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Definable Sets in Ordered Structures. I

Anand Pillay and Charles Steinhorn
Transactions of the American Mathematical Society
Vol. 295, No. 2 (Jun., 1986), pp. 565-592
DOI: 10.2307/2000052
Stable URL: http://www.jstor.org/stable/2000052
Page Count: 28
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Definable Sets in Ordered Structures. I
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Abstract

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the O-minimal structures. The definition of this class and the corresponding class of theories, the strongly 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 O-minimal ordered groups and rings. Several other simple results are collected in $\S3$. The primary tool in the analysis of 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 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 ℵ0-categorical O-minimal structures (Theorem 6.1).

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