On non-essential expansions of quite o-minimal theories
Beibut Kulpeshov
Authors: Beibut Kulpeshov and Sergey Sudoplatov
The present lecture concerns the notion of weak o-minimality originally studied by H.D. Macpherson, D. Marker and C. Steinhorn in [1]. A weakly o-minimal structure is a linearly ordered structure \(M=\langle M,=,<,\ldots \rangle\) such that any definable (with parameters) subset of \(M\) is a finite union of convex sets in \(M\).
Quite o-minimal theories were introduced in [2]. Let \(T\) be a weakly o-minimal theory, \(M\models T\), \(A\subseteq M\), \(p,q\in S_1(A)\) non-algebraic. We say that \(p\) is quite orthogonal to \(q\) (\(p\perp^q q\)) if there is no \(A\)–definable bijection \(f: p(M)\to q(M)\). We say that a weakly o-minimal theory is quite o-minimal if the notions of weak and quite orthogonality coincide for 1-types over arbitrary subsets of models of the given theory.
Theorem. Let \(T\) be a quite o-minimal Ehrenfeucht theory, \(M\) be a countable saturated model of \(T\). Then for every \(n<\omega\) and any \(\bar a=\langle a_1, \ldots, a_n\rangle \in M\) the theory \(T_1=Th(\langle M, \bar a\rangle)\) also is quite o-minimal and Ehrenfeucht. Moreover:
\((1)\) if each \(a_i\) is a realization of an isolated or quasirational 1-type over \(\emptyset\) then \(I(T_1,\omega)=I(T, \omega)\);
\((2)\) if there exist \(1\le s\le n\) and \(1\le i_1<i_2<\ldots <i_s\le n\) such that \(a_{i_t}\) is a realization of an irrational 1-type \(p_{i_t}\) over \(\emptyset\) for each \(1\le t\le s\) and the remaining \(a_w\) (i.e. \(w\ne i_t\) for each \(1\le t\le s\)) are realizations of isolated or quasirational 1-types over \(\emptyset\) then \(I(T_1, \omega)=6^{m_T-l}3^{k_T+2l}\), where \(l=\dim\{p_{i_1}, p_{i_2}, \ldots, p_{i_s}\}\).
This research has been funded by Science Committee of Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19674850), and in the framework of the State Contract of the Sobolev Institute of Mathematics, Project No. FWNF-2022-0012.
Bibliography
- H.D. Macpherson, D. Marker, and C. Steinhorn,shape Weaklyo-minimal structures and real closed fields, shape Transactionsof the American Mathematical Society, Vol. 352, No. 12 (2000), pp. 5435–5483.
- B.Sh. Kulpeshov, shape Convexity rank and orthogonality in weakly o-minimaltheories, shape News of the National Academy of Sciences of the Republic of Kazakhstan,physical and mathematical series, Vol. 227 (2003), pp. 26–31.