Definable sets in ordered structures. I

Pillay, Anand; Steinhorn, Charles

doi:10.1090/S0002-9947-1986-0833697-X

Definable sets in ordered structures. I
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by Anand Pillay and Charles Steinhorn

Trans. Amer. Math. Soc. 295 (1986), 565-592

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

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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 $\S 3$. 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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