Upper bounds on ideals in the computably enumerable Turing degrees

Abstract

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper σ40 ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no σ40 prime ideal in the c.e. Turing degrees. This answers a question of Calhoun (1993) [2]. Every proper σ30 ideal in the c.e. Turing degrees has a low. 2 upper bound. Furthermore, the partial order of σ30 ideals under inclusion is dense. © 2011 Elsevier B.V.