a ⪯_{X} a′ implies a ⪯_{f̂(Y)} a′. This is clearly equivalent to f̂(Y) ⊆ X.

(b) f preserves and reflects the radical relation if and only if the radical relations ⪯_{X} and ⪯_{f̂(Y)} of A coincide. Because of the assumption about A this means that X = f̂(Y).

As in the proposition, suppose that X = Spec(A, ⪯_{X}). By N(A, X) we denote the von Neumann regular ring ∏_{p∈X} A_{p}/pA_{p}, the canonical homomorphism A → N(A, X) is denoted by N_{A,X}. The map ε_{X} : X → Spec(N(A, X)) sending p ∈ X to the kernel of the projection map N(A, X) → A_{p}/pA_{p} is injective, hence we consider X also as a subset (not a subspace) of Spec(N(A, X)). Note that the subset X ⊆ Spec(N(A, X)) is dense. Since the composition Spec(N_{A,X}) ∘ ε_{X} is the inclusion of X into Spec(A), the image of Spec(N_{A,X}) is X. If ⪯ is the radical relation defined on N(A, X) by the entire spectrum then ⪯_{X} is the inverse image of ⪯ under N_{A,X}.

For model-theoretic purposes it is particularly important to study the situation where N_{A,X} is an embedding, i.e., where N_{A,X} is injective and preserves and reflects the radical relation. Evidently, this is the case if and only if A is reduced and X contains the minimal prime ideals of A. The radical relations having this property are characterized in the next result.

Proposition Suppose that A is a reduced ring with a radical relation ⪯, and let X = Spec(A, ⪯). Then the following conditions are equivalent:

(a) X contains the minimal prime ideals of A.

(b) If ab = 0 and a ≠ 0 then a ⪯̸ b.

Proof: (a)⇒(b) Since a ≠ 0 there is a minimal prime ideal p ∈ D(a). Because of ab = 0 it follows that p ∉ D(b). Using (a) we conclude that p ∈ X and D_{X}(a) ⊈ D_{X}(b), which means that a ⪯̸ b.

(b)⇒(a) If p ∉ X is a minimal prime ideal then there exists some s ∈ A such that p ∈ D(s) and D_{X}(s) = X ∩ D(s) = ∅. This latter condition implies that s ⪯ 0. Because s · 0 = 0 and s ≠ 0 this is a contradiction to (b).

Finally in this section, we return to real closed rings and summarize what we have learnt about their radical relations. We state the result for a larger class of rings since no additional effort is involved.

Theorem Let A be a reduced f-ring with the convexity condition. The radical relations of A satisfying the condition

∀a, b (ab = 0 ∧ a ≠ 0 ⇒ a ⪯̸ b)

are in bijective correspondence with the pro-constructible subsets of Spec(A) containing every minimal prime ideal. If ⪯ is such a radical relation then A together with the radical relation can be embedded into a von Neumann regular real closed ring as a ring, a partially ordered ring or an f-ring.

Proof: Only the statement about the embedding has not already been proved completely. So, let X be the pro-constructible subset of Spec(A) = Sper(A) belonging to the radical relation. Then the homomorphism σ_{A,X} : A → σ(A, X) is such an embedding.

5 Radical relations on von Neumann regular real closed rings

It was shown in Section 2 that the real closed rings are an elementary class with respect to each of the languages L, L^{≤}, L^{∧∨} and L^{≤∧∨}. Thus, if any one of them is augmented by the binary relation symbol ⪯ then the real closed rings with a radical relation are an elementary class for the new language. Axiomatizations are obtained from the axioms for real closed rings together with the axioms (1)–(6) for radical relations (Section 4). We consider only radical relations satisfying the additional condition

∀a, b (ab = 0 ∧ a ≠ 0 ⇒ a ⪯̸ b).

Thus, if a real closed ring with a radical relation happens to be von Neumann regular then it has only one radical relation, and that one is definable in the language of rings.

We shall use the following notation to designate the various theories that we shall encounter. Th is the theory of real closed rings, Th_{1} the theory of von Neumann regular real closed rings, Th_{2} the theory of von Neumann regular real closed rings without minimal idempotents. Superscripts will indicate the language we are using, e.g., Th_{1}^{≤∧∨⪯}. If ⪯ is among the superscripts then we are dealing with rings with radical relations.

Theorem The models of Th^{⪯...} have prime extensions in the class of models of Th_{1}^{⪯...} and in the class of models of Th_{2}^{⪯...}.

Proof: Suppose that (A, ⪯) is a model of Th^{⪯...}, let X ⊆ Spec(A) = Sper(A) be the pro-constructible subspace belonging to the radical relation. Then σ_{A,X} : A → σ(A, X) is an embedding with respect to the radical relation, the partial order and the lattice operations. Thus σ_{A,X} is an extension to a model of Th_{1}^{⪯...}. Now suppose that f : A → B is another extension to a model of Th_{1}^{⪯...}. In particular, the inverse image of the radical relation under f is the given radical relation of A, hence the image of Spec(f) is X. By [Sch-M], Theorem 7.18, there is a unique homomorphism \overline{f} : σ(A) → B of partially ordered rings such that f = \overline{f}σ_{A}. If we use Spec(σ_{A}) to identify Spec(σ(A)) and Spec(A) with respect to the constructible topologies then the image of Spec(\overline{f}) in Spec(σ(A)) is the closed subspace X. Setting I = ∩_{p∈X}p, let g : σ(A) → σ(A)/I be the canonical homomorphism. Because of the isomorphism σ(A)/I ≅ σ(A, X), the homomorphism \overline{f} factors uniquely as

\overline{f} : σ(A) ⟶^{g} σ(A, X) ⟶^{h} B

where h is an embedding. To start with, this is an embedding with respect to partial order and lattice operations. Because both rings are von Neumann regular it is also an embedding with respect to the radical relation. Thus, σ_{A,X} is indeed a prime extension in the model class of Th_{1}^{⪯...}.

By [Li], Theorem 2, every model of Th_{1}^{∧∨} has a prime extension in Th_{2}^{∧∨}. Every injective homomorphism of real closed rings is an embedding with respect to partial order and lattice operations. Therefore every model of Th_{1} has a prime extension in Th_{2}. Being a homomorphism of von Neumann regular rings, the prime extension is also an embedding with respect to the radical relation. It follows that every model of Th_{1}^{⪯...} has a prime extension in the model class of Th_{2}^{⪯...}. By composition of prime extensions we conclude that models of Th^{⪯...} have prime extensions among the models of Th_{2}^{⪯...}.

Theorem The theory Th_{2}^{⪯...} is the model-completion of the theory Th^{⪯...}.

Proof: First we show that the theories Th_{2}^{⪯...} are model-complete. Without radical relations this has been shown in section 3. For every model of Th_{2} the unique radical relation is definable, hence every L^{⪯...}-formula φ is equivalent to an L^{...}-formula ψ modulo Th_{2}^{⪯...}. If A ⊆ B is an extension of models of Th_{2}^{⪯...} then this is also an extension of models of Th_{2}^{...}. If X_{1}, …, X_{n} are the free variables of φ and ψ and if a_{1}, …, a_{n} ∈ A then the following assertions are equivalent:

(i) A ⊧ φ(a_{1}, …, a_{n}); (ii) A ⊧ ψ(a_{1}, …, a_{n}); (iii) B ⊧ ψ(a_{1}, …, a_{n}); (iv) B ⊧ φ(a_{1}, …, a_{n}).

Thus, the extension A ⊆ B is elementary, and this implies model-completeness. In particular, Th_{2}^{⪯...} is the model companion of Th^{⪯...}.

To finish the proof we need to check the following amalgamation property: Given two models B, C of Th_{2}^{⪯...}, and a joint substructure A ⊆ B, C (the inclusions are denoted by i_{B} and i_{C}) which is a model of Th^{⪯...}, there exists a model D of Th_{2}^{⪯...} together with embeddings j_{B} : B → D and j_{C} : C → D such the diagram

A ⟶^{i_{B}} B
↓^{i_{C}}  ↓^{j_{B}}
C ⟶^{j_{C}} D

is commutative.

Let i : A → A′ be the prime extension of A in the model class of Th_{2}^{⪯...}. Then we obtain the following commutative diagram:

A  ^{^{i_{B}}  \overline{}}B [figure: i_{C}; i; i′_{B}]

A′ [figure: i′_{C}]

C

Amalgamation for von Neumann regular f-rings yields a commutative diagram

A′ ⟶^{i′_{B}} B
↓^{i′_{C}}  ↓^{j′_{B}}
C ⟶^{j′_{C}} D′ .

If j : D′ → D is any extension to a model of Th_{2} then j_{B} = jj′_{B} and j_{C} = jj′_{C} are extensions of models of Th_{2}, hence they are extensions also as models of Th_{2}^{⪯...}. Because of j_{B}i_{B} = jj′_{B}i′_{B}i = jj′_{C}i′_{C}i = j_{C}i_{C} the proof is complete.

Corollary The theories Th_{2}^{⪯...} are complete.

Proof: It suffices to show that each of these theories has a prime model. But Th has a prime model, namely R_{0}, the field of real algebraic numbers, which is also a prime model of each Th^{⪯...}. The prime extension of R_{0} in the model class of Th_{2}^{⪯...} is the prime model of Th_{2}^{⪯...}. Explicitly, this is the ring of locally constant functions from the Cantor set to the field R_{0}.

Finally, we study the question whether any one of the theories Th_{2}^{⪯...} has quantifier elimination. We shall see that the answer to this question depends on the language we use.

Example Th_{2}^{≤⪯} does not have quantifier elemination We must show that the models of Th_{2}^{≤⪯} do not have amalgamation over substructures. Pick a partially ordered field (K, ≤) whose space of orderings is a fan (cf. [Lam], Section 10) of four elements, τ_{1}, τ_{2}, τ_{3}, τ_{4}. Let R_{i} be the real closure of the totally ordered field (K, τ_{i}). For every i we define A_{i} to be the ring of locally constant functions from the Cantor set to the field R_{i}. Let

φ_{i} : (K, ≤) → (K, τ_{i}) → R_{i} → A_{i}

be the canonical homomorphisms. Both

ψ_{1} : K → A_{1} × A_{3} × A_{4}, x ↦ (φ_{i}(x))_{i=1,3,4}

and

ψ_{2} : K → A_{2} × A_{3} × A_{4}, x ↦ (φ_{i}(x))_{i=2,3,4}

are embeddings of the partially ordered field (K, ≤) into models of Th_{2}^{≤}. Since all rings involved are von Neumann regular they are also embeddings for the radical relations. Thus, (K, ≤) with its radical relation is a common substructure of A_{1} × A_{3} × A_{4} and A_{2} × A_{3} × A_{4}. If amalgamation holds then the images of Sper(ψ_{1}) and Sper(ψ_{2}) in Sper(K, ≤) must coincide. But this is not the case: im(Sper(ψ_{1})) = {τ_{1}, τ_{3}, τ_{4}}, im(Sper(ψ_{2})) = {τ_{2}, τ_{3}, τ_{4}}, and it follows that amalgamation fails. □

Now we change the language and consider Th_{2}^{∧∨⪯} instead of Th_{2}^{≤⪯}. The difference is that the class of substructures is smaller now. We prove:

Theorem The theory Th_{2}^{∧∨⪯} has elimination of quantifiers.

Proof: Let B, C be models of Th_{2}^{∧∨⪯}, suppose that A ⊆ B, C is a common substructure. The inclusion homomorphisms are denoted by i_{B} : A → B, i_{C} : A → C. Being a substructure of B and C, A is a reduced f-ring with radical relation. The radical relation of A is the inverse image of the radical relations of B and C, hence it is defined by the pro-constructible subset

X = im(Spec(i_{B})) = im(Spec(i_{C})) ⊆ Spec(A).

Note that im(Spec(i_{B})) and im(Spec(i_{C})) are both pro-constructible and are contained in the set of real prime ideals. Since they define the same radical relation, they must coincide.

We show that X contains every minimal prime ideal of Spec(A). If p is minimal and p ∉ X then there exists some s ∈ A such that D(s) ∩ X = ∅ and s ∉ p. But then s ≠ 0 and i_{B}(s) = 0, i_{C}(s) = 0. This is impossible since i_{B} and i_{C} are embeddings.

The universal property of the real closure ρ_{A} : A → ρ(A) ([Sch5], Proposition I 4.2; [Sch-M], Section 12) yields unique homomorphisms \overline{i_{B}} : ρ(A) → B and \overline{i_{C}} : ρ(A) → C such that i_{B} = \overline{i_{B}}ρ_{A}, i_{C} = \overline{i_{C}}ρ_{A}. Both \overline{i_{B}} and \overline{i_{C}} map squares to squares, hence they preserve the partial order. Every order preserving homomorphism between reduced f-rings also preserves the lattice operations ([Dz-M], Lemma 2.2). To show that \overline{i_{B}} and \overline{i_{C}} are embeddings we consider the diagram:

Sper(ρ(A)) ⟶^{Sper(ρ_{A})} Sper(A)
↓^{supp_{ρ(A)}}  ↓^{supp}^{A}
Spec(ρ(A)) ⟶^{Spec(ρ_{A})} Spec(A) .

Both Sper(ρ_{A}) and supp_{ρ(A)} are known to be homeomorphisms ([Sch5], Section I.3), supp_{A} is a homeomorphism onto its image. Thus, Spec(ρ_{A}) is also a homeomorphism onto its image. Since every minimal prime ideal of A is contained in im(Spec(i_{B})) = im(Spec(i_{C})) we conclude that every minimal prime ideal of ρ(A) is contained in im(Spec(\overline{i_{B}})) = im(Spec(\overline{i_{C}})). Thus, \overline{i_{B}} and \overline{i_{C}} are both injective.

The inverse images of the radical relations of B and C under \overline{i_{B}} and \overline{i_{C}} are two radical relations of ρ(A) which define the same radical relation on A. But because Spec(ρ_{A}) is injective there is only one radical relation on ρ(A) whose inverse image under ρ_{A} is the radical relation defined by X. Thus, the two radical relations of ρ(A) coincide. With respect to this radical relation the maps \overline{i_{B}} and \overline{i_{C}} are embeddings, i.e., ρ(A) is a model of Th^{∧∨⪯} and a common substructure of B and C. Since Th_{2}^{∧∨⪯} is the model-completion of Th^{∧∨⪯} there is an amalgamation

ρ(A) ⟶^{\overline{i_{B}}} B
\sqrt{i_{C}}  ↓^{j_{B}}
C ⟶^{j_{C}} D

with D a model of Th_{2}^{∧∨⪯}. But then

j_{B}i_{B} = j_{B}\overline{i_{B}}ρ_{A} = j_{C} \overline{i_{C}}ρ_{A} = j_{C}i_{C},

and the proof is finished.

6 Radical relations on valuation rings

¿From now on we shall deal with real closed domains. As we shall see in Section 8, the model companion of real closed domains with a radical relation will consist, in particular, of valuation rings. Thus, we first investigate more closely the structure of radical relations on valuation rings.

Let V be a valuation ring, K its field of quotients, Γ its value group ordered by ≤, and v : K → Γ ∪ {∞} the corresponding valuation. Furthermore, let ⪯ be a radical relation on V satisfying in addition

V^{×} = {a ∈ V | 1 ⪯ a}

(where V^{×} is the group of units of V). Such radical relations will be called unitary. A unitary radical relation canonically induces a binary relation ⪯ on the value group Γ by taking:

v(a) ⪯ v(b) :⇔ a ⪯ b

for a, b ≠ 0.

Since ⪯ is invariant under multiplication by units of V, the relation ⪯ on Γ is well defined. One easily checks that ⪯ is a convexity relation on Γ, i.e., a binary relation on the non-negative elements α, β, γ of Γ satisfying:

α ⪯ α
α ⪯ β, β ⪯ γ ⇒ α ⪯ γ
α ⪯ β  ⇒ α + γ ⪯ β + γ
α ⪯ 2α
0 ⪯ α  ⇒ α = 0
α ⪯ 0

It is useful to note that these axioms imply

α ≤ β ⇒ β ⪯ α

In fact, 0 ≤ β − α yields β − α ⪯ 0 and hence β ⪯ α.

Conversely, a convexity relation ⪯ on Γ defines a unitary radical relation ⪯ on V by taking

a ⪯ b :⇔ v(a) ⪯ v(b)

for all non-zero a, b ∈ V and a ⪯ 0 for all a ∈ V. In fact, let a ⪯ c and b ⪯ c. We have to show that a + b ⪯ c. From min(v(a), v(b)) ≤ v(a + b) we obtain

(a + b ⪯ a or a + b ⪯ b),

which implies a + b ⪯ c.

The Representation Theorem 2.5 of [P-S1] now shows that there exists a nonempty subset P of SpecV defining ⪯ by:

a ⪯ b iff (b ∈ p ⇒ a ∈ p) for all p ∈ P,

where a, b are from V.

Since a prime ideal p of the valuation ring V corresponds to the convex subgroup Δ of Γ defined by

α ∈ Δ iff |α| < v(a) for all a ∈ p,

and every convex subgroup Δ conversely defines a prime ideal of V, we have proved the following

Theorem Let Γ be an ordered abelian group. Then every convexity relation ⪯ on Γ is obtained by taking a non-empty set C of convex subgroups of Γ and defining:

γ ⪯ δ iff (γ ∈ Δ ⇒ δ ∈ Δ) for all Δ ∈ C.

In fact, one simply takes V to be the valuation ring of the field ℝ((Γ)) of formal Laurent series with exponents in Γ and coefficients in ℝ. Now the unitary radical relations ⪯ on V are in one-to-one correspondence to the convexity relations on Γ.

In Section 7 we shall show that for divisible groups Γ (this is equivalent to saying that ℝ((Γ)) is real closed) the elementary type of (V, ⪯) is determined by the elementary type of the linear ordering obtained from the equivalence classes of values in Γ defined by the induced convexity relation. Before doing so, we have to investigate, however, the model theory of convexity relations on ordered abelian groups.

7 Model theory of groups with a convexity relation

In this section Γ always denotes a (non-zero) ordered abelian group and ⪯ denotes a convexity relation on Γ. Elements α, β ∈ Γ are called equivalent if α ⪯ β and β ⪯ α. We write [α] for the equivalence class of α. Clearly [α] is a convex subset of Γ. The equivalence classes are linearly ordered by taking

[α] ≤ [β] :⟺ α ⪯ β.

The equivalence class [0] of zero is maximal in this linear ordering, it contains only the element 0. The linearly ordered set of equivalence classes of positive elements of Γ will be denoted by L(Γ, ⪯).

The map γ ↦ [γ] actually defines a valuation on the ordered group Γ which is compatible with the linear ordering of Γ. Conversely, every compatible valuation of Γ defines canonically a convexity relation on Γ. We will not go into further details here. Instead we list the important properties of the above map that will be used in the sequel.

For positive elements α, β, α − β of Γ we have

(o) [α] < [β] ⇒ β < α

(i) min([α], [β]) ≤ [α ± β] (ii) [α] < [β] ⇒ [α − β] = [α] (iii) min([α], [β]) = [α + β]

The next lemma will be essential for the model theory of ordered abelian groups with a convexity relation.

Lemma Let (Γ, ⪯) ⊆ (Γ′, ⪯′), (Γ, ⪯) ⊆ (Γ′′, ⪯′′) be divisible ordered abelian groups with convexity relations. Assume further that: Γ′ is countable, (Γ′′, ⪯′′) is ℵ_{1}-saturated, and there exists an order-preserving monomorphism σ : L(Γ′, ⪯′) → L(Γ′′, ⪯′′) which is the identity on L(Γ, ⪯). Then for every x ∈ Γ′ \ Γ there exists an order-monomorphism ρ : Γ ⊕ ℚx → Γ′′ which extends the identity on Γ and respects the convexity relations ⪯′ on Γ′ and ⪯′′ on Γ′′, i.e., [ρ(γ)] = σ([γ]) for all γ ∈ Γ ⊕ ℚx.

Proof: Proving the claim amounts to finding some y ∈ Γ′′ such that

(1) \frac{n}{m}x + α >′ 0 iff \frac{n}{m}y + α >′′ 0 for all n, m ∈ ℤ, m > 0, α ∈ Γ, and

(2) \frac{n}{m}x + α] <′ [\frac{n_{1}}{m_{1}} x + β] iff [\frac{n}{m}y + α] <′′ [\frac{n_{1}}{m_{1}} y + β] for all n, m, n_{1}, m_{1} ∈ ℤ, m, m_{1} > 0, and α, β ∈ Γ.

Then we simply define ρ(\frac{n}{m}x + α) = \frac{n}{m}y + α.

Using divisibility of the groups and choosing x and y to be positive, the first condition reduces to

(α <′ x ⇒ α <′′ y) and (x <′ β ⇒ y <′′ β)

for all α, β > 0. To reduce the second condition we use the fact that [nα] = [α] and hence [\frac{n}{m}α] = [α] for all n, m ∈ ℕ, m ≠ 0 and α > 0. Thus in (2) we need only consider the classes

[x ± α], [β − x], [x], [γ]

with α, β, γ > 0.

The equivalences below will show that (2) reduces to the presentation of the following 9 cases:

[α] < [x] , [x] < [β] , [x] = [γ] ,

in case [x] = [α]:

[β − x] < [γ] , [β − x] = [γ] , [β − x] > [γ] ,

in case [x] = [β]:

[β − x] < [γ] , [β − x] = [γ] , [β − x] > [γ] ,

where α, β, γ, x − α, β − x are all positive.

In order to see this we equivalently replace [x + α] < [γ] by:

([x] = [α] ∧ [α] < [γ]) ∨ ([x] < [α] ∧ [x] < [γ]) ∨ ([α] < [x] ∧ [α] < [γ])

and similarly for [γ] < [x + α] and [γ] = [x + α]. In all three cases we have thus isolated [x].

For differences we will be able to isolate [x], or to combine [x−α] with [x] = [α]: e.g., replacing equivalently [x − α] < [γ] by:

([x] = [α] ∧ [x − α] < [γ]) ∨ ([x] < [α] ∧ [x] < [γ]).

Similarly we treat the cases [x−α] = [γ], [γ] < [x−α] and [β−x] < [γ], [β−x] = [γ], [β − x] > [γ].

The following 7 cases reduce to earlier ones: We replace equivalently [x − α] < [x − β] by:

([x − α] = [β − α] ∧ [β − α] < [x − β]).

Similarly we treat the cases [α−x] < [β −x], [α−x] < [x−β], and [x−α] < [β −x].

Finally, we replace equivalently (assuming β < α)

[x − α] = [β − x] by [α − β] = [x − α] ∧ [α − β] = [β − x]
[x − α] = [x − β] by ([α − β] = [x − α] ∧ [α − β] = [x − β]) ∨ [x − α] < [α − β] .

Similarly we treat [α − x] = [β − x].

Returning to (1) we replace the condition α < x by [x] < [α], if this stronger condition holds for x. If not, we add [x] = [α] to α < x. Similarly we treat x < β.

After having done all this we are finally able to find y ∈ Γ′′ satisfying all these conditions. We distinguish two cases.

Case 1: The case [x] = [γ] does not occur among our conditions on x. Thus, we are left only with conditions of the type [α] < [x] and [x] < [β]. In this case we may simply choose y arbitrarily from the equivalence class σ([x]). Now [y] occupies the same position as [x] with respect to L(Γ, ⪯).

Case 2: There is at least one condition of the type [x] = [γ]. In this case all other conditions are talking about the position of x inside the equivalence class of γ. They either say α < x and something about the distance [x − α], or x < β and something about [β − x], where α and β are in the class of γ.

In order to find y ∈ Γ′′ satisfying the same conditions as x, we now use the assumption that (Γ′′, ⪯′′) is ℵ_{1}-saturated. Since there are at most countably many conditions on x, it suffices to find a realizing element y ∈ Γ′′ for every finite subset of such conditions. Then ℵ_{1}-saturation of (Γ′′, ⪯′′) guarantees that there is a y ∈ Γ′′ satisfying all conditions simultaneously. Thus, let us assume that we have only finitely many conditions on x.

Let α be maximal among all α’s occuring in conditions of the type α < x and in differences [x − α]. Similarly let β be minimal among all conditions of the type x < β and in differences [β −x]. Note that there may be no such α or no such β. In case α and β occurs and [x − α] = [β − x], we may simply choose y to be \frac{1}{2}(α + β). If [x − α] < [β − x] we have [x − α] = [β − α]. Thus, by adding the condition [β − α] < [β − x] we may forget about the conditions with the α’s. Similarly we proceed in the case [x − α] > [β − x]. Thus, we are left with the following two subcases:

Subcase 1: We have x < β, [x] = [β] and finitely many conditions on the position of [β − x]. In this case we choose z ∈ Γ′′ such that z belongs to the class σ([β − x]). Then y := β − z clearly satisfies y < β, and [β − y] occupies the same position as [β − x]. If [β] < [β − x], we get in particular [β] < [β − y]. This clearly implies [y] = [β]. If, however, [β] = [β − x], we actually choose y = \frac{1}{2}β, observing that [β] = [\frac{1}{2}β].

Subcase 2: We have α < x, [x] = [α] and finitely many conditions on the position of [x − α]. In this case we choose z ∈ σ([x − α]) and take y = z + α. Clearly α < y and [y − α] occupies the same position as [x − α]. In particular we get [α] ≤ [y − α], which implies [α] = [(y − α) + α] = [y].

Remark: By induction the monomorphism ρ in the lemma may be extended to an embedding of (Γ′, ⪯′) in (Γ′′, ⪯′′).

As a first consequence we get an Elementary Classification of divisible ordered abelian groups with a convexity relation.

Theorem Let (Γ′, ⪯′) and (Γ′′, ⪯′′) be divisible ordered abelian groups with convexity relations. If L(Γ′, ⪯′) and L(Γ′′, ⪯′′) are elementarily equivalent as linear orderings, then (Γ′, ⪯′) and (Γ′′, ⪯′′) are elementarily equivalent, and conversely.

Proof: For simplicity we use here the Continuum Hypothesis^{1} ℵ_{1} = 2^{ℵ_{0}}. We may then assume w.l.o.g. that (Γ′, ⪯′) and (Γ′′, ⪯′′) are both ℵ_{1}-saturated and Γ′ and Γ′′ both have cardinality ℵ_{1}.

Since L(Γ′, ⪯′) and L(Γ′′, ⪯′′) are assumed to be elementarily equivalent and are both definable from (Γ′, ⪯′) and (Γ′′, ⪯′′), resp., they are order-isomorphic. Note that both are either finite or of cardinality ℵ_{1}.

Now let (x_{ν})_{ν<ℵ_{1}} and (y_{ν})_{ν<ℵ_{1}} be enumerations of Γ′ and Γ′′, resp. We then define an increasing sequence (ρ_{ν})_{ν<ℵ_{1}} of partial isomorphisms between (Γ′, ⪯′) and (Γ′′, ⪯′′). Eventually, the direct limit of the ρ_{ν}’s will give an isomorphism of (Γ′, ⪯′) with (Γ′′, ⪯′′). Thus, both are elementarily equivalent.

Let us start by taking Γ_{0} = {0} and ρ_{0} the identity on Γ_{0}. If ν is a limit ordinal, we let ρ_{ν} be the direct limit of all ρ_{μ} with μ < ν. Assume we already have defined a partial isomorphism ρ_{ν} : Γ′_{ν} ↔ Γ′′_{ν} and ν = λ + 2n with λ a limit ordinal or 0. We then let x be the x ∈ Γ′ with the smallest index such that x ∉ Γ′_{ν}. By the lemma we then obtain an embedding ρ_{ν+1} : Γ′_{ν+1} ↔ Γ′′_{ν+1} ⊂ Γ′′ extending ρ_{ν} and respecting the radical relations, where Γ′_{ν+1} = Γ′_{ν} ⊕ ℚx.

In case ν = λ+(2n+1) with λ a limit ordinal or 0, we let y ∈ Γ′′ be the element with lowest index such that x ∉ Γ′′_{ν}. Now we may extend ρ_{ν}^{−1} to ρ_{ν+1}^{−1} : Γ′′_{ν+1} ↔ Γ′_{ν+1} with Γ′′_{ν+1} = Γ′_{ν} ⊕ ℚy.

For the reverse implication of the theorem we only have to observe that L(Γ, ⪯) is definable from (Γ, ⪯).

Corollary Let (Γ′, ⪯′) be a non-zero divisible ordered abelian group with a convexity relation. Then:

(a) if Th(L(Γ′, ⪯′)) is decidable, so is Th(Γ′, ⪯′);

(b) if Th(L(Γ′, ⪯′)) is model-complete, so is Th(Γ′, ⪯′).

Proof: (a) A complete recursive axiomatization of L(Γ′, ⪯′) clearly translates to a recursive axiomatization of (Γ′, ⪯′) which is complete by the Classification Theorem.

(b) W.l.o.g. let (Γ′, ⪯′) be countable and let (Γ, ⪯) a submodel of (Γ′, ⪯′). Furthermore, let (Γ′′, ⪯′′) be an ℵ_{1}-saturated elementary extension of (Γ, ⪯). We have to show that every existential sentence φ with parameters from Γ which holds in (Γ′, ⪯′) also holds in (Γ′′, ⪯′′).

Since Th(L(Γ′, ⪯′)) is model complete, L(Γ, ⪯) is a submodel of L(Γ′, ⪯′) and L(Γ′′, ⪯′′) is also ℵ_{1}-saturated, there exists an embedding σ : L(Γ′, ⪯′) → L(Γ′′, ⪯′′) which is the identity on L(Γ, ⪯). The existence statement of φ is realized by a finite sequence of elements of Γ′. Thus an iterated application of our Lemma maps this realizing sequence into Γ′′.

The corollary applies, e.g., to the case where L(Γ′, ⪯′) is a densely ordered set without first and last elements. In this case we get even more:

Theorem The theory of non-zero divisible ordered abelian groups (Γ′, ⪯′) for which the convexity relation ⪯′ yields a densely ordered set L(Γ′, ⪯′) without first and last elements, admits Elimination of Quantifiers. This theory is the model completion of the theory of ordered abelian groups with convexity relations.

1It should be remarked that there are several ways to avoid the Continuum Hypothesis.

Proof: The proof for Elimination of Quantifiers is essentially the same as that of part (b) of the corollary above. Now (Γ′, ⪯′) and (Γ′′, ⪯′′) are both models of our theory, and (Γ, ⪯) is a common substructure. The existence of the embedding σ is obtained from the fact that the theory of L(Γ′, ⪯′) allows Elimination of Quantifiers. Clearly, Γ may again be assumed to be divisible.

It remains to show that every ordered abelian group with a convexity relation, (Γ, ⪯), can be embedded into some model (Γ′, ⪯′) of our theory.

We pass to an elementary card(Γ)^{+}-saturated extension Γ′ of the ordered group Γ. For every non-zero convex subgroup Δ of Γ we shall choose a countable sequence Δ′_{i} (i ∈ ℚ) of non-zero proper convex subgroups of Γ′ such that Δ′_{i} ∩ Γ = Δ, and such that the set {Δ′_{i} | i ∈ ℚ} has (under inclusion) the order type of ℚ, i.e., Δ′_{i} ⊂ Δ′_{j} if and only if i < j for all i, j ∈ ℚ.

Defining then the convexity relation with respect to all groups Δ′_{i}, where Δ runs through a defining system for ⪯, gives an extension ⪯′ of ⪯ to Γ′ which has the desired properties.

Given a non-zero convex subgroup Δ of Γ, in order to construct the set {Δ′_{i}|i ∈ ℚ} we look for elements γ′_{i} ∈ Γ′ (i ∈ ℚ) such that the following set of conditions is satisfied:

(i) δ < γ′_{i} for all δ ∈ Δ and i ∈ ℚ, (ii) nγ′_{i} < ε for all positive ε ∉ Δ, n ∈ ℕ, and i ∈ ℚ, (iii) nγ′_{i} < γ′_{j} for all n ∈ ℕ, i < j, i, j ∈ ℚ.

Since Δ is a non-zero convex subgroup of Γ, every finite subset of these conditions can be satisfied by suitable choices of γ′_{i} in Γ. The fact that Γ′ is card(Γ)^{+}- saturated enables us to find the desired γ′_{i} in Γ′, satisfying all conditions (i)-(iii). Finally we set

Δ′_{i} := {γ′ ∈ Γ′ | |γ′| ≤ nγ′_{i} for some n ∈ ℕ}.

8 Real closed valuation rings with a radical relation

As in Section 6, V always denotes a valuation ring, K its field of fractions, Γ its value group, and v : K → Γ ∪ {∞} the valuation corresponding to V. Thus V = {a ∈ K | v(a) ≥ 0}. Furthermore, let ⪯ be a unitary radical relation on V.

In Section 6 we showed that unitary radical relations ⪯ on V correspond bijectively to convexity relations ⪯ on Γ via

a ⪯ b iff v(a) ⪯ v(b).

In particular, the linear order L(Γ, ⪯) introduced in Section 6 represents the linear order of the equivalence classes on V obtained from the radical relation ⪯ on V. In fact, if we take (cf. Section 4)

[a] = {b ∈ V | a ⪯ b, b ⪯ a}

and define

[a] ≤ [b] iff a ⪯ b,

we have

[a] ≤ [b] iff [v(a)] ≤ [v(b)]

for all a, b ∈ V \ {0}.

Thus, we may as well write L(V, ⪯) for L(Γ, ⪯). In case V is a real closed valuation ring (i.e., K is real closed and V is convex in K), the elementary type of L(V, ⪯) completely determines the elementary type of (V, ⪯). This will be the content of the next theorem. Before we come to this classification theorem, let us make an important observation.

Assume that (V, ⪯) and (V′, ⪯′) are two valuation rings with unitary radical relations. Then

(V, ⪯) ⊆ (V′, ⪯′) ⇒ (K, V) ⊆ (K′, V′),

i.e., K ∩ V′ = V or, equivalently, v′ = K′ → Γ′ ∪ {∞} extends the valuation v : K → Γ ∪ {∞}. In fact, if a ∈ K is not in V, then a^{−1} is a non-unit of V. Thus 1 ⪯̸ a^{−1}, which implies 1 ⪯̸′ a^{−1}. Therefore a^{−1} is a non-unit of V′. Hence a ∉ V′.

Theorem (Elementary Classification) Let (V′, ⪯′) and (V′′, ⪯′′) be two real closed valuation rings with unitary radical relation. We then have

L(V′, ⪯′) ≡ L(V′′, ⪯′′) iff (V′, ⪯′) ≡ (V′′, ⪯′′).

The proof of this theorem as well as the proof of the following corollary run along the same lines as that of the corresponding statements in Section 7. The main ingredient is the following (see [P1], Lemma 4.24)

Embedding Lemma Let (K, v) ⊆ (K′, v′) and (K, v) ⊆ (K′′, v′′) be henselian valued fields. Assume that K is countable, the residue field \overline{K} of K has characteristic zero, and (K′′, v′′) is ℵ_{1}-saturated. Moreover, let ρ : Γ′ → Γ′′ be an embedding of the value group Γ′ into Γ′′ which is the identity on Γ, and let τ : \overline{K}′ → \overline{K}′′ be an embedding of residue fields inducing the identity on \overline{K}. Then, if Γ is pure in Γ′ we can find an embedding ε : K′ → K′′ over K which respects the valuation and induces ρ and τ.

Proof: (Elementary Classification): By the Classification Theorem of Section 7 the assumption L(V′, ⪯′) ≡ L(V′′, ⪯′′) implies (Γ′, ⪯′) ≡ (Γ′′, ⪯′′). Assuming w.l.o.g. that (V′, ⪯′) and (V′′, ⪯′′) are both ℵ_{1}-saturated, also (Γ′, ⪯′) and (Γ′′, ⪯′′) are ℵ_{1}-saturated, as they are definable from (V′, ⪯′) and (V′′, ⪯′′), resp. (Note that ⪯′ and ⪯′′ are unitary. Thus the groups of units and the maximal ideal are definable in (V′, ⪯′) and (V′′, ⪯′′).)

As in Section 7 we assume for simplicity ℵ_{1} = 2^{ℵ_{0}}. Thus V′, V′′, Γ′, Γ′′ all have cardinality ℵ_{1}. Since (Γ′, ⪯′) and (Γ′′, ⪯′′) are elementarily equivalent, there exists an isomorphism ρ : (Γ′, ⪯′) ↔ (Γ′′, ⪯′′). Moreover, there also exists an isomorphism τ : \overline{K}′ ↔ \overline{K}′′ of the residue fields. This follows from the fact that K′ and K′′ are real closed fields, definable from (V′, ⪯′) and (V′′, ⪯′′), resp. Thus, both are ℵ_{1}-saturated and of cardinality ℵ_{1}. Hence they are isomorphic.

Now we apply the back-and-forth method used in Section 7 to our situation. The Embedding Lemma above then yields an isomorphism ε : (K′, v′) ↔ (K′′, v′′) where K′ and K′′ are the fields of fractions of V′ and V′′, resp., and v′ and v′′ are the corresponding valuations. As a substructure (K, v) to start the back-and-forth procedure, we use K = ℚ with the trivial valuation.

Finally we restrict ε to V′ and observe that ε respects the radical relations. In fact, a ⪯′ b is equivalent to v′(a) ⪯′ v′(b). Applying ρ, this is equivalent to v′′(ε(a)) ⪯′′ v′′(ε(b)) which is equivalent to ε(a) ⪯′ ε(b).

As in Section 7 we get some model-theoretic consequences from the Embedding Lemma and the Classification Theorem. Note that item (c) in the following corollary is a generalization of the main theorem in [Ch-D].

Corollary Let (V′, ⪯′) be a real closed valuation ring with a unitary radical relation.

(a) If the theory of the linear ordering L(V′, ⪯′) is decidable, so is the theory of (V′, ⪯′).

(b) If Th(L(V′, ⪯′)) is model-complete, so is Th(V′, ⪯′).

(c) If L(V′, ⪯′) is densely ordered without first and last element, then (V′, ⪯′, |) has Quantifier Elimination, where | is the divisibility relation in V′.

Proof: (a) By the Classification Theorem, a complete recursive axiomatization of L(V′, ⪯′) yields a complete recursive axiomatization of (V′, ⪯′).

(b) W.l.o.g. we may assume that (V, ⪯) ⊆ (V′, ⪯′) are two countable real closed valuation rings with unitary radical relations, where L(V, ⪯) is an elementary substructure of L(V′, ⪯′). As explained at the beginning of this section, we find (K, V) ⊆ (K′, V′). Next let (V′′, ⪯′′) be an elementary ℵ_{1}-saturated extension of (V, ⪯). Then also (K′′, V′′) is an elementary extension of (K, V) and is ℵ_{1}- saturated. Moreover, L(V′′, ⪯′′) is an elementary ℵ_{1}-saturated extension of L(V, ⪯). Thus by the corollary of Section 7, (Γ′′, ⪯′′) is an elementary (ℵ_{1}-saturated) extension of (Γ, ⪯). Therefore we find an embedding ρ : (Γ′, ⪯′) → (Γ′′, ⪯′′) inducing the identity on (Γ, ⪯).

Applying now the Embedding Lemma, we obtain an embedding ε : (K′, V′) → (K′′, V′′), which induces ρ. Thus we get an embedding ε : (V′, ⪯′) → (V′′, ⪯′′) as explained in the proof of the Classification Theorem. Hence, every existential sentence with parameters from V transfers from (V′, ⪯′) to (V′′, ⪯′′) and then down to (V, ⪯).

(c) The main difference compared with (b) is that (V, ⪯) need not be a real closed valuation ring, but rather a common substructure of (V′, ⪯′) and (V′′, ⪯′′). However, we have added the divisibility relation now. Thus, (V, ⪯, |) being a substructure of (V′, ⪯′, |), implies that again (K, V) ⊆ (K′, V′) and similarly (K, V) ⊆ (K′′, V′′). Since the real closures of K together with the induced valuations inside (K′, V′) and (K′′, V′′) are isomorphic (cf. [P1], Section 4.5) we may even assume that K is real closed. In fact, passage to the real closure will make the value group Γ of V divisible. This, however, does not change the radical relation. Thus we may actually assume that (V, ⪯) is a real closed valuation ring. Comparing with (b), now L(V, ⪯) need not be an elementary substructure of L(V′, ⪯′) or of L(V′′, ⪯′′). Nevertheless, Section 7 yields an embedding ρ : (Γ′, ⪯′) → (Γ′′, ⪯′′). As in (b), we conclude that (V′, ⪯′) can be embedded into (V′′, ⪯′′) over (V, ⪯).

We now return to the model companion of real closed domains.

Theorem The model companion of the theory of real closed domains with radical relation is the theory of real closed valuation rings (V, ⪯) with unitary radical relation such that the linear ordering L(V, ⪯) is dense without extreme points.

Proof: It remains to show that every real closed domain (A, ⪯) with radical relation ⪯ can be embedded into a real closed valuation ring (V′, ⪯′) where ⪯′ is unitary and L(V′, ⪯′) is dense without extreme points.

Let (A, ⪯) be a real closed domain with a radical relation. Since {0} ⊆ A is a prime ideal, K = qf(A) is a real closed field and A is integrally closed in K. The partial order ≤ of A is linear and is the restriction of the total order of K. Every prime ideal p ⊆ A is convex, i.e., 0 < a < b ∈ p implies a ∈ p for all a, b ∈ A. Considering the prime ideals p ∈ Spec(A, ⪯), there is a largest one, lets call it p_{m}, among them.

Let V be a convex valuation ring of K, dominating p_{m}, i.e., V contains A and M ∩ A = p_{m} where M is the maximal ideal of V. Such a valuation ring V can be obtained by taking the convex hull of the localization A_{p_{m}} in K (cf. [P2], Lemma 4.3). In order to extend ⪯ from A to V, for every prime ideal p ∈ Spec(A, ⪯) we choose some prime ideal P in V such that P ∩ V = p. Then by [PS1], Lemma 2.7, the collection of such prime ideals P defines an extension of ⪯. For P we simply choose the convex hull of p in K. Since A is closed under radicals of positive elements, P is again prime. Thus we have (A, ⪯) ⊆ (V, ⪯). In order to make ⪯ also unitary, we need to know that the convex hull of p_{m} is actually the maximal ideal M of V. This is guaranteed if we choose for V the maximal valuation ring of K, dominating p_{m}.

At this point we use the fact that the valued real closed field (K, V) has a real closed residue field R, and a divisible value group Γ. Therefore (K, V) can be embedded (as a valued field) into the field R((Γ)) of formal Laurent series with coefficients from R and exponents from Γ (cf. [PC], Satz 21). Let R[[Γ]] denote the canonical valuation ring of R((Γ)). As in Section 6, the radical relation ⪯ on V corresponds to a convexity relation ⪯ on Γ, which in turn gives a radical relation ⪯ on R[[Γ]], extending that of V.

Now we can use the last result of Section 7 and extend (Γ, ⪯) to a divisible ordered abelian group with a convexity relation (Γ′, ⪯′) such that L(Γ′, ⪯′) is dense and has no extreme points. The corresponding radical relation ⪯′ on V′ := R[[Γ′]] then yields the desired object (V′, ⪯′).

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