scholarly journals Model theory of operator algebras III: elementary equivalence and II1factors

2014 ◽  
Vol 46 (3) ◽  
pp. 609-628 ◽  
Author(s):  
Ilijas Farah ◽  
Bradd Hart ◽  
David Sherman
Keyword(s):  
Model Theory ◽  
Operator Algebras ◽  
Elementary Equivalence

Some model theory for almost real closed fields

1996 ◽  
Vol 61 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
Françoise Delon ◽  
Rafel Farré
Keyword(s):  
Model Theory ◽  
Residue Field ◽  
Elementary Equivalence ◽  
First Order ◽  
Valuation Rings ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields ◽  
Elementary Inclusion ◽  
Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

The model theory of finitely generated finite-by-abelian groups

1984 ◽  
Vol 49 (4) ◽  
pp. 1115-1124
Author(s):  
Francis Oger
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Elementary Embeddings

AbstractIn [O1], we gave algebraic characterizations of elementary equivalence for finitely generated finite-by-abelian groups, i.e. finitely generated FC-groups. We also provided several examples of finitely generated finite-by-abelian groups which are elementarily equivalent without being isomorphic.In this paper, we shall use our previous results to describe precisely the models of the theories of finitely generated finite-by-abelian groups and the elementary embeddings between these models.

scholarly journals Model theory of operator algebras I: stability

2013 ◽  
Vol 45 (4) ◽  
pp. 825-838 ◽  
Author(s):  
Ilijas Farah ◽  
Bradd Hart ◽  
David Sherman
Keyword(s):  
Model Theory ◽  
Operator Algebras

scholarly journals Model theory of operator algebras II: model theory

2014 ◽  
Vol 201 (1) ◽  
pp. 477-505 ◽  
Author(s):  
Ilijas Farah ◽  
Bradd Hart ◽  
David Sherman
Keyword(s):  
Model Theory ◽  
Operator Algebras

The Keisler–Shelah theorem for $\mathsf{QmbC}$ through semantical atomization

2018 ◽  
Vol 28 (5) ◽  
pp. 912-935 ◽  
Author(s):  
Thomas Macaulay Ferguson
Keyword(s):  
Model Theory ◽  
Classical Model ◽  
Isomorphism Theorem ◽  
Elementary Equivalence

AbstractIn this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues’ call for a ‘wider model theory.’ This call demands that we align the practices and techniques of model theory for logics of formal inconsistency (LFIs) as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, i.e. that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś’ theorem and introduce a useful technique of model-theoretic ‘atomization’ in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well.

scholarly journals ENFORCEABLE OPERATOR ALGEBRAS

2019 ◽  
pp. 1-33 ◽  
Author(s):  
Isaac Goldbring
Keyword(s):  
Positive Solution ◽  
Model Theory ◽  
Operator Algebras ◽  
Continuous Model ◽  
Continuous Functions ◽  
Embedding Problem ◽  
Prime Model ◽  
Canonical Operator ◽  
Classical Notion ◽  
Building Models

We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II$_{1}$factor is an enforceable II$_{1}$factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian$\text{C}^{\ast }$-algebras and use this to show that it is the prime model of its theory.

Tensor product and theories of modules

1999 ◽  
Vol 64 (2) ◽  
pp. 617-628 ◽  
Author(s):  
Mike Prest
Keyword(s):  
Tensor Product ◽  
Model Theory ◽  
Representation Type ◽  
General Situation ◽  
Elementary Equivalence ◽  
Primary Motivation ◽  
Complete Theories

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

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