An example of a $P$-minimal structure without definable Skolem functions

TL;DR

This paper demonstrates the existence of intermediate P-minimal structures that lack definable Skolem functions, highlighting limitations in cell decomposition within certain minimal structures.

Contribution

It introduces intermediate P-minimal structures without definable Skolem functions, expanding understanding of the diversity and properties of P-minimal structures.

Findings

Existence of intermediate P-minimal structures without Skolem functions

Such structures do not admit classical cell decomposition

Highlights limitations in P-minimal structure theory

Abstract

We show there are intermediate $P$-minimal structures between the semi-algebraic and sub-analytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are $P$-minimal structures which do not admit classical cell decomposition.