Mitotic recursively enumerable sets

Abstract

A recursively enumerable (r.e.) set is mitotic if it is the disjoint union of two r.e. sets both of the same degree of unsolvability. A. H. Lachlan has shown in [3] that there exists a nonmitotic r.e. set. In this paper we make an initial investigation into the class of mitotic sets.The following results are proved, (i) An r.e. set is mitotic if and only if it is auto-reducible, (ii) There is a nonmitotic r.e. set of degree 0′, (iii) If d is an arbitrary non-recursive r.e. degree then there exists a nonmitotic r.e. set of degree ≤d. (iv) There exists a maximal set which is mitotic and a maximal set which is nonmitotic.Albert R. Meyer had independently proved (ii) and (iii) for nonautoreducible sets before (i) was known.