| Abstract: | + of ultrafiliters on (0,1) that converge to 0 is a semigroup under the restriction of the usual operation + on BetaR d, the Stone-Cech compactification of the discrete semigroup (R d,+). It is also a subsemigroup of Beta((0,1) d,·). The interaction of these operations has recently yielded some strong results in Ramsey Theory. Since (0 +,·) is an ideal of Beta((0,1) d,·), much is known about the structure of (0 +,·). On the other hand, (0 +,+) is far from being an ideal of ( BetaR d,+) so little about its algebraic structure follows from known results. We characterize here the smallest ideal of (0 +,+), its closure, and those sets "central" in (0 +,+), that is, those sets which are members of minimal idempotents in (0 +, +). We derive new combinatorial applications of those sets that are central in (0 +,+). |
