Constructing Minimal Pairs of Degrees

We prove that there exist sets of natural numbers A and B such that A and B form a minimal pair with respect to Turing reducibility, enumeration reducibility, hyperarithmetical reducibility and hyperenumer-ation reducibility. Relativized versions of this result are presented as well. 1. Introduction In the present paper we consider four kinds of reducibilities among sets of natural numbers: Turing reducibility (T), enumeration reducibility (e), hyperarithmeti-cal reducibility (h) and hyperenumeration reducibility (he). The rst three of those reducibilities are well known. The hyperenumeration reducibility is introduced by Sanchis in 5] and further studied in 6]. It is a kind of positive reducibility which relates to hyperarithmetical reducibility as enumeration reducibility relates to Turing reducibility.