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.

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 variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (04) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z 2} and {Y(u), u (Z 2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z 2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

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.

scholarly journals DIFFERENTIATION IN P-MINIMAL STRUCTURES AND A p-ADIC LOCAL MONOTONICITY THEOREM

2014 ◽  
Vol 79 (4) ◽  
pp. 1133-1147 ◽  
Author(s):  
TRISTAN KUIJPERS ◽  
EVA LEENKNEGT
Keyword(s):  
Wide Class ◽  
Local Version ◽  
First Order ◽  
Almost Everywhere ◽  
Local Monotonicity ◽  
Definable Functions

AbstractWe prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K → K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.

Undecidability in diagonalizable algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 79-116 ◽  
Author(s):  
V. Yu. Shavrukov
Keyword(s):  
Wide Class ◽  
Formal Theory ◽  
First Order ◽  
Unary Operator ◽  
Diagonalizable Algebra

AbstractIf a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

On the variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (4) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z2} and {Y(u), u (Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

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 ${\bf{No}}$.

scholarly journals Order in Affine and Projective Geometry

1963 ◽  
Vol 6 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Joe Lipman
Keyword(s):  
Projective Geometry ◽  
Affine Geometry ◽  
Skew Field ◽  
Ordered Fields ◽  
Skew Fields

In this note we give a characterisation of ordered skew fields by certain properties of the negative elements.Properties of negative elements of a skew field K are then interpreted as statements about "betweenness" in any affine geometry over K, or as statements about "separation" in any projective geometry over K.In this way, our axioms for ordered fields permit of immediate translation into postulates of order for affine or projective geometry (over a field). The postulates so obtained seem simple enough to be of interest, (cf. [4], p. 22)

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