author={Anand Pillay and Charles Steinhorn},
  journal={Transactions of the American Mathematical Society},
This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ¢minimal structures. The definition of this class and the corresponding class of theories, the strongly dLminlmal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of 41-minimal ordered groups and rings. Several other simple results are collected in §3. The… Expand
Dense pairs of o-minimal structures
The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays aExpand
On linearly Ordered Structures of finite Rank
A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here, which include all ordered structures definable (as subsets of n-tuples of the universe) in o-minimal structures. Expand
The Boolean Spectrum of an o-Minimal Theory
It is shown that, for every theory T of linearly ordered structures, SpecT contains the isomorphism type of the countable atomic Boolean algebra Bo with the property that for every infinite b G Bo, there exist infinite bxb2 G Bo such that bx v b2 = b and b{ Λb2 = 0. Expand
An introduction to $C$-minimal structures and their cell decomposition theorem
Developments in valuation theory, specially the study of algebraically closed valued fields, have used the model theory of C-minimal structures in different places (specially the work ofExpand
First Order Topological Structures and Theories
  • A. Pillay
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1987
The notion of a first order topological structure is introduced, and various possible conditions on the complexity of the definable sets in such a structure are considered, drawing several consequences thereof. Expand
Introduction to O-minimality
  • 2008
An introduction to o-minimality (sometimes referred to as order-minimality) is provided in this paper. An o-minimal structure is a model-theoretic structure containing a dense linear order where theExpand
A trichotomy theorem for o-minimal structures
Let M ˆ kM; <; . . . l be a linearly ordered structure. We de®ne M to be o-minimal if every de®nable subset of M is a ®nite union of intervals. Classical examples are ordered divisible abelian groupsExpand
Vaught's Conjecture for o-Minimal Theories
The analysis of the countable models of an o-minimal theory to prove a strong form of Vaught's conjecture for these theories, and shows that if there is a nonisolated type which is not simple then T has 2' countablemodels. Expand
Quasi-o-minimal structures
It is shown that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1, and a technique to investigate quasi-O-minimal ordered groups is developed. Expand
Structure theorems in tame expansions of o-minimal structures by a dense set
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\calExpand


Introduction to stability theory
  • A. Pillay
  • Mathematics, Computer Science
  • Oxford logic guides
  • 1983
The aim of this course is to give an introduction to stability theory and some of its generalizations, and to survey some applications and connections to other subjects such as set theory, algebra and combinatorics. Expand
An Isomorphism Theorem for Real-Closed Fields
A classical theorem of Steinitz [12, p. 125] states that the characteristic of an algebraically closed field, together with its absolute degree of transcendency, uniquely determine the field (up toExpand
On Strongly Minimal Sets
The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal. Expand
Uniqueness and Characterization of Prime Models over Sets for Totally Transcendental First-Order Theories
  • S. Shelah
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1972
A conclusion of the theorem is the uniqueness of the prime differentially closed field over a differential field, which is the total transcendency of the theory of differentiallyclosed fields. Expand
Remarks on Tarski's problem concerning (R, +, *, exp)
Publisher Summary The chapter presents the elementary theory of the structure (R , + , .), and the results could be extended to the structure (R, +, ., exp). Some aspects of on (R , + , .) areExpand
Some Theorems About the Sentential Calculi of Lewis and Heyting
This paper shall prove theorems about some systems of sentential calculus, by making use of results established elsewhere regarding closure algebras' and Brouwerian albegras, and the Lewis system and the Heyting system. Expand
Stratifications and Mappings
Publisher Summary This chapter discusses stratifications and mappings. One of the fundamental objectives of the theory of singularities of mappings is to study the local structure of a smoothExpand
Elementary Structure of Real Algebraic Varieties
A real (or complex) algebraic variety V is a point set in real n-space R n (or complex n-space C n ) which is the set of common zeros of a set of polynomials. The general properties of a real V as aExpand
Saturated Model Theory
Ordinals and Diagrams Similarity Types of Structures Elementary Equivalence Model Completeness Skolemization of Structures Saturated Structures Omitting a Type Homogeneous Structures Inverse SystemsExpand
Partially Ordered Rings and Semi-Algebraic Geometry
Introduction 1. Partially Ordered Rings 2. Homomorphisms and Convex Ideals 3. Localization 4. Some Categorical Notions 5. The Prime Convex Ideal Spectrum 6. Polynomials 7. Ordered Fields 8. AffineExpand