Closed left ideal decompositions of G*1

Abstract

Let G be a countably in nite discrete group and let G be the Stone- Cech compacti cation of G. The group operation of G extends to G making it a compact right shift continuous semigroup. The semigroup G has import applications to Ramsey theory and to topological dynamics. It has long been known that remainder G = GnG can be decomposed into 2c left ideals of G (E. van Dowen 1980-s, D. Davenport and N. Hindman 1991). In 2005 I. Protasov strengthened this theorem by proving that G can be decomposed into 2c closed left ideals of G such that the corresponding quotient space is Hausdor . Let I denote the nest decomposition of G into closed left ideals of G with the property that the corresponding quotient space of G is Hausdor and let I0 denote the nest decomposition of G into closed left ideals of G. If p 2 G is a P-point then ( G)p 2 I. We show that it is consistent with ZFC, the system of usual axioms of set theory, that if G can be algebraically embedded into a compact group, then every I 2 I contains 2c maximal principal left ideals of G, in particular, neither member of I is a principal left ideal of G. We also show that there is a dense subset of points p 2 G such that ( G) p 2 I0 n I, in particular, I0 is ner than I