Some properties of r-maximal sets and Q _1,N -reducibility

Abstract

We show that the c.e. ${Q_{1,N}}$-degrees are not an upper semilattice. We prove that if M is an r-maximal set, A is an arbitrary set and ${M \equiv{}_ {Q_{1,N}}A}$, then ${M\leq{}_{m} A}$. Also, if M ₁ and M ₂ are r-maximal sets, A and B are major subsets of M ₁ and M ₂, respectively, and ${M_{1}{\setminus} A\equiv{}_{Q_{1,N}}M_{2}{\setminus} B}$, then ${M_{1}{\setminus}A\equiv{}_{m}M_{2}{\setminus} B}$. If M ₁ and M ₂ are r-maximal sets and ${M_{1}^{0},\,M_{1}^{1}}$ and ${M_{2}^{0},\,M_{2}^{1}}$ are nontrivial splittings of M ₁ and M ₂, respectively, then ${M_{1}^{0} \equiv{}_{Q_{1,N}}M_{2}^{0}}$ if and only if ${M_{1}^{0} \equiv{}_{1}M_{2}^{0}}$. From this result follows that if A and B are Friedberg splitting of an r-maximal set, then the ${Q_{1,N}}$-degree of ${A\,(Q_{1,N}}$-degree of B) contains only one c.e. 1-degree.