Homomorphisms, ideals and commutativity in the Stone-Čech compactification of a discrete semigroup

In this paper we consider the Stone-Čech compactifications of discrete semigroups which are countable, commutative and cancellative. We show that, if S and T are two such semigroups, then any continuous injective homomorphism from T∗ into S∗ arises from a homomorphism mapping a cofinite subsemigroup of T into S. We also consider the structure of semiprincipal left ideals of βS, and show that any nonminimal ideal of this kind lies immediately above 2c others for a certain order relationship on these ideals. We also show that it belongs to a reverse well-ordered chain of such ideals of type ω1∗, each maximal subject to being strictly less than all its predecessors in this ordering. We also show that each nonminimal idempotent of βS lies immediately above 2c other idempotents for the order relation defined on idempotents by stating that α ⩽ β if α + β = β + α = α. Finally, if η is any nonminimal idempotent in βS, we show that the centre of the semigroup η + βS + η is contained in G + η, where G denotes the group generated by S.