A generalized type semigroup and dynamical comparison

In this paper, we construct a semigroup associated to an action of countable discrete group on a compact Hausdorff space, that can be regarded as a higher dimensional generalization of the type semigroup. Using this generalized type semigroup we obtain a new characterization of dynamical comparison. This answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal.