In set forcing every partial order has a Boolean completion given by the regular open sets. In class forcing this fails: We show that the existence of a Boolean completion is equivalent to the forcing theorem and, moreover, Boolean completions need not exist in class forcing. On the other hand, we prove that in Kelley-Morse class theory, every partial order has a Boolean completion. Furthermore, we characterize both pretameness and the Ord-chain condition in terms of Boolean completions.
This is joint work with Peter Holy, Philipp Luecke, Ana Njegomir and Philipp Schlicht.