Expansion of a model of a weakly o-minimal theory by a family of unary predicates (Q2758066)

A structure \((M,<,\dots)\) is said to be weakly o-minimal if every definable subset of \(M\) is a finite union of convex subsets. The structure is called uniformly weakly o-minimal if every model of its theory is weakly o-minimal. Every o-minimal structure is uniformly weakly o-minimal; an easy example of a uniformly weakly o-minimal but not o-minimal structure is the expansion of the ordered set of the rationals by any interval with irrational endpoints. The notion of weak o-minimality was introduced in 1985 by M. Dickmann. General properties and examples of weakly o-minimal structures were studied by \textit{D. Macpherson, D. Marker} and \textit{C. Steinhorn} [Trans. Am. Math. Soc. 352, 5435-5483 (2000; Zbl 0982.03021)]; a preprint version of that paper appeared in 1993. In the preprint an open question attributed to G. Cherlin had been formulated: is the expansion of an o-minimal structure by a convex subset weakly o-minimal? In 1994, B. Baizhanov announced a positive solution to the problem: the expansion of an o-minimal structure by any collection of convex subsets is uniformly weakly o-minimal. In 1995, Y. Baisalov and B. Poizat proved a result which easily implied the latter theorem: if \(N\) is a sufficiently saturated elementary extension of an o-minimal structure \(M\) then the expansion of \(M\) by the \(M\)-traces of all definable relations on \(N\) admits quantifier elimination. Moreover, the quantifier elimination is uniform in parameters: for any formula \(\varphi(\bar x,y,\bar u)\) there is a formula \(\psi(\bar x,\bar v)\) such that, for any \(\bar a\) in \(N\), the \(\bar x\)-projection of the \(M\)-trace of \(\varphi(\bar x,y,\bar a)\) is the \(M\)-trace of \(\psi(\bar x,\bar b)\), for a suitable \(\bar b\) in \(N\). A short and clear proof of this result had been published by \textit{Y. Baisalov} and \textit{B. Poizat} in J. Symb. Log. 63, 570-578 (1998; Zbl 0910.03025). NEWLINENEWLINENEWLINEThe paper under review is an attempt to prove that the expansion of a uniformly weakly o-minimal structure by any collection of convex subsets is uniformly weakly o-minimal. A first version of a proof appeared in a preprint in 1996. The author's approach is to prove an analog of Baisalov-Poizat's quantifier elimination result in the weakly o-minimal context. The author considers the following as the main result of the paper: if \(N\) is a sufficiently saturated elementary extension of a uniformly weakly o-minimal structure \(M\) then for any \(\bar a\) in \(N\) there is \(\bar b\) in \(N\) such that for any formula \(\varphi(\bar x,y,\bar u)\) there is a formula \(\psi(\bar x,\bar v)\) such that the \(\bar x\)-projection of the \(M\)-trace of \(\varphi(\bar x,y,\bar a)\) is the \(M\)-trace of \(\psi(\bar x,\bar b)\). NEWLINENEWLINENEWLINEThe paper is messy and difficult to read. The reviewer didn't manage to overcome the conglomeration of cumbersome notation and numerous definitions, and couldn't understand the proof. Recently N.~Suzuki found an error in Theorem~29 which is a step in the proof of the main result; however, it seems the author suggested a satisfactory correction. Nevertheless, the reviewer believes that the theorem (which certainly is interesting) is waiting for a clear and readable proof.