Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

On Dedekind complete o-minimal structures

1987 ◽  
Vol 52 (1) ◽  
pp. 156-164
Author(s):  
Anand Pillay ◽  
Charles Steinhorn
Keyword(s):  
Complete Model ◽  
Boolean Combination ◽  
Minimal Theory

AbstractFor a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model of T is sequentially complete if and only if ≺ for some Dedekind complete model . We also prove that if T has a Dedekind complete model of power greater than , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory—namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables—that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

Non Σn axiomatizable almost strongly minimal theories

1989 ◽  
Vol 54 (3) ◽  
pp. 921-927 ◽  
Author(s):  
David Marker
Keyword(s):  
Natural Number ◽  
Algebraic Closure ◽  
Predicate Symbol ◽  
Elementary Extension ◽  
Minimal Set ◽  
Categorical Theory ◽  
Minimal Theory ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Infinite Power

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

Dedekind Completeness and the Algebraic Complexity of o-Minimal Structures

1992 ◽  
Vol 44 (4) ◽  
pp. 843-855 ◽  
Author(s):  
Alan Mekler ◽  
Matatyahu Rubin ◽  
Charles Steinhorn
Keyword(s):  
Algebraic Complexity ◽  
Elementary Extension ◽  
Ordered Structure ◽  
Complete Model ◽  
Boolean Combination ◽  
Free Variables ◽  
Definable Subset ◽  
Dedekind Completeness

AbstractAn ordered structure is o-minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o-minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary.) We prove that if a theory is not binary then it is not rc-ary for any n. We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T, and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T, then there is an interval I in so that if is a Dedekind complete elementary extension of .

scholarly journals Generic derivations on o-minimal structures

2020 ◽  
pp. 2150007
Author(s):  
Antongiulio Fornasiero ◽  
Elliot Kaplan
Keyword(s):  
Complete Model ◽  
Minimal Theory ◽  
Elementary Formula ◽  
Ordered Fields ◽  
Differential Fields ◽  
Model Completion ◽  
Open Core ◽  
Theory Formula

Let [Formula: see text] be a complete, model complete o-minimal theory extending the theory [Formula: see text] of real closed ordered fields in some appropriate language [Formula: see text]. We study derivations [Formula: see text] on models [Formula: see text]. We introduce the notion of a [Formula: see text]-derivation: a derivation which is compatible with the [Formula: see text]-definable [Formula: see text]-functions on [Formula: see text]. We show that the theory of [Formula: see text]-models with a [Formula: see text]-derivation has a model completion [Formula: see text]. The derivation in models [Formula: see text] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [Formula: see text]-substructure of [Formula: see text]. If [Formula: see text], then [Formula: see text] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [Formula: see text] has [Formula: see text] as its open core, that [Formula: see text] is distal, and that [Formula: see text] eliminates imaginaries. We also show that the theory of [Formula: see text]-models with finitely many commuting [Formula: see text]-derivations has a model completion.

AN ELEMENTARY PROOF OF A THEOREM OF LEE

1991 ◽  
Vol 11 (3) ◽  
pp. 356-360 ◽  
Author(s):  
Jia'an Yan
Keyword(s):  
Elementary Proof

Analyzing the development of approaches to company performance measurement

2020 ◽  
Vol 19 (6) ◽  
pp. 1173-1188
Author(s):  
A.B. Kogan ◽  
A.N. Pristavka
Keyword(s):  
Comparative Analysis ◽  
Performance Measurement ◽  
Business Environment ◽  
Theoretical Research ◽  
Effectiveness Evaluation ◽  
Economic Science ◽  
Company Performance ◽  
Financial Indicators ◽  
Methodological Framework ◽  
Simple Set

Subject. The article presents various definitions of the efficiency concept, their changes as the economic science evolves, and describes various methods to measure company performance efficiency, ranging from a simple set of financial indicators to comprehensive systems for effectiveness evaluation. Objectives. The purpose of the study is to systematize the said definitions and identify a category that will meet the current condition of business environment. Methods. The study rests on the retrospective and comparative analysis of interpretations of the efficiency concept in the economic science. We also employ the historical and logical methods of general theoretical research. Results. We identify three approaches to the interpretation of the efficiency concept. Within the selected approaches, we consider the main methods for company performance measurement that have emerged since 1914. The paper formulates criteria, which were used to carry out the comparative analysis of these methods. The analysis enabled to trace all changes in the methods. Conclusions. We propose to use the term Integrated Company Efficiency and to develop methodological framework for measuring the comprehensive efficiency of companies operating in various industries.

Concern, Respect, and Cooperation

2018 ◽  
Author(s):  
Garrett Cullity
Keyword(s):  
Moral Theory ◽  
Fundamental Principle ◽  
Simple Set ◽  
Collective Activity ◽  
Complexity Results ◽  
Concern For Others ◽  
Moral Theories ◽  
Moral Responsibilities

Three things often recognized as central to morality are concern for others’ welfare, respect for their self-expression, and cooperation in worthwhile collective activity. When philosophers have proposed theories of the substance of morality, they have typically looked to one of these three sources to provide a single, fundamental principle of morality—or they have tried to formulate a master-principle for morality that combines these three ideas in some way. This book views them instead as three independently important foundations of morality. It sets out a plural-foundation moral theory with affinities to that of W. D. Ross. There are major differences: the account of the foundations of morality differs from Ross’s, and there is a more elaborate explanation of how the rest of morality derives from them. However, the overall aim is similar. This is to illuminate the structure of morality by showing how its complex content is generated from a relatively simple set of underlying elements—the complexity results from the various ways in which one part of morality can derive from another, and the various ways in which the derived parts of morality can interact. Plural-foundation moral theories are sometimes criticized for having nothing helpful to say about cases in which their fundamental norms conflict. Responding to this, the book concludes with three detailed applications of the theory: to the questions surrounding paternalism, the use of others as means, and our moral responsibilities as consumers.

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