Abstract. The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets.

Abstract.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 ©-minimal 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 C-minimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¿¡-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an (5-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0-categorical ¿¡¡-minimal structures (Theorem 6.1).

The paper introduces the notion of definable compactness and within the context of o‐minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o‐minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o‐minimal structures. The main result proved is that any infinite definable group in an o‐minimal structure that is not definably compact contains a definable torsion‐free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o‐minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Abstract. A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

Abstract. A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula ϕ(x,ȳ), whereȳ = (y 1 , . . . , y m ):(i) There is a positive constant C and a finite set E ⊂ R >0 such that forOne-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.