Reducts of the Generic Digraph

TL;DR

This paper characterizes all reducts of the countable homogeneous digraph known as the generic digraph, by analyzing the lattice of closed groups between its automorphism group and the full symmetric group.

Contribution

It determines the lattice of reducts of the generic digraph, providing a complete classification of its definable relations and automorphism groups.

Findings

Lattice of reducts of the generic digraph is fully characterized.

Identifies all closed groups between Aut$(D,E)$ and Sym$(D)$.

Provides a classification of all reducts based on definable relations.

Abstract

The generic digraph $(D,E)$ is the unique countable homogeneous digraph that embeds all finite digraphs. In this paper, we determine the lattice of reducts of $(D,E)$, where a structure $\mathcal{M}$ is a reduct of $(D,E)$ if it has domain $D$ and all its $\emptyset$-definable relations are $\emptyset$-definable relations of $(D,E)$. As $(D,E)$ is $\aleph_0$-categorical, this is equivalent to determining the lattice of closed groups that lie in between Aut$(D,E)$ and Sym$(D)$.