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

Abstract

An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a \cup c = b$. 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. In fact, we show that if $\mathbf b$ is a sufficiently large honest elementary degree, then there is a non-zero honest elementary degree $\mathbf a <_{\mathrm E} \mathbf b$ that does not cup to $\mathbf b$. For comparison, we modify a result of Cai to show that in several versions of the related degrees of relative provability the preceding property holds for all non-zero $\mathbf b$, not just sufficiently large $\mathbf b$.