Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Definable sets in ordered structures. I

Authors: Anand Pillay and Charles Steinhorn
Journal: Trans. Amer. Math. Soc. 295 (1986), 565-592
MSC: Primary 03C45; Secondary 03C40, 03C50, 06F99
MathSciNet review: 833697
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Abstract: This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the $ \mathcal{O}$-minimal structures. The definition of this class and the corresponding class of theories, the strongly $ \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 $ \mathcal{O}$-minimal ordered groups and rings. Several other simple results are collected in $ \S3$. The primary tool in the analysis of $ \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 $ \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 $ {\aleph _0}$-categorical $ \mathcal{O}$-minimal structures (Theorem 6.1).

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Article copyright: © Copyright 1986 American Mathematical Society