  We characterise the groupoid $C^*$-algebras associated to the transformation
groupoids of injective actions of discrete countable Ore semi-groups on compact
topological spaces in terms of the reduced crossed product from the dual
actions, and characterise the continuous orbit equivalence for injective
actions by means of the transformation groupoids, as well as their reduced
groupoid $C^*$-algebras. Finally, we characterize the injective action of
semi-group on its compactifications.
