Multiplications in additive compactifications of N and Z

There is a natural action (n,x)↦nx of (N,·) on any semigroup (S,+). When S is compact, there is always an extension to a map βN×S→S. When S has additional properties, there are extensions to other familiar semigroup compactifications of N, for example, wapN, apN and sapN≅bZ (the Bohr compactification of Z). Special cases of these extensions yield multiplications ∗ on these compactifications. The properties of multiplication, and its relationships with the natural addition, in each compactification are discussed. In particular, (bZ,+,∗) is a ring and its properties can often be pulled back to the other structures. The final section is devoted to the enveloping semigroups (which are in fact rings) of the actions of Z on compact groups. There turn out to be few possibilities: for example, if the group is not totally disconnected, then the enveloping ring for the action n↦nx is just (bZ,+,∗).