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

DOI:
https://doi.org/10.1090/S0002-9947-1986-0833697-X

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 -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 -minimal ordered groups and rings. Several other simple results are collected in . 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 -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 -categorical -minimal structures (Theorem 6.1).

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0833697-X

Article copyright:
© Copyright 1986
American Mathematical Society