We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).

We generalize Buium’s notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form

$$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$

for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).

We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot ,\mathbb {N})$ . Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$ . We also show that any given expansion of $(\mathbb {R}, <, +,\mathbb {N})$ by subsets of $\mathbb {N}^n$ (n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

Given a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type.

If the theory admits no nonsimple types, then we use Mayer’s characterization of isomorphism for such theories to compute invariants for countable models. If the theory is small, then the invariant is real-valued, and therefore its isomorphism relation is smooth. If not, the invariant corresponds to a countable set of reals, and therefore the isomorphism relation is Borel equivalent to F2.

Combining these two results, we conclude that $\left( {{\rm{Mod}}\left( T \right), \cong } \right)$ is either maximally complicated or maximally uncomplicated (subject to completely general model-theoretic lower bounds based on the number of types and the number of countable models).

We let R be an o-minimal expansion of a field, V a convex subring, and (R0,V0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R0 is model complete relative to quantifier elimination in R, provided that kR (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).

An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.

Let be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S 1, …, Sl of its realization ∣K∣ in R we prove that there exist a subdivision K′ of K and a definable triangulation φ′: ∣K′∣ → ∣K∣ of ∣K∣ partitioning S 1, …, Sl with φ′ definably homotopic to id ∣K∣. As an application of this result we obtain the semialgebraic Hauptvermutung.

Let ${\cal M} =\langle M,<,\ldots\rangle$ be alinearly ordered structure. We define ${\cal M}$ to be{\em o-minimal} if every definable subset of $M$ is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal ${\cal M}$ can induce on a neighbourhood of any $a$in $M$. Roughly said, one of the following holds: \begin{enumerate}\item[(i)] $a$ is trivial (technical term), {\em or}\item[(ii)] $a$ has a convex neighbourhood on which ${\cal M}$ induces the structure of an ordered vector space, {\em or}\item[(iii)] $a$ is contained in an open interval on which ${\cal M}$ induces the structure of an expansion of a real closed field. \end{enumerate} The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods.

1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.