Orbits of computably enumerable sets: low sets can avoid an upper cone Original Research Article

We investigate the orbit of a low computably enumerable (c.e.) set under automorphisms of the partial order View the MathML source of c.e. sets under inclusion. Given an arbitrary low c.e. set A and an arbitrary noncomputable c.e. set C, we use the New Extension Theorem of Soare to construct an automorphism of View the MathML source mapping A to a set B such that View the MathML source. Thus, the orbit in View the MathML source of the low set A cannot be contained in the upper cone above C. This complements a result of Harrington, who showed that the orbit of a noncomputable c.e. set cannot be contained in the lower cone below any incomplete c.e. set.