Speaker
Beibut Kulpeshov
Kazakh British Technical University
Talks at this conference:
Friday, 17:20, J222 
On nonessential expansions of quite ominimal theories 
Authors: Beibut Kulpeshov and Sergey Sudoplatov The present lecture concerns the notion of weak ominimality originally studied by H.D. Macpherson, D. Marker and C. Steinhorn in [1]. A weakly ominimal 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 ominimal theories were introduced in [2]. Let \(T\) be a weakly ominimal theory, \(M\models T\), \(A\subseteq M\), \(p,q\in S_1(A)\) nonalgebraic. 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 ominimal theory is quite ominimal if the notions of weak and quite orthogonality coincide for 1types over arbitrary subsets of models of the given theory. Theorem. Let \(T\) be a quite ominimal 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 ominimal and Ehrenfeucht. Moreover: \((1)\) if each \(a_i\) is a realization of an isolated or quasirational 1type 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 1type \(p_{i_t}\) 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 1types over \(\emptyset\) then \(I(T_1, \omega)=6^{m_Tl}3^{k_T+2l}\), 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. FWNF20220012. Bibliography
