Title: A model theoretic Rieffel's theorem of quantum 2-torus,
Abstract: We defined a notion of quantum 2-torus $T_\theta$ in "Masanori Itai and Boris
Zilber, Notes on a model theory of quantum 2-torus $T_q^2$ for generic $q$,
arXiv:1503.06045v1 [mathLO]" and studied its model theoretic property. In this
note we associate quantum 2-tori $T_\theta$ with the structure over ${\mathbb
C}_\theta = ({\mathbb C}, +, \cdot, y = x^\theta),$ where $\theta \in {\mathbb
R} \setminus {\mathbb Q}$, and introduce the notion of geometric isomorphisms
between such quantum 2-tori.
We show that this notion is closely connected with the fundamental notion of
Morita equivalence of non-commutative geometry. Namely, we prove that the
quantum 2-tori $T_{\theta_1}$ and $T_{\theta_2}$ are Morita equivalent if and
only if $\theta_2 = {\displaystyle \frac{a \theta_1 + b}{c \theta_1 + d}}$ for
some $ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)
\in {\rm GL}_2({\mathbb Z})$ with $|ad - bc| = 1$. This is our version of
Rieffel's Theorem in "M. A. Rieffel and A. Schwarz, Morita equivalence of
multidimensional noncummutative tori, Internat. J. Math. 10, 2 (1999) 289-299"
which characterises Morita equivalence of quantum tori in the same terms.
The result in essence confirms that the representation $T_\theta$ in terms of
model-theoretic geometry \cite{IZ} is adequate to its original definition in
terms of non-commutative geometry.