Model Complete Theory Research Articles

Model companions of theories with an automorphism

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.

Nearly Model Complete Theories

A theory T of a language L is 1-model complete (nearly model complete) iff for every formula ρ of L there is a formula ϕ (χ) of L which is a ∀∃-formula (a Boolean combination of universal formulas) such that T ⊨ ∀x[ϕθ]. The main results of the paper give characterizations of nearly model complete theories and of 1-model complete theories. As a consequence we obtain that a theory T is nearly model complete iff whenever 𝔅 is a model of T and 𝔄⊆1𝔅, then T ∪ Δ1𝔄 is a complete L(A)-theory, where Δ1𝔄 is the 1-diagram of 𝔄. We also point out that our main results extend to (n + l)-model complete and nearly ra-model complete theories for all n > 0.

Model complete theories of trees

Model complete theories of trees

Homogenizable relational structures

We consider certain classes of relational structures which do not have the Amalgamation Property, and hence do not have a universal homogeneous structure, but which can be made to satisfy the Amalgamation Property by the imposition of some additional structure. We thus obtain a class which is the age of a homogeneous structure, and which has the original class as «underlying structures». We show that the underlying structure of the countable universal homogeneous structure has a model complete theory, so is a model comparison for its universal theory

Model-complete theories of bounded unars

Model-complete theories of bounded unars

William H. Wheeler. Model-complete theories of pseudo-algebraically closed fields. Annals of mathematical logic, vol. 17 (1979), pp. 205–226.

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

Model-complete theories of e-free Ax fields

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.

On ℵ 0-categorical class two groups

We exhibit some widely differing examples of class 2 ℵ 0-categorical groups. Certain central roducts of infinitely many copies of a given finite group are shown to be ℵ 0-categorical. For each prime p, we construct a family of p-groups of class 2, many of which are ℵ 0-categorical and have model-complete theories. These two examples are contrasted with Boolean powers of finite class 2 groups.

Model-complete theories of pseudo-algebraically closed fields

The model-complete, complete theories of pseudo-algebraically closed fields are characterized in this paper. For example, the theory of algebraically closed fields of a specified characteristic is a model-complete, complete theory of pseudo-algebraically closed fields. The characterization is based upon the algebraic properties of the theories' associated number fields and is the first step towards a classification of all the model-complete, complete theories of fields.A field F ispseudo-algebraically closed if whenever I is a prime ideal in a polynomial ring F[x1...xm]=F[x] and F is algebraically closed in the quotient field of F[x]/l, then there is a homorphism from F[x]/l into F which is the identity on F. The field F can be pseudo-algebraically closed but not perfect; indeed, the non-perfect case is one of the interesting aspects of this paper. Heretofore, this concept has been considered only for a perfect field F, in which case it is equivalent to each nonvoid, absolutely irreducible F-variety's having an F-rational point. The perfect, pseudo-algebraically closed fields have been prominent in recent metamathematical investigations of fields [1, 2, 3, 11, 12, 13, 14, 15, 28]. Reference [14] in particular is the algebraic springboard for this paper.A field F has bounded corank if F has only finitely many separable algebraic extensions of degree n over F for each integer n⩾2.A field F will be called an B-field for an integral domain B if B is a sabring of F.

Notes on the stability of separably closed fields

AbstractThe stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].

Model Complete Theories with a Distinguished Substructure

The properties of a structure of “infinite dimension” over a substructure are investigated.

Model complete theories with a distinguished substructure

Model complete theories with a distinguished substructure

A characterization of companionable, universal theories

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.

Complete and model-complete theories of monadic algebras

Complete and model-complete theories of monadic algebras

The undecidability of intuitionistic theories of algebraically closed fields and real closed fields

Let T be a set of axioms for a classical theory TC (e.g. abelian groups, linear order, unary function, algebraically closed fields, etc.). Suppose we regard T as a set of axioms for an intuitionistic theory TH (more precisely, we regard T as axioms in Heyting's predicate calculus HPC).Question. Is TH decidable (or, more generally, if X is any intermediate logic, is TX decidable)? In [1] we gave sufficient conditions for the undecidability of TH. These conditions depend on the formulas of T (different axiomatization of the same TC may give rise to different TH) and on the classical model theoretic properties of TC (the method did not work for model complete theories, e.g. those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories TH for a classically model complete TC remained open. An undecidability result in this direction, for dense linear order was obtained by Smorynski [4]. The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.One more remark before we start. There are several possible formulations for an intuitionistic theory of, e.g. fields, that correspond to several possible axiomatizations of the classical theory. Other formulations may be given in terms of the apartness relation, such as the one for fields given by Heyting [5]. The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5].

Existentially closed structures

One of the major problems of model theory is the spectrum problem, i.e. the development of structure theorems for the spectrum of models of a given theory. I hope that this paper will make a (small) contribution to the solution of this problem.Broadly speaking, in this paper we take a fixed (but arbitrary) theory T and consider a particular class ℰ of models of T. The structures in ℰ (which are known as existentially closed structures) are connected with the model complete extensions of T. These structures have already appeared several times in the literature.In §1 we survey the notation and terminology that we use, as well as the well known facts that we require. We also consider several concepts which are not new here but which may not be well known.In §2 we define, and give the basic results concerning existentially closed structures. For most theories, T, the class ℰ is not elementary and hence is not directly amenable to a model theoretic study. Because of this we consider a smaller class of structures—the uniformly existentially closed structures. These are discussed in §3.In §4 we look at model complete theories via existentially closed structures. This section is essentially a refinement of parts of [6].In §5 we look at model companions of theories via existentially closed structures.Finally, in §6 we make some remarks on the relevance of existentially closed structures to the spectrum problem.

Some model theory of abelian groups

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.