This talk is concerned with the interaction between the semigroup of invariant types and the preorder of domination, i.e. small-type semi-isolation. In the superstable case, the induced quotient semigroup, which goes under the name of "domination monoid", parameterises "finitely generated saturated extensions of U" and how they can be amalgamated independently. In general, the situation is much wilder, and the domination monoid need not even be well-defined. Nevertheless, this object has been used to formulate AKE-type results, can be computed in various natural examples, and there is heuristic evidence that well-definedness may hold under NIP. I will give an overview of the subject and present some results on these objects from my thesis.