ALMOST 1-TRANSITIVITY IN LINEARLY ORDERED STRUCTURES

The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally deeply studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization of the notion of o-minimality introduced by A. Pillay and C. Steinhorn in series of joint papers. As is known, the ordered field of real numbers is an example of an o-minimal structure. We continue studying model-theoretic properties of o-minimal and weakly o-minimal structures. In particular, we introduce the notion of almost 1-transitivity in linearly ordered structures and study tits properties. Almost 1-transitive o-minimal and weakly o-minimal linear orderings have been described. It has been established that an almost 1-transitive weakly o-minimal linear ordering is isomorphic to a finite number of concatenations of almost 1-transitive o-minimal linear orderings. Properties of expansions of families of almost 1-transitive linearly ordered theories are studied. Rank values for families of almost 1-transitive o-minimal and weakly o-minimal linear orderings have been found. A criterion for preserving both the almost 1-transitivity and weak o-minimality has been found at expanding an almost 1-transitive weak o-minimal theory by an arbitrary unary predicate. Dense ordering of an almost 1-transitive weakly o-minimal theory that is almost omega-categorical has been established.

1. Macpherson H.D., Marker D. and Steinhorn C. (2000) Weakly o-minimal structures and real closed fields // Transactions of the American Mathematical Society, volume 352, no. 12, pp. 5435–5483.

2. Sudoplatov S.V. (2021) Ranks for families of theories and their spectra // Lobachevskii Journal of Mathematics, volume 42, no. 12, pp. 2959–2968.

3. Ikeda K., Pillay A., Tsuboi A. (1998) On theories having three countable models // Mathematical Logic Quarterly, volume 44, no. 2, pp. 161–166.

4. Sudoplatov S.V. (2018) Klassifikacija schetnyh modelej polnyh teorij. – Novosibirsk: NGTU, chasti 1 i 2. (In Russian)

5. Baizhanov B.S. (2001) Expansion of a model of a weakly o-minimal theory by a family of unary predicates // The Journal of Symbolic Logic, volume 66, no. 3, pp. 1382–1414.