Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups

1996 ◽  
Vol 61 (1) ◽  
pp. 277-284 ◽  
Author(s):  
Anatole Khelif
Keyword(s):  
Open Problem ◽  
General Framework ◽  
Direct Proof ◽  
Model Companion ◽  
Standard Methods ◽  
Division Rings ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Constructible Sets

AbstractWe prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.Our main result uses constructible sets and can be put in an abstract general framework.Comparison with the standard methods which use forcing (cf. [4]) is sketched.

Finitely generic models of TUH, for certain model companionable theories T

1985 ◽  
Vol 50 (3) ◽  
pp. 604-610
Author(s):  
Francoise Point
Keyword(s):  
Discriminator Variety ◽  
Generic Models ◽  
Elementary Class ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Closed Element ◽  
Starting Point ◽  
Joint Embedding ◽  
Joint Embedding Property ◽  
Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Un principe d'ax-kochen-ershov pour des structures intermédiates entre groupes et corps valués

1999 ◽  
Vol 64 (3) ◽  
pp. 991-1027 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Abelian Group ◽  
Cross Section ◽  
Additive Group ◽  
Ordered Group ◽  
Unary Predicate ◽  
Valued Fields ◽  
Valued Field ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Image Position

AbstractAn Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields.We will consider structures that we call valued B-groups and which are of the form 〈G, B, *, υ〉 where– G is an abelian group,– B is an ordered group,– υ is a valuation denned on G taking its values in B,– * is an action of B on G satisfying: ∀x ϵ G ∀ b ∈ B υ(x * b) = ν(x) · b.The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications:1. Assume that υ(x) = υ(nx) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[B] ∖ {0} embeds in the automorphism group of G. Then 〈G, B, *, υ〉 is decidable if and only if B is decidable as an ordered group.2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure:where k((B))+ is the additive group of k((B)), S is a unary predicate interpreting {Tb ∣ b ϵB}, and ×↾k((B))×S is the multiplication restricted to k((B)) × S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable.3. A valued B–group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

scholarly journals Existentially closed models of the theory of artinian local rings

1999 ◽  
Vol 64 (2) ◽  
pp. 825-845 ◽  
Author(s):  
Hans Schoutens
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Model Companion ◽  
Closed Model ◽  
Gorenstein Rings ◽  
Existentially Closed ◽  
Finite Set ◽  
Residue Fields

AbstractThe class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

scholarly journals FIELDS WITH SEVERAL COMMUTING DERIVATIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 1-19 ◽  
Author(s):  
DAVID PIERCE
Keyword(s):  
Natural Number ◽  
Geometric Characterization ◽  
Model Companion ◽  
First Order ◽  
Existentially Closed ◽  
Differential Fields ◽  
Theory Of Fields ◽  
Differential Varieties

AbstractFor every natural numberm, the existentially closed models of the theory of fields withmcommuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

A note on existentially closed difference fields with algebraically closed fixed field

2001 ◽  
Vol 66 (2) ◽  
pp. 719-721 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Model Companion ◽  
Fixed Field ◽  
Existentially Closed ◽  
Difference Fields

AbstractWe point out that the theory of difference fields with algebraically closed fixed field has no model companion.

scholarly journals Model-Free Model Reconciliation

2019 ◽  
Author(s):  
Sarath Sreedharan ◽  
Alberto Olmo Hernandez ◽  
Aditya Prasad Mishra ◽  
Subbarao Kambhampati
Keyword(s):  
Decision Making ◽  
Open Problem ◽  
General Framework ◽  
Sequential Decisions ◽  
User Models ◽  
Model Free ◽  
Task Model ◽  
Free Model ◽  
The Common

Designing agents capable of explaining complex sequential decisions remains a significant open problem in human-AI interaction. Recently, there has been a lot of interest in developing approaches for generating such explanations for various decision-making paradigms. One such approach has been the idea of explanation as model-reconciliation. The framework hypothesizes that one of the common reasons for a user's confusion could be the mismatch between the user's model of the agent's task model and the model used by the agent to generate the decisions. While this is a general framework, most works that have been explicitly built on this explanatory philosophy have focused on classical planning settings where the model of user's knowledge is available in a declarative form. Our goal in this paper is to adapt the model reconciliation approach to a more general planning paradigm and discuss how such methods could be used when user models are no longer explicitly available. Specifically, we present a simple and easy to learn labeling model that can help an explainer decide what information could help achieve model reconciliation between the user and the agent with in the context of planning with MDPs.

Existentially closed algebras and boolean products

1988 ◽  
Vol 53 (2) ◽  
pp. 571-596
Author(s):  
Herbert H. J. Riedel
Keyword(s):  
Sufficient Condition ◽  
Subdirectly Irreducible ◽  
Universal Class ◽  
Model Companion ◽  
Universal Horn Class ◽  
Existentially Closed ◽  
Boolean Product ◽  
Boolean Products ◽  
Simple Algebras ◽  
Subdirectly Irreducible Algebras

AbstractA Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

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.

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