Consider an action of a discrete group G on a compact, Hausdorff, 0-dimensional space X. Say that two clopen subsets A,B of X are equidecomposable if there is a clopen partition A_1,...,A_n of A and elements g_1,...,g_n of G such that g_1 A_1,..., g_n A_n form a partition of B. The clopen type semigroup of the action is the semigroup obtained from the free semigroup generated by the nonempty clopen subsets of X after modding out by the equidecomposability relation. This is an analogue of a classical object introduced by Tarski; it was recently observed (notably by Kerr, then Ma) that this object encodes whether an action on the Cantor space has the dynamical comparison property (I will define this property and try to motivate its study). This actually holds also in the non-metrizable case, and I will describe some other properties of the action that one can study via the clopen type semigroup (notably, the existence of a dense locally finite subgroup of the topological full group) and outline some consequences for generic properties in the space of minimal actions of a given countable group on the Cantor space. There are many related open problems and I hope to have time to discuss some of them.