On bQ1-degrees of c.e. sets

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 <=(bQ1) B, then there exists a simple set C such that C <=(1) A and C <=(1) B. (2) the c.e. bQ(1)-degrees (b(Q1)-degrees) do not form an upper semilattice. (3) The c.e. b(Q1)-degrees are not dense, but are upwards dense. (4) The b(Q1)-degrees are not dense.

Automated summary

In this paper, we study the bQ(1)-degrees of c.e. sets using properties of simple sets. We prove several results, including the existence of a simple set C such that C <=(1) A and C <=(1) B when A is a simple set and A <=(bQ1) B, the non-upper semilattice property of c.e. bQ(1)-degrees, the non-density of c.e. b(Q1)-degrees, and the upward density of c.e. b(Q1)-degrees.