Complete and incomplete Boolean degrees

We study the partially ordered set of Boolean $P_2$-degrees. We introduce the notions of complete and incomplete Boolean degrees. We show that for each complete $P_2$-degree there exist both a countable decreasing chain of $P_2$-degrees and a countable antichain of $P_2$-degrees. We prove that above each incomplete $P_2$-degree there is a continuum of $P_2$-degrees. Thus, in total we show that in the partially ordered set of $P_2$-degrees there are no maximal elements.