The goal of this paper is to generalise Alex Rennet's proof of the non‐axiomatizability of the class of pseudo‐o‐minimal structures. Rennet showed that if is an expansion of the language of ordered fields and is the class of pseudo‐o‐minimal ‐structures (‐structures elementarily equivalent to an ultraproduct of o‐minimal structures) then is not computably axiomatizable. We give a general version of this theorem, and apply it to several classes of structures.

We study a reduct ${\mathcal{L}_*}$ of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the ${\mathcal{L}_*}$ -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, ${\mathcal{L}_*}$ ) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From (...) this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions. (shrink)

We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...) language. (shrink)

We develop a dimension theory for D-semianalytic sets over an arbitrary non-Archimedean complete field. Our main results are the equivalence of several notions of dimension and a theorem on additivity of dimensions of projections and fibers in characteristic 0. We also prove a parameterized version of normalization for D-semianalytic sets.

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function, where and, that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.