Computably Enumerable Sets that are Automorphic to Low Sets

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 low$_2$ sets. Using work of Miller, we can see that every set with semilow complement is $\Delta^0_3$ 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 semilow$_{1.5}$ complement, having the the outer splitting property, and having a semilow$_2$ complement. We show that in every non low \ce degree, there are sets with semilow$_{1.5}$ complements without semilow complements as well as sets with semilow$_2$ complements and the outer splitting property that do not have semilow$_{1.5}$ complements. We also address the question of which sets are automorphic to low$_2$ sets.