On b Q 1 $bQ_1$ -degrees of c.e. sets

Abstract

Using properties of simple sets we study ${bQ}_1$-degrees of c.e. sets. In particular, we prove: (1) If $A$ and $B$ are c.e. sets, $A$ is a simple set and $A{\le }_{{bQ}_1}B$, then there exists a simple set $C$ such that $C{\le }_{1}A$ and $C{\le }_{1}B$. (2) the c.e. ${bQ}_1$-degrees (${bQ}_1$-degrees) do not form an upper semilattice. (3) The c.e. ${bQ}_1$-degrees are not dense, but are upwards dense. (4) The ${bQ}_1$-degrees are not dense.