Say that an incomplete d.r.e. degree has almost universal cupping property, if it cups all the r.e. degrees not below it to 0'. In this paper, we construct such a degree d, with all the r.e. degrees not cupping d to 0' bounded by some r.e. degree strictly below d. The construction itself is an interesting 0''' argument and this new structural property can be used to study final segments of various degree structures in the Ershov hierarchy.
"An almost-universal cupping degree." J. Symbolic Logic 76 (4) 1137 - 1152, December 2011. https://doi.org/10.2178/jsl/1318338843