Computably Enumerable Sets that are Automorphic to Low Sets

TL;DR

This paper investigates which computably enumerable sets are automorphic to low or low2 sets within the structure of c.e. sets ordered by inclusion, exploring automorphy under various lowness conditions and extending previous results.

Contribution

It demonstrates the existence of sets without semilow complements that are effectively automorphic to low sets and analyzes automorphy to low2 sets under different lowness notions.

Findings

Sets without semilow complements can be effectively automorphic to low sets.

In non low c.e. degrees, there are sets with semilow_{1.5} and semilow_2 complements with specific properties.

The paper extends understanding of automorphy to low and low2 sets under various lowness conditions.

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.