Basics of DTS quasigroups: algebra, geometry and enumeration

A directed triple system can be defined as a decomposition of a complete digraph to directed triples $⟨x,y,z⟩$. By setting $xy=z$, $yz=x$, $xz=y$ and $uu=u$ we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.