We show that the top of any diamond with bottom 0 in the r.e. degrees is also the top of a stack of n diamonds with bottom 0.

The project was partially supported by a NSF grant of China. The author was grateful to Professor S. Lempp for his encouragement and suggestion while the author was visiting the Department of Mathematics at the University of Wisconsin.

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] $\prec$ [b] $\prec$ [a] there exists [c] ∈ R/M such that [c] $\prec$ [a] and [a] = [b] ∨ [c].