Algebraic and topological equivalences in the Stone-Čech compactification of a discrete semigroup

We consider the Stone-Čech compactification βS of a countably infinite discrete commutative semigroup S. We show that, under a certain condition satisfied by all cancellative semigroups S, the minimal right ideals of βS will belong to 2c homeomorphism classes. We also show that the maximal groups in a given minimal left ideal will belong to 2c homeomorphism classes. The subsets of βS of the form S + e, where e denotes an idempotent, will also belong to 2c homeomorphism classes. e left ideals of βN of the form βN + e, where e denotes a nonminimal idempotent of βN, will be different as right topological semigroups. If e denotes a nonminimal idempotent of βZ, e + βZ will be topologically and algebraically isomorphic to precisely one other principal right ideal of βZ defined by an idempotent: −e + βZ. The corresponding statement for left ideals is also valid.