Tame extension of almost o-minimal structure

Abstract

We consider an almost o-minimal expansion of an ordered group $\mathcal{M}=(M,<,+,0,\dots )$ and its tame extension $\mathcal{N}=(N,<,+,0,\dots )$. We demonstrate that the subset $\{x\in M^n|\mathcal{N}\models \Phi (x,a)\}$ of $M^n$ defined by a formula $\Phi (x,y)$ with $\mathcal{M}$-bounded parameters $a$ in $\mathcal{N}$ is $\mathcal{M}$-definable. We also introduce its corollaries.