Plongement dense d'un corps ordonné dans sa clôture réelle

1991 ◽  
Vol 56 (3) ◽  
pp. 974-980 ◽  
Author(s):  
Françoise Delon
Keyword(s):  
Complete Theory ◽  
Unary Predicate ◽  
Ordered Field ◽  
Ordered Fields ◽  
Elementary Equivalent ◽  
Real Closure

AbstractWe study the structures (K ⊂ Kr), where K is an ordered field and Kr its real closure, in the language of ordered fields with an additional unary predicate for the subfield K. Two such structures (K ⊂ Kr) and (L ⊂ Lr) are not necessarily elementary equivalent when K and L are. But with some saturation assumption on K and L, then the two structures become equivalent, and we give a description of the complete theory.

scholarly journals Ordered fields dense in their real closure and definable convex valuations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lothar Sebastian Krapp ◽  
Salma Kuhlmann ◽  
Gabriel Lehéricy
Keyword(s):  
Ordered Fields ◽  
Real Closure

Abstract In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

On the notion of algebraic closedness for noncommutative groups and fields

1971 ◽  
Vol 36 (3) ◽  
pp. 441-444 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Wide Class ◽  
First Order ◽  
Division Rings ◽  
Ordered Field ◽  
Ordered Fields ◽  
Skew Fields

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

Model completeness of o-minimal structures expanded by Dedekind cuts

2005 ◽  
Vol 70 (1) ◽  
pp. 29-60 ◽  
Author(s):  
Marcus Tressl
Keyword(s):  
Valuation Ring ◽  
Ordered Set ◽  
Real Closed Field ◽  
Complete Theory ◽  
Closed Field ◽  
Model Completeness ◽  
Unary Predicate ◽  
Minimal Structure ◽  
Totally Ordered Set ◽  
Dedekind Cuts

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.

Transfer principles for pseudo real closed e-fold ordered fields

1986 ◽  
Vol 51 (4) ◽  
pp. 981-991 ◽  
Author(s):  
Şerban A. Basarab
Keyword(s):  
Elementary Theory ◽  
Field Extension ◽  
Equivalent Model ◽  
Ordered Field ◽  
Ordered Fields ◽  
Order Language ◽  
Irreducible Variety ◽  
Famous Paper ◽  
Definition Of ◽  
Geometric Definition

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.

Generic computation of the real closure of an ordered field

1996 ◽  
Vol 42 (4-6) ◽  
pp. 541-549 ◽  
Author(s):  
Zenon Ligatsikas ◽  
Renaud Rioboo ◽  
Marie Françoise Roy
Keyword(s):  
The Real ◽  
Ordered Field ◽  
Real Closure ◽  
Generic Computation

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II

2018 ◽  
Vol 83 (2) ◽  
pp. 617-633
Author(s):  
PHILIP EHRLICH ◽  
ELLIOT KAPLAN
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Integer Part ◽  
Ordered Field ◽  
Number Systems ◽  
Ordered Fields ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Models ◽  
Surreal Numbers

AbstractIn [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.

A transfer theorem for Henselian valued and ordered fields

1993 ◽  
Vol 58 (3) ◽  
pp. 915-930 ◽  
Author(s):  
Rafel Farré
Keyword(s):  
Residue Field ◽  
Valued Fields ◽  
Transfer Theorem ◽  
Ordered Field ◽  
Existentially Closed ◽  
Henselian Valued Fields ◽  
Ordered Fields ◽  
Elementary Embeddings ◽  
Residue Fields ◽  
Natural Way

AbstractIn well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

A note on computable real fields

1970 ◽  
Vol 35 (2) ◽  
pp. 239-241 ◽  
Author(s):  
E. W. Madison
Keyword(s):  
Algebraic Closure ◽  
Algebraic Extension ◽  
Real Field ◽  
Ordered Fields ◽  
Real Closure ◽  
Carry Over

It is well known that every field (formally, real field ) has an algebraic closure (real-closure ). This is to say is an algebraic extension of which is algebraically closed (real-closed). Of course, certain properties of carry over to . In particular, M. O. Rabin has proved in [3] that the algebraic closure—which is of course unique up to isomorphism—of a computable field is computable. The purpose of this note is to establish an analogue of Rabin's theorem for formally real fields. It is clear that a direct analogue can be formulated only in the case of ordered fields, for otherwise there may be many (nonisomorphic) such .

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