The theorem of the complement for nested sub-Pfaffian sets

Abstract

Let $R$ be an o-minimal expansion of the real field, and let $L_{\mathrm{nest}}(R)$ be the language consisting of all nested Rolle leaves over $R$. We call a set nested sub-Pfaffian over $R$ if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over $R$. Assuming that $R$ admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over $R$ is again a nested sub-Pfaffian set over $R$. As a corollary, we obtain that if $R$ admits analytic cell decomposition, then the Pfaffian closure $P(R)$ of $R$ is obtained by adding to $R$ all nested Rolle leaves over $R$, a one-stage process, and that $P(R)$ is model complete in the language $L_{\mathrm{nest}}(R)$.