부산대학교 도서관

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 and M are r-maximal sets, A and B are major subsets of M and M, 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 and M are r-maximal sets and $${M_{1}^{0},\,M_{1}^{1}}$$ and $${M_{2}^{0},\,M_{2}^{1}}$$ are nontrivial splittings of M and M, 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. [ABSTRACT FROM AUTHOR]