The notion of cupping/noncupping has played an essential role in the study of various degree structures in the Ershov hierarchy. As an approach to refute Shoenfield conjecture, Yates (see [2]) proved the existence of a nonzero noncuppable r.e. degree, a degree cupping no incomplete r.e. degree to 0'. In contrast to this, Arslanov proved in [1] that nonzero noncuppable degrees do not exist in the structure of d.r.e. degrees, which shows that the structures of r.e. degrees and d.r.e. degrees are not elementary equivalent. On the other hand, Posner and Robinson proved in [7] and [8] that any degree below 0' has a complement in the Δ_2~0 degrees. Slaman and Steel [9] improved this result by showing that such complements can be 1-generic degrees. This implies the existence of a Δ_2~0 degree such that 0 and 0' are the only r.e. degrees comparable with it - such degrees are called Yates degrees, as introduced by Wu in [11]. Say that an incomplete degree has universal cupping property if it cups every nonzero r.e. degree to 0'. By Lachlan's observation that every nonzero n-r.e. degree bounds a nonzero r.e. degree, no universal cupping degree can be n-v.e. In [5], Li, Song and Wu proved that in terms of the Ershov hierarchy, universal cupping degrees can be ω-r.e. In this paper, we consider those degrees with almost universal cupping property. Here an incomplete degree d has almost universal cupping property if it cups every nonzero r.e. degree to 0', except for those degrees below d. In [4], Cooper, Harrington, Lachlan, Lempp and Soare showed the existence of an incomplete maximal d.r.e. degree. Obviously, such maximal d.r.e. degrees have the almost universal cupping property.
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