Reducts of the Generic Digraph

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 $\varnothing$-definable relations are $\varnothing$-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$(D)$.