Binary ℵ_{0}–categorical Weakly O-minimal Theories

Kulpeshov Beibut Sh.

Let M = ⟨M, <, …⟩ be a linearly ordered structure. A subset B of M is convex if whenever a < x < b and a, b ∈ B, we have x ∈ B. A weakly o-minimal structure is a linearly ordered structure M = ⟨M, <, …⟩ such that any parametrically definable subset of M is a finite union of convex sets. We say that a complete theory is weakly o-minimal if all its models are weakly o-minimal. Here we prove some properties of binary ℵ_{0}–categorical weakly o-minimal theories. For the obtaining of these results, we use Baizhanov’s technique which he has elaborated for classification of one-types in weakly o-minimal theories.

References:

1. B.Sh. Kulpeshov, Weakly o-minimal structures and some of their properties, The Journal of Symbolic Logic, vol. 63 (1998), pp. 1511–1528.

2. A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565–592.

3. B.S. Baizhanov, Classification of one-types in weakly o-minimal theories, 1996, preprint.

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E-mail: kulpesh@usa.net