On quasi-reducibility for c.e. sets Part I. The structure of the <i>Q</i> -degrees and the <i>sQ</i> -degrees

Abstract We study the structure of the c.e. $Q$- and $sQ$-degrees. For both structures, we show that there are join irreducible degrees but no Ahmad pairs. We show that the structures are not distributive, and that the lattice $N_{5}$ embeds in both structures. On the other hand the lattice $M_{5}$ cannot be embedded in the c.e. $sQ$-degrees, but a critical triple can be embedded. Finally, we show that no initial segment of the c.e. $sQ$-degrees or the c.e. $Q$-degrees is a lattice.