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Although the quantum torus algebra can be thought of as the dual object of a group (the quotient of the circle by integer multiples of an irrational rotation), it does not admit a Hopf structure. This mystery can be solved by introducing the notion of a "hopfish structure" by generalizing, in the spirit of Morita theory, the structure maps of a Hopf structure to bimodules. I will first explain the concept of hopfish algebras in general and then show how to construct it for the quantum torus algebra. The hopfish counit and comodule define a unital and associative tensor product on isomorphism classes of representations, which can be calculated explicitly for an interesting class of modules. The resulting monoid is found to contain the original group structure of orbits of irrational rotations. This is joint work with Xiang Tang and Alan Weinstein.