Computably enumerable sets that are automorphic to low sets

Article type: Research Article

Authors: Cholak, Peter^a | Epstein, Rachel^b

Affiliations: [a] Department of Mathematics, University of Notre Dame, South Bend, IN, USA. cholak@nd.edu. http://www3.nd.edu/~cholak/ | [b] Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA, USA. rachel.epstein@swarthmore.edu. https://www.swarthmore.edu/NatSci/repstei1/

Abstract: We work with the structure consisting of all computably enumerable (c.e.) sets ordered by set inclusion. The question we will partially address is which c.e. sets are autormorphic to low (or low2) sets. Using work of R. Miller, we can see that every set with semilow complement is Δ30 automorphic to a low set. While it remains open whether every set with semilow complement is effectively automorphic to a low set, we show that there are sets without semilow complement that are effectively automorphic to low sets. We also consider other lowness notions such as having a semilow1.5 complement, having the outer splitting property, and having a semilow2 complement. We show that in every nonlow c.e. degree, there are sets with semilow1.5 complements without semilow complements as well as sets with semilow2 complements and the outer splitting property that do not have semilow1.5 complements. We also address the question of which sets are automorphic to low2 sets.

Keywords: 03D25, computably enumerable, low

DOI: 10.3233/COM-160053

Journal: Computability, vol. 6, no. 1, pp. 23-45, 2017