scholarly journals Model companions of theories with an automorphism

2000 ◽  
Vol 65 (3) ◽  
pp. 1215-1222 ◽  
Author(s):  
Hirotaka Kikyo
Keyword(s):  
Amalgamation Property ◽  
Complete Theory ◽  
Model Companion ◽  
Independence Property ◽  
Model Completion

AbstractFor a theory T in L, Tσ is the theory of the models of T with an automorphism σ. If T is an unstable model complete theory without the independence property, then Tσ has no model companion. If T is an unstable model complete theory and Tσ has the amalgamation property, then Tσ has no model companion. If T is model complete and has the fcp, then Tσ has no model completion.

A characterization of companionable, universal theories

1978 ◽  
Vol 43 (3) ◽  
pp. 402-429 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Amalgamation Property ◽  
Finiteness Condition ◽  
Model Companion ◽  
Coherence Property ◽  
Finite Presentation ◽  
First Order Theory ◽  
Closed Fields ◽  
Inductive Theory ◽  
Model Completion ◽  
Finitely Presented

A first-order theory is companionable if it is mutually model-consistent with a model-complete theory. The latter theory is then called a model-companion for the former theory. For example, the theory of formally real fields is a companionable theory; its model-companion is the theory of real closed fields. If a companionable, inductive theory has the amalgamation property, then its model-companion is actually a model-completion. For example, the theory of fields is a companionable, inductive theory with the amalgamation property; its model-completion is the theory of algebraically closed fields.The goal of this paper is the characterization, by “algebraic” or “structural” properties, of the companionable, universal theories which satisfy a certain finiteness condition. A theory is companionable precisely when the theory consisting of its universal consequences is companionable. Both theories have the same model-companion if either has one. Accordingly, nought is lost by the restriction to universal theories. The finiteness condition, finite presentation decompositions, is an analogue for an arbitrary theory of the decomposition of a radical ideal in a Noetherian, commutative ring into a finite intersection of prime ideals for the theory of integral domains. The companionable theories with finite presentation decompositions are characterized by two properties: a coherence property for finitely generated submodels of finitely presented models and a homomorphism lifting property for homomorphisms from submodels of finitely presented models.

The structure of algebraically and existentially closed Stone and double Stone algebras

1989 ◽  
Vol 54 (2) ◽  
pp. 363-375 ◽  
Author(s):  
David M. Clark
Keyword(s):  
Finite Algebra ◽  
Amalgamation Property ◽  
Explicit Construction ◽  
Structural Description ◽  
Model Companion ◽  
Existentially Closed ◽  
Boolean Spaces ◽  
Countably Infinite ◽  
Model Completion ◽  
Primitive Recursive

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

The model-companion of a class of structures

1972 ◽  
Vol 37 (3) ◽  
pp. 546-556 ◽  
Author(s):  
G. L. Cherlin
Keyword(s):  
Order Theory ◽  
Model Companion ◽  
Elementary Substructure ◽  
Closed Model ◽  
First Order ◽  
First Order Theory ◽  
Logical Equivalence ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Model Completion

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:(i) Σ* is the class of models of some first order theory K*.(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

Generic variations of models ofT

2002 ◽  
Vol 67 (3) ◽  
pp. 1025-1038 ◽  
Author(s):  
Andreas Baudisch
Keyword(s):  
Complete Theory ◽  
Model Companion

AbstractLetTbe a model-complete theory that eliminates the quantifier ∃∞xForTwe construct a theoryT+such that any element in a model ofT+determines a model ofT. We show thatT+has a model companionT1. We can iterate the construction. The produced theories are investigated.

scholarly journals The Theory of Tracial Von Neumann Algebras Does Not Have A Model Companion

2013 ◽  
Vol 78 (3) ◽  
pp. 1000-1004 ◽  
Author(s):  
Isaac Goldbring ◽  
Bradd Hart ◽  
Thomas Sinclair
Keyword(s):  
Positive Solution ◽  
Von Neumann Algebras ◽  
Quantifier Elimination ◽  
Complete Theory ◽  
Embedding Problem ◽  
Model Companion ◽  
Von Neumann

AbstractIn this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II1 factors.

scholarly journals The strict order property and generic automorphisms

2002 ◽  
Vol 67 (1) ◽  
pp. 214-216 ◽  
Author(s):  
Hirotaka Kikyo ◽  
Saharon Shelah
Keyword(s):  
Complete Theory ◽  
Order Property ◽  
Model Companion

AbstractIf T is a model complete theory with the strict order property, then the theory of the models of T with an automorphism has no model companion.

Partial orderings of fixed finite dimension: Model companions and density

1981 ◽  
Vol 46 (4) ◽  
pp. 789-802 ◽  
Author(s):  
Alfred B. Manaster ◽  
Jeffrey B. Remmel
Keyword(s):  
Finite Dimension ◽  
Partial Ordering ◽  
Common Language ◽  
Model Companion ◽  
Elementary Substructure ◽  
Undecidable Theory ◽  
Natural Notion ◽  
Partial Orderings ◽  
Definition Of ◽  
Model Completion

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

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].

scholarly journals MODEL THEORETIC PROPERTIES OF METRIC VALUED FIELDS

2014 ◽  
Vol 79 (3) ◽  
pp. 655-675
Author(s):  
ITAÏ BEN YAACOV
Keyword(s):  
Projective Spaces ◽  
Order Logic ◽  
First Order Logic ◽  
Model Companion ◽  
Valued Fields ◽  
First Order ◽  
Valued Field ◽  
Metric Structures ◽  
Model Completion

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic.For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KPn, as bounded metric structures.We show that the class of (projective spaces over) metric valued fields is elementary, with theory MVF, and that the projective spaces Pn and are Pm biinterpretable for every n, m ≥ 1. The theory MVF admits a model completion ACMVF, the theory of algebraically closed metric valued fields (with a nontrivial valuation). This theory is strictly stable (even up to perturbation).Similarly, we show that the theory of real closed metric valued fields, RCMVF, is the model companion of the theory of formally real metric valued fields, and that it is dependent.

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