Prime e.c. commutative rings in characteristic n ≥ 2

1999 ◽  
Vol 64 (2) ◽  
pp. 629-633
Author(s):  
Dan Saracino
Keyword(s):  
Commutative Rings ◽  
Order Theory ◽  
Prime Model ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Building Models ◽  
Open Questions ◽  
A Chain ◽  
Mathematics Department

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.

scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

An Example of a Complete First-Order Theory with All Models Algorithmically Trivial but Without Locally Finite Models

1982 ◽  
Vol 5 (3-4) ◽  
pp. 313-318
Author(s):  
Paweł Urzyczyn
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
Program Schema ◽  
Locally Finite ◽  
First Order ◽  
Finite Models ◽  
First Order Theory

We show an example of a first-order complete theory T, with no locally finite models and such that every program schema, total over a model of T, is strongly equivalent in that model to a loop-free schema. For this purpose we consider the notion of an algorithmically prime model, what enables us to formulate an analogue to Ryll-Nardzewski Theorem.

Some consequences of the axiom of power-set

1965 ◽  
Vol 30 (3) ◽  
pp. 293-294 ◽  
Author(s):  
Alexander Abian ◽  
Samuel Lamacchia
Keyword(s):  
Predicate Symbol ◽  
Order Theory ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Power Set ◽  
Binary Predicate ◽  
Definition Of

In this paper we prove:Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.The proof of Theorem 2 will follow from the lemmas which we establish below.We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.Clearly, in every theory of sets we have:(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),(2) (x = y) ↔ (P(x) = P(y)),(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),(4) ⋃P(x) = x.In view of (2), (3) and the definition of equality, we have:Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.We have also, in view of (4):Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.

Prime models and almost decidability

1986 ◽  
Vol 51 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
Order Theory ◽  
The Other ◽  
Prime Model ◽  
First Order ◽  
Decidable Theory ◽  
Complete Extension ◽  
First Order Theory ◽  
Additional Constant ◽  
Open Question ◽  
Countable Models

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

Prime models of finite computable dimension

2009 ◽  
Vol 74 (1) ◽  
pp. 336-348
Author(s):  
Pavel Semukhin
Keyword(s):  
Order Theory ◽  
Prime Model ◽  
Computable Model ◽  
Undirected Graphs ◽  
Integral Domains ◽  
Computable Model Theory ◽  
First Order ◽  
First Order Theory ◽  
Computable Dimension ◽  
Open Question

AbstractWe study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

Locally finite theories

1986 ◽  
Vol 51 (1) ◽  
pp. 59-62 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Main Idea ◽  
Order Theory ◽  
Interpolation Theorem ◽  
Point Of View ◽  
Finite Model ◽  
Consistent Theory ◽  
Locally Finite ◽  
First Order ◽  
First Order Theory ◽  
Finite Theory

We say that a first order theoryTislocally finiteif every finite part ofThas a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theoryTa locally finite theory FIN(T) which is syntactically (in a sense) isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of [5], [6] and [7]. This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.)The first mathematically strong locally finite theory, called FIN, was defined in [1] (see also [2]). Now we get much stronger ones, e.g. FIN(ZF).From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilbert's second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects.In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.The results of this paper were announced in [3].

SEPARABLE MODELS OF RANDOMIZATIONS

2015 ◽  
Vol 80 (4) ◽  
pp. 1149-1181 ◽  
Author(s):  
URI ANDREWS ◽  
H. JEROME KEISLER
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Order Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
Separable Models ◽  
Homogeneous Models ◽  
Countable Models

AbstractEvery complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of TR is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T. This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.

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