Definable sets in ordered structures. I
Authors:
Anand Pillay and Charles Steinhorn
Journal:
Trans. Amer. Math. Soc. 295 (1986), 565592
MSC:
Primary 03C45; Secondary 03C40, 03C50, 06F99
MathSciNet review:
833697
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Abstract: This paper introduces and begins the study of a wellbehaved 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 orderpreserving 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:
http://dx.doi.org/10.1090/S0002994719860833697X
PII:
S 00029947(1986)0833697X
Article copyright:
© Copyright 1986
American Mathematical Society
