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?”.

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.

Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

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 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 of abelian groups

1972 ◽  
Vol 37 (2) ◽  
pp. 335-342 ◽  
Author(s):  
Paul C. Eklof
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Existential Sentence ◽  
Complete Theories

AbstractWe study the relations between abelian groups B and C that every universal (resp. universal-existential) sentence true in B is also true in C, and give algebraic criteria for these relations to hold. As a consequence we characterize the inductive complete theories of abelian groups and prove that they are exactly the model-complete theories.

scholarly journals COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

2014 ◽  
Vol 20 (1) ◽  
pp. 39-79 ◽  
Author(s):  
JOHN T. BALDWIN
Keyword(s):  
Formal Methods ◽  
Model Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Specific Content ◽  
First Order ◽  
Complete Theories ◽  
The Stability ◽  
Second Order Logic

AbstractWe propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

scholarly journals Method of the rheostat for studying properties of fragments of theoretical sets

2020 ◽  
Vol 100 (4) ◽  
pp. 152-159
Author(s):  
Aibat Yeshkeyev ◽  
Keyword(s):  
Model Theory ◽  
Complete Theory ◽  
Existentially Closed ◽  
Complete Theories ◽  
Syntactic Similarity ◽  
Definition Of

In this article discusses the model-theoretical properties of fragments of theoretical sets and the rheostat method. These two concepts: theoretical set and rheostat are new. The study of this topic in the framework of the study of Jonsson theories, the Jonsson spectrum, classes of existentially closed models of such fragments is a new promising class of problems and their solution is closely related to many problems that once defined the classical problems of model theory. The purpose of this article is to determine the rheostat of the transition from complete theory to Jonsson theory, which will be consistent with the corresponding concepts for any α and any α-Jonsson theory. For this we define a theoretical set. On the basis of research by the author formulated a model-theoretical definition of the concept of a rheostat in the transition from complete theories to ϕ(x)-theoretically convex Jonsson sets. Also was formulated an application of h-syntactic similarity to α-Jonsson theories.

scholarly journals On the representation type of tensor product algebras

1994 ◽  
Vol 144 (2) ◽  
pp. 143-161 ◽  
Author(s):  
Zbigniew Leszczyński
Keyword(s):  
Tensor Product ◽  
Representation Type
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