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A theorem of the complement and some new o-minimal structures. (English) Zbl 0948.03037
A prestructure is a sequence $$(S_n)$$ where $$S_n$$ is a collection of subsets of $$\mathbb{R}^n$$. It is a structure if the $$S_n$$ are Boolean algebras, contain the semi-algebraic subsets of $$\mathbb{R}^n$$, and for $$m>n$$ the projection of $$S_m$$ is contained in $$S_n$$. A structure is o-minimal if the boundary of every set in $$S_1$$ is finite.
It is known that $$\mathbb{R}_{\text{an}}$$, the structure of globally analytical sets, is o-minimal. Another example of an o-minimal structure is $$\mathbb{R}_{\text{exp}}$$, which is generated by the sets of the form $$f^{-1}(0)$$, where $$f:\mathbb{R}^n\to\mathbb{R}$$ is an exponential polynomial.
The author investigates $$\mathbb{R}_{\text{Pfaff}}$$, the prestructure of the sets $$f^{-1}(0)$$ with $$f:\mathbb{R}^n\to \mathbb{R}$$ Pfaffian. He shows that the structure generated by $$\mathbb{R}_{\text{Pfaff}}$$ is o-minimal. It is not known whether $$\mathbb{R}_{\text{Pfaff}}$$ is closed under complementation.
Reviewer: M.Weese (Potsdam)

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 03C10 Quantifier elimination, model completeness and related topics 14P10 Semialgebraic sets and related spaces
##### Keywords:
Pfaffian function; o-minimal structure; prestructure
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