On the function depth in an o-stable ordered group of a finite convexity rank
DOI:
https://doi.org/10.70474/6tx4zj76Keywords:
O-minimal theory, NIP theory, piecewise monotonicity, local monotonicity, o-stable theory, the convexity rankAbstract
We investigate the monotonicity properties of unary functions definable in ordered groups whose elementary theories are o-stable and have finite convexity rank. The notion of o-stability, combining o-minimality and stability, ensures tameness of types around cuts. Prior work established piecewise or local monotonicity of definable functions in weakly o-minimal structures, with key contributions by Pillay, Steinhorn, Wencel, and others. We build on these results by focusing on local monotonicity, $n$-tidiness, and the depth of definable functions. In particular, we show that any such function is piecewise $n$-tidy for some finite $n$, extending the theory of monotonicity beyond weakly o-minimal structures to a broader o-stable context.
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