Essential kinds of 1-types over sets of models of weakly o-minimal theories
DOI:
https://doi.org/10.70474/ckvn1r44Keywords:
o-minimal theories, weakly o-minimal theories, convex set, quasisolitary type, quasirational type, irrational typeAbstract
We explore the properties of 1-types over sets in weakly o-minimal theories. We introduce and analyze six essential classes of non-algebraic 1-types (isolated, quasirational, irrational, quasisolitary, solitary, and social), highlighting the ways in which these types capture key convexity behaviors and definability conditions. Building on parallels with classical o-minimal theories, we establish lemmas and propositions that illustrate how properties extend from the o-minimal to the weakly o-minimal context. In particular, we demonstrate that one-types in these theories can often be decomposed into finitely many convex subsets or cuts, thereby preserving important one-dimensional geometric intuitions.
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Baizhanov B.S., Verbovskiy V., Turekhanova G. Неортогональность 1‑типов в слабо о‑минимальных структурах конечной глубины. Algebra and Model Theory. Collection of papers (eds. Pinus & Ponomaryov), Novosibirsk State Technical University, 2003, pp. 7–14.
Baizhanov B.S. Classification of one‑types in weakly o‑minimal theories and its corollaries. Preprint, 1996.
Baizhanov B.S. Expansion of a model of a weakly o‑minimal theory by a family of unary predicates. J. Symbolic Logic 66:1382–1414 (2001). DOI: https://doi.org/10.2307/2695114
Baizhanov B.S. Expansion of an o‑minimal model by unary convex predicates. Researches in Theory of Algebraic Systems, Karaganda, 1995, pp. 3–23.
Baizhanov B.S. One‑types in weakly o‑minimal theories. Proc. Informatics and Control Problems Institute, Almaty, 1996, pp. 77–90.
Baizhanov B.S. Orthogonality of one‑types in weakly o‑minimal theories. Algebra and Model Theory. Collection of papers (eds. Pinus & Ponomaryov), Novosibirsk State Technical University, 1999, pp. 5–28.
Baizhanov B.S. Types in weakly o‑minimal theories. 1st Congress of Kazakhstan Mathematicians, Shymkent, 1996, p. 177.
Baizhanov B.S., Kulpehsov B.Sh. On behaviour of 2‑formulas in weakly o‑minimal theories. Mathematical Logic in Asia, 2006.
Baizhanov B.S., Sudoplatov S.V., Verbovskiy V.V. Conditions for non‑symmetric relations of semi‑isolation. Siberian Electronic Math. Reports 9:161–184 (2012).
Baldwin J. Fundamentals of stability theory. Springer‑Verlag, 1988.
Dickmann M.A. Elimination of quantifiers for ordered valuation rings. In: Proc. 3rd Easter Model Theory Conference, Gross Köris, Berlin, 1985.
Van den Dries L. Remarks on Tarski’s problem concerning (ℝ, +, ·, exp). In: Logic Colloquium ’82 (eds. Lolli, Longo & Marcja), Stud. Logic Found. Math. 112, North‑Holland, 1984, pp. 97–121. DOI: https://doi.org/10.1016/S0049-237X(08)71811-1 DOI: https://doi.org/10.1016/S0049-237X(08)71811-1
Kulpehsov B.Sh. Weak o‑minimality of a linearly ordered structure. Researches in Theory of Algebraic Systems, Karaganda, 1995, pp. 61–67.
Laskovski M., Steinhorn Ch. On o‑minimal expansions of Archimedean ordered groups. J. Symbolic Logic 60:817–878 (1995). DOI: https://doi.org/10.2307/2275758
Macpherson D., Marker D., Steinhorn Ch. Weakly o‑minimal structures and real closed fields. Trans. Amer. Math. Soc. 352(12):5435–5483 (2000). DOI: https://doi.org/10.1090/S0002-9947-00-02633-7
Marker D. Omitting types in o‑minimal theories. J. Symbolic Logic 51:63–74 (1986). DOI: https://doi.org/10.2307/2273943
Marker D., Steinhorn Ch. Definable types in o‑minimal theories. J. Symbolic Logic 59:185–198 (1994). DOI: https://doi.org/10.2307/2275260 DOI: https://doi.org/10.2307/2275260
Shelah S. Classification theory and the number of non‑isomorphic models. North‑Holland, 1978.
Verbovskiy V. On formula depth of weakly o‑minimal structures. In: Algebra and Model Theory (eds. Ponomarev & Pinus), Novosibirsk, 1997, pp. 209–224.
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