CLOPEN TYPE SEMIGROUPS OF ACTIONS ON 0-DIMENSIONAL COMPACT SPACES

The paper proves the equivalence rigorously: [[α]] has a dense locally finite (indeed ample) subgroup if and only if T(α) is unperforated and cancellative, via Lemma 2.26 and the Ara–Goodearl machinery leading to Proposition 2.27 and Theorem 2.28. The candidate solution claims the same equivalence but its forward direction hinges on an incorrect identification of finite-level type monoids as N^k (it should be one N per orbit, not per atom) and on an unsubstantiated claim that cancellativity and unperforation are preserved under arbitrary inductive limits in this setting. Its reverse direction omits the key tameness/stable-finiteness bridge used in the paper and replaces the crucial conjugation step (Appendix B) with a vague “marker/patching” argument. Hence, while the conclusion matches the paper, the proposed proof contains substantial gaps and errors.