Honest elementary degrees and degrees of relative provability without the cupping property

An element a of a lattice cups to an element b > a if there is a c < b such that a∪c = b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann [17]. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0 <E a <E b that does not cup to b. For comparison, we also modify a result of Cai [8] to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees.