Abstract:

In this paper we study the question of the existence of exact pairs for ideals in upper semilattices of computable numberings and prove that all proper $\Sigma^0_3$-ideals in the semilattices of computable numberings of the families of all partial recursive functions, r.e. sets, left-r.e. reals, and Martin-Löf random left-r.e. reals have exact pairs.