Definable Sets in Ordered Structures. I

Abstract

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the <!-- MATH $\mathcal{O}$ --> -minimal structures. The definition of this class and the corresponding class of theories, the strongly <!-- MATH $\mathcal{O}$ --> -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 <!-- MATH $\mathcal{O}$ --> -minimal ordered groups and rings. Several other simple results are collected in . The primary tool in the analysis of <!-- MATH $\mathcal{O}$ --> -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 <!-- MATH $\mathcal{O}$ --> -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 <!-- MATH ${\aleph _0}$ --> -categorical <!-- MATH $\mathcal{O}$ --> -minimal structures (Theorem 6.1).