Notes on a model theory of quantum 2-torus for generic q

TL;DR

This paper explores the model theory of quantum 2-tori, revealing their categoricity, interpretability of arithmetic, and stability properties of certain reducts, advancing understanding of their logical complexity.

Contribution

It introduces a model-theoretic analysis of quantum 2-tori, showing their categoricity, interpretability of arithmetic, and stability of quantum line bundle reducts.

Findings

Quantum 2-tori have uncountably categorical L_omega1,omega-theory.

The first-order theory interprets arithmetic and is unstable and undecidable.

Quantum line bundles are superstable.

Abstract

We describe a structure over the complex numbers associated with the non-commutative algebra Aq called quantum 2-tori. These turn out to have uncountably categorical L_omega1,omega-theory, and are similar to other pseudo-analytic structures considered by the second author. The first-order theory of a quantum torus for generic q interprets arithmetic and so is unstable and undecidable. But certain interesting reduct of the structure, a quantum line bundle, is superstable.