scholarly journals A VALUATION THEORETIC CHARACTERIZATION OF RECURSIVELY SATURATED REAL CLOSED FIELDS

2015 ◽  
Vol 80 (1) ◽  
pp. 194-206 ◽  
Author(s):  
PAOLA D’AQUINO ◽  
SALMA KUHLMANN ◽  
KAREN LANGE
Keyword(s):  
Abelian Group ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Theoretic Characterization ◽  
Image Position ◽  
Recursively Saturated

AbstractWe give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization for κ-saturation, for a cardinal $\kappa \ge \aleph _0 $. Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

ON THE CLASSIFICATION OF CENTRALLY FINITE ALTERNATIVE DIVISION RINGS SATISFYING ALGEBRAIC CLOSURE CONDITIONS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250088
Author(s):  
RICCARDO GHILONI
Keyword(s):  
Algebraic Closure ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Division Rings ◽  
Closed Fields ◽  
Closure Conditions ◽  
The One ◽  
Algebraically Closed Fields

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

COMPARING TWO VERSIONS OF THE REALS

2016 ◽  
Vol 81 (3) ◽  
pp. 1115-1123
Author(s):  
G. IGUSA ◽  
J. F. KNIGHT
Keyword(s):  
Residue Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Computing Power ◽  
Ordered Field ◽  
Image Position ◽  
Recursively Saturated

AbstractSchweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

Relative randomness and real closed fields

2005 ◽  
Vol 70 (1) ◽  
pp. 319-330 ◽  
Author(s):  
Alexander Raichev
Keyword(s):  
Real Number ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Closed Fields ◽  
Master's Thesis ◽  
National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

A note on valuation definable expansions of fields

1998 ◽  
Vol 63 (2) ◽  
pp. 739-743 ◽  
Author(s):  
Deirdre Haskell ◽  
Dugald Macpherson
Keyword(s):  
Valuation Ring ◽  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Valued Field ◽  
Closed Fields ◽  
Binary Predicate ◽  
Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

scholarly journals THE NON-AXIOMATIZABILITY OF O-MINIMALITY

2014 ◽  
Vol 79 (01) ◽  
pp. 54-59 ◽  
Author(s):  
ALEX RENNET
Keyword(s):  
Function Symbol ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Fields ◽  
Closed Fields

Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory To−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L), is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-minimal.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

Algebraic theories with definable Skolem functions

1984 ◽  
Vol 49 (2) ◽  
pp. 625-629 ◽  
Author(s):  
Lou van den Dries
Keyword(s):  
Nonempty Subset ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
First Order ◽  
Definable Subset ◽  
Skolem Functions ◽  
Algebraic Theories ◽  
Closed Fields ◽  
Definitional Extension

(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member.(1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D, as follows: D is a union of finitely many singletons and intervals (a, b) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval (a, b) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that (a, b) ⊂ D. Four cases have to be distinguished:(i) a = − ∞ and b = + ∞; then we choose 0;(ii) a = − ∞ and b ∈ R; then we choose b − 1;(iii) a ∈ R and b ∈ = + ∞; then we choose a + 1;(iv) a ∈ R and b ∈ R; then we choose the midpoint (a + b)/2.It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions.

Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

1972 ◽  
Vol 37 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Simple Proof ◽  
Finite Sequence ◽  
The Other ◽  
Closed Field ◽  
Prime Model ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Transcendental Theory

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this followsIf T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

scholarly journals Models of true arithmetic are integer parts of models of real exponentation

2021 ◽  
Vol 13 ◽  
Author(s):  
Merlin Carl ◽  
Lothar Sebastian Krapp
Keyword(s):  
Multiplicative Group ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
Integer Part ◽  
Real Numbers ◽  
The Real ◽  
Positive Elements ◽  
Closed Fields ◽  
Schanuel's Conjecture

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

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