Definable Sets in Ordered Structures. I

Abstract

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).

Weakly o-minimal structures and real closed fields

“…Theorem 7.2 shows that, for a weakly o-minimal structure, several conditions are equivalent to the condition that algebraic closure does not have the exchange property. We also prove that a weakly o-minimal theory does not have the independence property, generalizing a result of [22]. This gives an alternative proof of the result of Mathews [20] that real closed rings do not have the independence property.…”

“…Example 2.6.1 provides us with a structure whose theory is weakly o-minimal and which has a definable (unary) function that is locally constant and everywhere continuous but not piecewise constant. This shows that the Monotonicity Theorem of [22] fails. It also implies (see Theorem 7.2) that the algebraic closure operator does not in general have the Exchange Property, which hinders the development of a good dimension theory (centripetal contraction groups also provide examples of this).…”

“…In Section 3 we present the best analogues of the monotonicity theorems of [22] that we were able to prove. These theorems are for definable functions f : M −→ M .…”

“…Throughout the paper we assume familiarity with basic results of o-minimality as contained [22] and [14] (see also [9]). …”

Formal language properties of hybrid systems with strong resets

“…In order to define the notion of o-minimal hybrid system as introduced in [36], we first recall the notion of o-minimality [41]. In an o-minimal structure, the definable subsets of M are thus the simplest possible: the ones which are definable with parameters in M, < .…”