In [1], Haskell, Hrushovski and Macpherson define a notion of domination equivalence between invariant types, inspired by the classical notion of domination in stable theories. At least in some cases, the tensor product is well-defined on the quotient, and endows it with a monoid structure. In my PhD project, carried under the supervision of D. Macpherson and V. Mantova, I am studying these monoids, what information they encode about the theory they are computed in, and how they change if we modify the relevant equivalence relation. 

[1] D. Haskell, E. Hrushovski and D. Macpherson. Stable Domination and Independence in Algebraically Closed Valued Fields, volume 30 of Lecture Notes in Logic. Cambridge University Press (2008).