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We investigate some properties of the clopen type semigroup of an action of acountable group on a compact, $0$-dimensional, Hausdorff space X. We discusssome characterizations of dynamical comparison (most of which were alreadyknown in the metrizable case) in this setting; and prove that for a Cantorminimal action $\alpha$ of an amenable group the topological full group of$\alpha$ admits a dense, locally finite subgroup iff the corresponding clopentype semigroup is unperforated. We also discuss some properties of clopen typesemigroups of the Stone-\v{C}ech compactifications and universal minimal flowsof countable groups, and derive some consequences on generic properties in thespace of minimal actions of a given countable group on the Cantor space.

Clopen type semigroups of actions on $0$-dimensional compact spaces