scholarly journals Generic derivations on o-minimal structures

2020 ◽  
pp. 2150007
Author(s):  
Antongiulio Fornasiero ◽  
Elliot Kaplan
Keyword(s):  
Complete Model ◽  
Minimal Theory ◽  
Elementary Formula ◽  
Ordered Fields ◽  
Differential Fields ◽  
Model Completion ◽  
Open Core ◽  
Theory Formula

Let [Formula: see text] be a complete, model complete o-minimal theory extending the theory [Formula: see text] of real closed ordered fields in some appropriate language [Formula: see text]. We study derivations [Formula: see text] on models [Formula: see text]. We introduce the notion of a [Formula: see text]-derivation: a derivation which is compatible with the [Formula: see text]-definable [Formula: see text]-functions on [Formula: see text]. We show that the theory of [Formula: see text]-models with a [Formula: see text]-derivation has a model completion [Formula: see text]. The derivation in models [Formula: see text] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [Formula: see text]-substructure of [Formula: see text]. If [Formula: see text], then [Formula: see text] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [Formula: see text] has [Formula: see text] as its open core, that [Formula: see text] is distal, and that [Formula: see text] eliminates imaginaries. We also show that the theory of [Formula: see text]-models with finitely many commuting [Formula: see text]-derivations has a model completion.

Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

Model-complete theories of formally real fields and formally p-adic fields

1983 ◽  
Vol 48 (4) ◽  
pp. 1130-1139
Author(s):  
William H. Wheeler
Keyword(s):  
Galois Groups ◽  
General Situation ◽  
Complete Model ◽  
Yield Model ◽  
Valued Fields ◽  
Subsequent Investigation ◽  
Ordered Fields ◽  
Closed Fields ◽  
Complete Theories ◽  
Algebraically Closed Fields

The complete, model-complete theories of pseudo-algebraically closed fields were characterized completely in [11]. That work constituted the first step towards determining all the model-complete theories of fields in the usual language of fields. In this paper the second step is taken. Namely, the methods of [11] are extended to characterize the complete, model-complete theories of pseudo-real closed fields and pseudo-p-adically closed fields.In order to unify the treatment of these two types of fields, the relevant properties of real closed ordered fields and p-adically closed valued fields are abstracted. The subsequent investigation of model-complete theories of fields is based entirely on these properties. The properties were selected in order to solve three problems: (1) finding universal theories with the joint embedding property, (2) finding first order conditions in the usual language of fields which are necessary and sufficient for a polynomial over a field to have a zero in a formally real or formally p-adic extension of that field, and (3) finding subgroups of Galois groups whose fixed fields are formally real or formally p-adic.This paper is related to, and uses in §1 but not in the other sections, parts of K. McKenna's work [8] on model-complete theories of ordered fields and p-valued fields. However, the results herein are not direct consequences of his work, both because these results apply to a more general situation and because they use a different formal language. Concerning the latter point, in some instances, such as real closed ordered fields and p-adically closed valued fields, model-complete theories in expanded languages do yield model-complete theories of ordinary fields other than theories of pseudo-algebraically closed fields. However, in other cases, such as differentially closed fields, this is not so.

scholarly journals DEFINABLE AND INVARIANT TYPES IN ENRICHMENTS OF NIP THEORIES

2017 ◽  
Vol 82 (1) ◽  
pp. 317-324 ◽  
Author(s):  
SILVAIN RIDEAU ◽  
PIERRE SIMON
Keyword(s):  
Sufficient Condition ◽  
Image Position ◽  
Differential Fields ◽  
Model Completion ◽  
Formula Type

AbstractLet T be an NIP ${\cal L}$-theory and $\mathop T\limits^\~ $ be an enrichment. We give a sufficient condition on $\mathop T\limits^\~$ for the underlying ${\cal L}$-type of any definable (respectively invariant) type over a model of $\mathop T\limits^\~$ to be definable (respectively invariant). These results are then applied to Scanlon’s model completion of valued differential fields.

On Dedekind complete o-minimal structures

1987 ◽  
Vol 52 (1) ◽  
pp. 156-164
Author(s):  
Anand Pillay ◽  
Charles Steinhorn
Keyword(s):  
Complete Model ◽  
Boolean Combination ◽  
Minimal Theory

AbstractFor a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model of T is sequentially complete if and only if ≺ for some Dedekind complete model . We also prove that if T has a Dedekind complete model of power greater than , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory—namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables—that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

scholarly journals Recursive functions and existentially closed structures

2019 ◽  
Vol 20 (01) ◽  
pp. 2050002
Author(s):  
Emil Jeřábek
Keyword(s):  
Model Theory ◽  
Recursive Functions ◽  
Existentially Closed ◽  
Model Completion ◽  
The Relationship ◽  
Theory Formula

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory [Formula: see text] in which all partially recursive functions are representable, yet [Formula: see text] does not interpret Robinson’s theory [Formula: see text]. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of [Formula: see text] theories interpretable in existential theories in the process.

scholarly journals Model theory of Steiner triple systems

2019 ◽  
Vol 20 (02) ◽  
pp. 2050010
Author(s):  
Silvia Barbina ◽  
Enrique Casanovas
Keyword(s):  
Model Theory ◽  
Triple System ◽  
Steiner Triple System ◽  
Quantifier Elimination ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Limit Formula ◽  
Model Completion ◽  
Theory Formula

A Steiner triple system (STS) is a set [Formula: see text] together with a collection [Formula: see text] of subsets of [Formula: see text] of size 3 such that any two elements of [Formula: see text] belong to exactly one element of [Formula: see text]. It is well known that the class of finite STS has a Fraïssé limit [Formula: see text]. Here, we show that the theory [Formula: see text] of [Formula: see text] is the model completion of the theory of STSs. We also prove that [Formula: see text] is not small and it has quantifier elimination, [Formula: see text], [Formula: see text], elimination of hyperimaginaries and weak elimination of imaginaries.

Prime model extensions for differential fields of characteristic p ≠ 0

1974 ◽  
Vol 39 (3) ◽  
pp. 469-477 ◽  
Author(s):  
Carol Wood
Keyword(s):  
Prime Model ◽  
Perfect Field ◽  
Characteristic P ◽  
Forthcoming Book ◽  
Closed Fields ◽  
Differential Fields ◽  
Model Completion

The main purpose of this paper is to show that there exists a prime differentially closed extension over each differentially perfect field. We do this in a roundabout manner by first giving new and simple axioms for the theory of differentially closed fields (in the manner of Blum [1] for characteristic 0) and by proving that this theory is the model completion of the theory of differentially perfect fields. This paper can be read independently from [10], where we gave more complicated axioms for the same theory (in the manner of Robinson [6] for characteristic 0).I am indebted to E. R. Kolchin for answering many questions and for making the manuscript of his forthcoming book [2] available to me.

A generalized model companion for a theory of partially ordered fields

1979 ◽  
Vol 44 (4) ◽  
pp. 643-652
Author(s):  
Werner Stegbauer
Keyword(s):  
General Model ◽  
Order Theory ◽  
Model Companion ◽  
Generalized Model ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Model Completion

The notion of a model companion for a first-order theory T was introduced and discussed in [1] and [2] as a generalization of the concept of a model completion of a theory. Both concepts reflect, on a general model theoretic level, properties of the theory of algebraically closed fields. The literature provides many examples of first-order theories with and without model companions—see [3] for a survey of these results. In this paper, we give a further generalization of the notion of a model companion.Roughly speaking, we allow instead of embeddings more general classes of maps (e.g. homomorphisms) and we allow any set of formulas which is preserved by these maps instead of existential formulas. This plan is worked out in detail in [5], where we discuss also several examples. One of these examples is given in this paper.In order to clarify the model theoretic background, we now introduce the relevant concepts and theorems from [5].

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