The Semigroup of Ultrafilters Near 0.

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