Self-full ceers and the uniform join operator

TL;DR

This paper investigates the properties of self-full computably enumerable equivalence relations (ceers), introduces hereditarily self-full ceers, and explores their closure properties and relationships with other classes of ceers.

Contribution

It shows that the class of hereditarily self-full ceers is closed under uniform join and properly intermediate between dark ceers and all infinite self-full ceers.

Findings

Existence of self-full ceers whose uniform join is non-self-full.

Hereditary self-full ceers are closed under uniform join.

Every non-universal ceer is bounded by a hereditarily self-full ceer.

Abstract

A computably enumerable equivalence relation (ceer) $X$ is called self-full if whenever $f$ is a reduction of $X$ to $X$ then the range of $f$ intersects all $X$-equivalence classes. It is known that the infinite self-full ceers properly contain the dark ceers, i.e. the infinite ceers which do not admit an infinite computably enumerable transversal. Unlike the collection of dark ceers, which are closed under the operation of uniform join, we answer a question from \cite{joinmeet} by showing that there are self-full ceers $X$ and $Y$ so that their uniform join $X\oplus Y$ is non-self-full. We then define and examine the hereditarily self-full ceers, which are the self-full ceers $X$ so that for any self-full $Y$, $X\oplus Y$ is also self-full: we show that they are closed under uniform join, and that every non-universal degree in $\textrm{Ceers}_{/{\mathcal{I}}}$ have infinitely many incomparable hereditarily self-full strong minimal covers. In particular, every non-universal ceer is bounded by a hereditarily self-full ceer. Thus the hereditarily self-full ceers form a properly intermediate class in between the dark ceers and the infinite self-full ceers which is closed under $\oplus$.