11 problems

Let $T$ be the theory of divisible ordered abelian groups, equivalently ordered $\mathbb{Q}$-vector spaces, or more generally the theory of ordered $R$-vector spaces for an ordered…

Let $n\in\mathbb{N}$. An expansion of a divisible ordered Abelian group is a structure obtained by adding definable relations to such a group. For a definable discrete set…

Let $\mathcal{R}$ be an expansion of a divisible ordered Abelian group of dp-rank $2$. Consider infinite definable sets that are discrete, dense and codense in an interval, or Cant…

The conjecture. Spectrality and spectral resolutions for the unit interval $[0,u]$ should be equivalent to the respective notions in its partially ordered unital abelian group $G$…

Let $G$ be an ordered abelian group. Distal expansion conjecture. Every ordered abelian group admits a distal expansion. This proposes that distality can always be obtained by expa…

Irrational-slope trace-definability conjecture. A divisible ordered abelian group cannot trace define

Let $(R,<,+)$ be a dense archimedean ordered abelian group, let ${\mathscr{R}}$ be a strongly dependent and noiseless expansion, and let ${\mathscr{N}}$ be a highly saturated eleme…

Let $(R,<,+)$ be a dense archimedean ordered abelian group, let ${\mathscr{R}}$ be an expansion of it, and let ${\mathscr{R}}^{\square}$ be the completion structure used in the pap…

Finite-singular-primes conjecture. If $G$ has only finitely many singular primes, then $G$ is strongly dependent and, in fact, has finite dp-rank.

Non-definability conjecture. In no nontrivial precontraction group $(\Gamma,\chi)$ can one define, even allowing parameters, a function $\psi:\Gamma^{\neq}\to\Gamma$ such that…

Let $n$ be a natural number, and let $(G,G^+)$ be an ordered, directed, unperforated, rank $n$ free abelian group with Riesz interpolation. An inductive-system realization conjectu…